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、、••• E,,
r
nk ,,EE--··
飞、
Combinatorial Identities
H. W. Gould
Copyright @ Henry W. Gould 1972
Printed by Morg:mtown Printing and Binding Co.
To my wife Josephine
for her never-failing
support and enthusiasm
"How do 1 love thee? Let me count the ways."
Combinatorial Identities
A STANDARDIZED SET OF TABLES
LISTING 5∞ BINOMIAL COEFFICIENT SUMMA TIONS
"Sc仿ηtia η.on habet inimicum nisi ign.orantiam"
HENRY W. Go皿D
Pr时essor of M athematics
West Virginia University
Revised Edition
Morgantown, W. Va. 1972
INTRODUCTION
Anyone who has taken the time to read far into the vast
1iterature of mathematics wi11 be aware of the fact that summation
formulas for binomial coefficients are very wide1y scattered in
books and journa1s. 咀le situation is para11e1 wi世1 what we
shou1d have if no tab1e of integra1s existed for our dai1y use
and convenience. It is our object here to atternpt to fi11 this
gap 由ld we present what we be1ieve is the beginning of a st由ldard
set of tab1es for binomia1 coefficient summations. 、 ..
It wou1d, of course , be a hope1ess task to 1ist a11 known
。r a11 possib1e formulas , and so the best which an aspiring tab1e-
maker may hope to achieve is to 1ist at a given date those formulas
which seem of frequent occurrence, or which are interes1址ng for
some specia1 reason, or which are very genera1 in nature. It might
be possib1e to produce some type of minima1 1ist of formulas
together with a set of operating ru1es whereby other formulas
could be obtained, yet we sha11 not attempt this as such an att四Ipt
wou1d destroy the uti1ity of the kind of tab1es we propose.
Certain 1ists of identities which exist in the 1iterature
deserve specia1 mention here. 世le 1ists of Hagen, Netto, Schwatt,
Dougal1, and Bateman seem very iu飞portant. 咀le Bateman Manuscript
Project, which resu1ted in the series on specia1 到mctions , is a
monumenta1 achievement in bringing about order in a chaotic fie1d.
Bateman himse1f had projected a book on binomia1 coefficients apart'
from their connection with the hypergeometric functions. The author
has worked his "shoe box" notes into pub1ishab1e form. When
a tru1y comp1ete index of formu1as is conternp1ated, such notes and
materia1 w土工1 be taken into account.. See bib1iography be1ow.
- i -
Binom1a1 coeff1c1ent su回nat10nø heve a1weys proved a popu1af
Bubject 1n prob1em departments of mathemat1ca1 journals. Much
lnterest was stirred up a1so after A. C. D1xon f1rst found a
c108ed expression for the sum of an a1ternatlng eeries of cube8
of b1nom1e1 coeff1cients. Thi8 1ed to the great contribut10n8 of
::' ~'ranel , and Maclt.ahon on the eummetion of powers of the
",.omial coeff1cients. Meny newcomers to this work are , however , unawere of Frane1's paper 1n l' Inhrm(diaire 些主 Mathémat1ciens.
The author' s 1nterest in th1s work began- twenty-seven years ag。
with the study of prob1ems 1n the !m盯1can Mathemat1ca1 坠且且1..
!t would be 1mposs1b1e to g1ve complete references for the tab1es
of formu1as he has collected , and 1n some caees extended,for he would
have to 1iet v1rtual1y a11 the journa1s which have been at his
disposa1 , and this wou1d inc1ude journa1s and books at theUniversity
of V1rg1nia , Duke Univere1ty , Univers1ty of North Oar011na , the Library
of Congress , and West Virgin1a University. The author was warned
10ng ago that he might we1l f1 0under in formu1as , 1f on1y 1n one
8ma1l field , such as St1r11ng numbers , or Bernou111 numbere , not to
ment10n this vast subject of b1Dom1al coefficient summations , and
that is one reason he w111 feel some 8enee of accomplishmen"t if
even a beginning 1s made in the present tab1es.
It must be poinhd out that we omit proofs 1n the present work.
In most 1nstances the author ha8 worked out proofs wh1ch dl何时 from
the known proofs; however,to 1nc1ude proofs in the present tex乞 wou1d
magn1fy the 1ength of thie volume by about ten. The 8uthor's per80nal
interest 1s not the mere tabu1ation of for皿11a8 , but th18 emphaeis 1_
deslred in the tab1es we g1ve here.
- 11幽
PROOFS
As far as proofs are concerned , it is important to indicate
the most 缸nportant methods. These seem to be any one or a cornbination
。f the fo110wing: c。η飞parison of coefficients in series expansions,
use of differentiation or integration , finite àifference rnethods ,
solution of difference equations or recurrence f,。ηnu1as , mathernatica1 ,1
induction, enurneration of 1attice points , theory of combinations
an卢 permutations , and generalserics transformations. The use of
the finite Taylor's series for po1ynomials is rather 却nport田lt a1so.
1n the last part of our ihdex wi11 be founct some useful transformations
which are especial1y important in handling the algebra of series.
Many of these wi11 be found 1n intricate detai1 1n Schwatt's Introduction
to the Operations wi th Serie~ [72 J NOTAT10NS AND DEFINIT10NS
1n the body of our tables , we have used the 1etters x ,y ,z for
real nurnbcrs except 1n a few instances ",here some note to the contrary
appears. 哑le letters j ,m,n , r , s generally have been reserved for non-
negat1ve intcgers. 1n most cases k has been used for the dummy variab1e
of summation and appears in no other usage. The bitiorr归1 coeff1cient (X J 飞 n.l,
is defined by the fo110wing:
(:)=川);;…4打 (1 -and we define x! = r (x + 1) in a ;few cases , but general工y we
feel it best to avoid use of the standard gamma function. The gamma
function of Euler i6 only one of an infinity of ways of extending
the factorial concept. -ii1-
By extensive use of re1etions such e8
、飞l
』/
xn //tII、、
1lt/
x-mH -x-Z/fttmv
\fh/ nxn
x-fIt--
飞
EE
/flt飞=、
lll-/
、11ftffιln
xn· /i\μ1\
(-OE(-1)nf:-) n ny
nJ』
hEl--/ n
n
叫ζ
,,,『tz飞
n 、‘.,,
aEE E
\}/ Aεn
fftt\
many veriations of the listed formules We give msy be found. A few of
the more usef、u1 binoruia1 coefficient relations of this type ere col1ected >>
together 1n our lsst tab1e.
Although we do not include formu1as which sum seriea involving
Bernoul1i numbers , Stir11ng numbers , Eu1er numbers , etc" We do need to
define the Stirling numbers ineswuch ss many sums involving binomiel
coefficients can on1y be given in terms of these. We sha11 define
thE Stirlinz 凹旦坦王旦旦主旦旦旦旦主坠旦立 by theeoeff1eient cZin the
expansion n
C)- ~_ C~ k x
The Stir 1in旦出业旦旦旦生旦旦旦旦主丛丛 we define by the expension
n
xn - .2; 11111
,J
XK ,,,ZEE-
nk nu
k-=O
See the auth町 's papers(sec。nd bipli。graphy)U汀,巳 8] , [3吁, (41) ,巳吨,(621 for discussions of notation and other properties. A better
nota tion S 1 (n ,时, S2(n ,k) was introduced in[11].
-l.V-
工n particular it followa that
and
(*)
哇=叫 :ω( k \ j嚣'J=olj /
AIK+a1IK-n 、时 CB=(-1)k y l l l lJ-dd
L 1,. _ .: I l ,. , .: I j! - j j::O 'k - j I 飞 k + j I '
the latter being perhaps the simpleat expression known for the
Stirling numbers of the first kind , given by L. Sch工Iff工i and O.
SchlBmilch. The reader should consult [2汀, [3才, [3吁,也才, [61J
for details regarding the Stirling numbers. 工n[3司 the author
proved a formula inverse to (*) above.
工terated summations of terms have been omitted , since their
inclusion would necessitate a tabulation of much greater length.
The receþtion accorded the first printing (1959-60) of these
tables has been so overwhelminglyfavorable and continued,‘ thælt
the demand for copies has long since exhausted the supply. This
corrected reprint has been undertaken to meet the need for such
tables. Rather than retype the entire manuscript and run the risk
of introducing new errors , it has been decided to correct those
misprints which are easy to change. At the end will be found a set
。 f errata tables , bringing'together those errata which have been
ca工led to the attent工on of the author in the intervening years.
These lists will be useful to anyone who has access to one of the
。riginal printings of these tab工es. The bibliography has not been
altered; however a 工ist of the author's pàpers is included so that
the reader can consu工 t those items for a general idea of the trend
of research since 1960. Most ofthis work has been supported by
research grants from the Nationa工 Science Foundation since 1960.
A few new formulas have been introduced , just those which seem
to be absolutely essential. Al工 binomial formulas are kept on
file in a card catalogue , and proofs are kept in running volumes
of notes now covering several thousand pageß_ 工 t is abundantly
clear that this reprint will serve a usefu工 purpose until the
author completes a manuscr土pt in preparation for a comp工ete工y new
treatise on the subject of combinatorial 土dentities.
-v-
Sucha new treatise will offer many new identities , a
complete revision of the present tables , model proofs , hundreds
of references to the literature and a whole chapter on the
generalized Vandermonde and Abel convolutions which have occupied
much of the author's attention.
The author is obl土ged to Drs. T. A. Botts and Reed Dawson
who originally suggested the compilation of the tables. He must
express his deepest thanks to the following who have made many
suggestions and 土n many cases supplied corrections to the tables
or suggested new formulas~ L. Carlitz , John Riordan , Donald Knuth , Eldon R. Hansen , R. E. Greenwood , Michael P. Drazin , Verner E.
H‘'oggatt , Jr. , Julius Lieblein , Emory P. Starke , and many others
too numerous to mention.
The author would be pleased to hear from users of the tables , and welcomes any summat土on prob工em involving binomial coefficients
whether the user knows the sum or not. Al工 suggest土。ns for
improvement will be careful工y studied.
The attent土。n of the reader is called to a very important recent book COMB工NATOR工AL 工DENTITIES , by John Riordan , published
by John Wi工ey and Sons , New York , 1968. This is the first book-
length treatise deal土ng entire工y w土 th the theory of combinatoria工
土dentities. The subject matter covers: recurrences , inverse
relations (two chapters) , generating functions , partitionp。工ynomials ,
and operators. Anyone dea工ing with our subject matter should consult
Riordan's book. Some of this wr土ter's work on inverse binomial
series re工ations is treated in detail in Riordan's book. There has
been a great flood of books recently dea工ing with combinatorial
theory , but the subject of comb土natorial 立dentit主es remains
scattered i口 the 1土 terature of several hundred journals , in portions
of books , in unpub工ished manuscripts and files , etc. 工 t is
expected that this situation will shortly change. There 土s a need
for pedagogical treatises on the subject. 工n TIlE ART OF COMPUTER
PROGRAMMING , by Donald Knuth , Addison-Wesley , 1968 , V。工 .1 , pp.53-73 , the reader will find a good ,工eisurely account of techniques for
attacking a prob工em of summing a binomial series.
-vi-
A NOTE ABαJT THE SYSTEM OF INDEXING
It is importent 1n 8 compi1etion such 88 this tomake use of
8 system of 1ndexing ~hich ~i11be e8sy to use 8nd flexib1e enough
to 8110~ the eddition of e fe~ formulss from time to time without
complete1y destroying the old ordering.
We therefore drew Up e sequence of tebles of formu 1es , numbered
。, 1 , 2 ,}, etc. Eech teb1e lists formulSs of the type S:p/q for
certe1n ve1ues of p end q , where p denotes the number of binomie1
coeff1c1ents eppeering in the genere1 numeretor term of the summetlon , 8nd q hes the S8me me8n1ng for the generel denom1netor term.
The tebles ere to be found errenged 1n the ordering
。/0 , 1/0 , 0/1 , 2/0 , 1/1 , 0/2 , }/O , 2/1 , 1/2 , O/~ , 4/0 ,当V1 , .• .etc.
It 18 eesy to prove thet the teb1e number , n , essigned to eech
type formu18 , S:p/q , 1s given uniquely eccording to the formu1a p+ q
D-q+L k-q+灿 q)(p +什1 )
k-O
Thus , in this system , e11 formules for sums of form s:4/} ehou1d be
11sted in teb1e number ~1. The proof ~f th1s is releted to the usuel
proof of the countebi1ity of the r8t1one1 numbers.
Conversely , if We desire to find out whet type Qf s\皿 18 tobe
found in e teble of number n we mey determine' th1s from the formu1as
p- 告(N-1)(N+2) - n q&n 回去N(N-1) ,
where N_G1+巧有J 丁J ' end where [寸 meens the
1ntegre 1 pert of x. The proof depends on the fect thet every odd squere
is of the form 80 + 1. Thus for teble ~1 , N = 8 , so p-= 告(7)( 10) - ~1
-= }5 - ~1 - 4 , eod q = ~1 -部的(7) = }1 - 28 - }.
-v工工-
In this sec∞d eòition of our formulary , the followlng
tebles Bre 11sted Bnd the indiceted eporoximete number of
summBtions of specified type Bre to be found within eech
+.8 ble •
• :",_ 7.E NR. Form S:p/q listed Approx. nr. sums listed
t 1/0 135 2 。/1 26
岛主 2/0 180 1/1 29
5 。/2 2 6 ;/0 52 7 2/1 48 8
A1//2 O 10 9 11 ;/1 3 12 2/2 8 16 4/1 3 17 当/2 5 21 6/0 22 5/1 2
225 1+ 4/2
75A6////冒F 5 6 ;1 2 71 97 x p/O 15 z Generel theorems 30
TOTALI 555
As might be expected , the 6ums of form 5:2/0 , 5:1/0 , 5.2/1 ,
end S:;lo are most well known. 1n general 6ums of the form 5.0/哇'
with lerge veluee of q , ere relatively rare in the literature.
An important fector to keep in mind 18 that there 18 no unique
form Srp/q in which e formula of Some bapic type 1s elwsy8 wr1tten ,
bec8use by use of elementsry binomiel coefficient identitie8 we may
rewrite a given formula uelng more or fewer binomial coefficienta.
Thus there will slweys exist 8 large degree of whet i8 really
cross-classification. However we do not propose to draw up a minimal
list of formulss. An axiomatic study is planned later.
-viii-
TABLE 1
SUMMATIONS OF THE FORM Sr1/0
00 c--飞 IX 飞
(1.1) )‘ f l zk a(1+z)X ‘, Binomiel Theorem
zz。、 k I Velid for erbitrary complex x , end cemplex Iz'<:1 , where the principal val~e 。可 (1 +吟。 is taken.
Converp:ence is irrelevant when wet-hink of this- as a. gener8ting function. -00
寸1 .k / X 飞(1.2) ). ( -1 ) ~ ( )- 0 , R( x) > 0 ,
ιJ 飞 kJ
k=O
00
( 1.~) Z (γ) .k _ l xl <. 1 , (1 _ x)n + 1
k-O
n
(1.4 ) Z K(x)Yx-1) 叫C: 丁,("。 k , (-1)a -1 十k-e
、
EES''
x-k /'''飞
nI川
飞
1112/
-n XA ,fat-、、
n 飞EEl-
XK IF-『E飞
bι
nz… 民
J
'
" r
艺川(:) kr -L (_l)k(:) (:::)
r (1.6) Bk '
k=O k-=O
n - 1 x-r 2 川C)kr(1.7) 11m n - (r - x) r (-x) ' a• 00
k-O (Prob. 句51 , Amer.Meth.Monthly ,195"p.482)
n n
(1.8) ~_ C ~1) zk L C-:十 k) zk z"(1+z) ' k
k-O k-O
(1. 9) 艺 C) yk 帽 Z fj(1JKN)k ,
k=O k 黯 o
(1. 10) /f'EEt飞
…γι
n-k-1 x -z c ~:)川)k' ,
::1)ZO
n a哩,
'-bh-
、
J-k
aE ••
"
+x-,,‘、- '
k- 。
= L (: ~:) n - k
n r+k
x
r
"的 Z(:)r~k
Z n 1- k (x 手-1) - 1
n t- k k=O k -= 1
(r ~ 1)
(1.1~) L (-1盯) k l -= 0 , 0ζj <:::.. n
a(-1)n rEF , j z n
(11A)27(jch-k)n+1E2x;~ (n+1)! ,
n xn『
ζ
a哥,-
--肉,』
x坷,』白
唱,
-b叶
+-2 n-tT一
呵/』
nE ZJ-
(n+2)! ' (1.15 )
呵J』午
'k x
飞飞tt
』/
nk J''taEK K Z k-O
n 飞112/
-ba x fIt--
nυ
j?C Brzfj
n -f- k ' (1. 16)
+ n k x \
飞11/
nk fFSEEbh
艺k-O
-2-
Let sjz L(叫:) "十 3k •
Theo
s , (-1)n 咱nn 、,,,四
.‘.·· +一刊
n-
自 (-1)n(n+j)! z C ::y:) ·唱4
+ bAb--
nu
--川
'
(1. 17 )
., ··EE-
n n ‘‘,, aE, ' ,
,.、
.. 饵,
S B CO
CH
-A a '-e e p eu
stE( ,1)nh+1月÷'
(1 .18) n (0 + 2)! • o(~n + 1)
S2 - (-1) ' 2 12
(n + ~)' • 2
(1.19) n n'(n+1) s - (-1)
当 6 8 '
n (n + 4)! n(15♂ +50n2+Pn-2) (1.20 ) 84 - (-1)
24 240 '
(1.21 ) 85 _ (_1)n (n +- 5)! 02(如~ + 10n2 + 50 - 2) .
' 120 96
(1.22 ) n (0 + 6)! n(6'n5 + ~15n乌+ ~15n~ - 9102 - 420 + 16)
86 - (-1) 61 40~2 . (Velues of S当 Bnd S7prioted in[72)were incorrect , see page 102 that text.)
。岖,
(11.2~ ) L (n: k) 2-k n + 1
- 2 k 翩 O
n
}叫二三 C) - 2. ,
k-O -~-
、飞tt'/
nk fftt飞
b民
nZ山
区J
『ζ
o n ~ d ,
12 n EO
n
( 1.26) 二 C) 008 k> nx 乎号_) n
"'" 2 'COB 2'l COS ' k 自 O
n
(1.27) Z C) 810 k> M 乎-r) n - 2 .s1n 寸r "l C08 ' k-O
( 1 约)三。(叫)ωabEHf2千叶ncoa 忖),
(1.29) Z (叫:)…= (-1)口川in 云)山 n(x:时,
(1 列)艺 C)叫'仰 , 2j oh +-2
,
(1.~1) 三 (2: )ωs lcx -斗
x ba n --、
EEEF/
nk
qι
,,,Itt 川z
…同ζ
XJ zr-1:叫
-4-
x ‘‘,, a冒-
4' k 句ζ
,,‘、
s o e 飞飞ZE
,,/
B+
』凰
吨'''
//飞
飞l丁J
幽
tre
e--.
,回
-F吨ζ-、,'!-E
nEF
飞/回
-句,
ι
』且
rtlL
、•• , 写J
ZJ ... ,,‘、
n+ 阳2-叶ω) e…)·(-1Mh/川叮X))
(1.~4) øin(2k + 1)x
k o::: O
=叮cJ川in(叶阶1♂(x/2) 毗 n{?)l
(1.~5 )
岛、飞,,,/飞飞
t
』/
knK
Ir--飞
Jftz飞
阳2
吊2
十+(n~2) " …
-2 '
4'
(1.~6) 1 飞( -1L(斗) ,
k...O
(l.m 2: (:) k:1. a唱··"
'-
1-x 4'
一-η
n--T ‘‘.,," 1-n
-,,E、
牛'一
x-a,‘、- '
〔妇 n t 1
L C:) n t- 1
(1 .~8) )' ( -- \ x - (x + 1) - (1 - x) 2k / 2k + 1 2(n +1)x '
k c:: O
P二3 nr1 n+-1
(1.~9) Z (2k: 1)
2k x (x + 1) . + (1 - x) - 2 .:
k+1 (n + 1)x2 ' k-O
ω2ω
+-f x-z
斗
r-H
>
例-T肺、
?一-K 1
一十
EZ
飞、11
』/
XK I''st、
k
- z C:j 叫
-5-
yruedothis::f4;斗。f form 511/1
(1.41)
n?L
k=O
n?L川
、‘,,
ZJ tu『
,,,‘、
、‘,,
L
斗L斗
.... ,,,、‘
飞、11
,,,r
nk /111飞
.K ‘、.,,
a吨,,
,,‘、
、
1111/
n-K /III-
‘
k 、‘.,,
,,‘‘、
飞飞』』JJ
xn
n/l
飞
‘‘.,,, 4 •••
‘、,,,,
--un ,,.‘、.-x ,
,.‘、
'因…E
X - k
二 ('1)k(:::)(kf/γ 三
( 1 勾引 L
ak+ 1
n nz ( -1叫 :)t
k = 1 k .配 1
The following surn 18 en 5:1/0 or e special instence of an 5:1/1 1
(1. 46) L (叫:)1
( 1.47)
( 1.48)
k==O 1 + - 1 • = n(n+1)
2 (;7) {η 2(n+~)
n
kj j-1
Z 川(:) x ( j 豆豆 n ) - (-1)
C!J , x+k k-O
n
z r:7=(":::7· 乙:',) ' k-O
-6-
什j/
+ nn + x /IIltt\ 飞
11j/
k 41k x /if--\ n勺/乙
ny LUτ
(1.50 ) L ('~丁k "'" 1
x +k
「fAVJ
、
IlIJ/
n 卡ta
x -r''『lk
「飞
}J飞lttz、
17
、、tlsf/
al?
x
/fit飞
、飞tlt
,
ff
aE‘唱,
++ nx /IIIEE飞
\lII/ -KX /ltl飞
nz… 良
/
-l,。
\ht』
J/
,唱,.a
『••
丰1tT
na
、d
-f/''111
‘飞
a \、111/
k-J rIt--、
nZ川
、、.,,
叫/』
EJ AY--,,,‘、
(1.5~) L \11ls/
b
n4'
且
//l\
where ~也
\ivz nu
川l
时
//I飞
/fIL
阳z…由2
功另
k = 0
a 卡 krx ~+Z (UJr j)'a(1 千去电Jr j户 ,
j = 1
2 1ri/r z::: e
(r-1 ~ a) (8 = pOS. or neg. integer) rz
乎 COB~j)n cos (0-2;)3 7γ r
j - 1 (n ~ 8 主 0 , r-1 注 8)
Z (1叫j)n__= 三 Z(叫nms平j ... 1 j .. 1
(156)Z。 {Us 刊 2nHmi号'
-7-
[号.!J
(157)24:1)→F十2W(?j'
( 1 归)Z (;)→ {2n十川μ抖,在
( 1δ9) z C:) k=O
(1. 60 ) L(叫n : k) K n - ~lC (xy)-(x + y) a:
n 4- 1 n 沪 1X 皿 y
' x - y k a: O
(1.61) L C: k) n - 2k k 2 z ...
n 斗 1 n + 1 x- y
x - y
wh町et x = l+F口, y a: 1- ,r;:+可
(1. 62) L k=O
(叫 ni 丁 (2COSX)n-2K E"::::1)X
[亏7
Z n - 2k ( 1.6~)
k (":丁 (2 COB x) 2 (-1) t k -= -- 008 nx ' n - k n
k 嚣。
[+J C :k) n ~ k (+)
n n
2 x + y (1.64 ) . -= n-1 ,
n 2 x+y
k=O here r x = 1 +.俨亨丁, y-1- -V耳可
-8-
(1.65) L (-1 )k C :丁k-=O
( 1 担(-1川k-=O
(1672]('1)k(n:)
( 1 岱~k c= O
2"-2k 2 2" + 1 (" ~1) ..
" - k n
b且
肉,ι
n-A叫一+
B-k oCF nζF
"+2 (2 008 X) -2 e08(n 千 2)X
' "+2
k
b一+
句ζEL且
a唱··
+n nζ
aE, ‘"
专n h
斗-= ' n 千 2
(叫(-1)n÷ , if' 个'
n (Cambridge Msth. Tripos , 19究; Hardy , Pu re Math. , p9ge 445.)
(-1 )叫 L , if 才 n '
l n : k).k a
(1.71罢仁γ=
\i/ /it飞
m艺
k-O
♂+ (_1)"2n , (n 杀 1)
n
(2Z + 1全J亨丁正 ) 十 (-1) (2z) 忖n+-1
川 n/22 户 (2叶 1 -t-{ï工石)" -(地斗 1 圣J丰石)
'
k=O
( 1 叫z
k a:: O
\(1.72)L
n -x 二十
千一门
n-x-
x
E二-x z_ ( 1 千 x)2
, where I '
(叫n : k) 2叫 =n 十 1 ,
-9-
叭l/
/IIt飞
ε…
2n + 1 -圄--22n
,
( 1拦 C: 丁= ( 1+ .J雪)n 十 1.(1-4吉)
2n + 1 -ý歹
E 凰n E n-th Fibonaccinumber , Where aos1·la1'
and a -= 8 + a n+1 n n-1
(1.75 )艺片~ [:巳~]
(-1) + (-1) , (?1)kfi 丁 E2
k-O
(1.76)三 (γ) - ZC"k-) aa2rEusing (1·7A)
k-O k-O
b且
川l/
叶位
rjt1、
n
「也… 1
/〉飞
J
K X
+ n x + b
且x 十n x
r飞
4JtL
\ll/ k +k n /ftt飞
PZ… 只
U吁t
(1.79)L c:γ 自 2n ' k-O
吨'』
k 叭l/
4,n
/,
1····\ ∞汇川
o au ,
-10-
'
b且同
ζ
\III-/ 』民
-n n 『ζ
,,ttE飞飞·nu
n
勺乙h
au _2n - ~
、
11l』/
b且
缸十
//-11、
甲Zau
1llIFf d -n n' 内ζ
ft1h\
十叫ζ
n 吨ζ
内ζ
E , (n 注 1)
k = 0
(1.8, ) Z Cn: 1)_ / ,
k== 。
>/n 竹il/
-n n 同ζ
/,JI--\
+、
叫ζ
n 句ζnd
叭1I/
-L
且
n nt /Its-、
nv
nγLh
L
斗nu
‘ •. , \// n
叭1·j/
-n n nζ
//ti\
斗tn 吨
'』同
ζE
叫l
,
ν
呵'』
/lt·\ nz… ‘‘,,
良J
nu
(1 附 2、川l/
-n n 句丘
/III-1飞
n 、‘,,
••• ,,‘、
= 飞
-1j/
nk 吨ζ/
/『t11、
b民、
‘,,
a唱··
,,‘、
, k c: O
k x
\111/ nk
肉,』
//I11飞
「lJnv吁VLh
RU
n n (1 +-v'X) + (1 -吗rx)
2
(1.88) Z n 2
n …-2
k~O
肉,』
J'' n ‘、,,
aE· ,,.‘、
E
、
1111JF
nk 呵'』吨
ζ
/tt『lt1飞
b且‘ •. ,,
4·· ,,,‘、
( 1.89) z 乙:) ' (n 主 1)n-1
- 2
-11-
'
'
日斗 I n 飞 ,.. . .、n ,.、n1.则~ (_1)k (2:) =: (1 +1) :~' ~ 1) _ (-v2 )n C08 芋,
(1.91) 艺 C:) - L Lk:nl) , (n 兰 1)
(1.97) Z L川 -2n-1'
k-O k...O
(1.92 )
mz… 2 十
、{之主 j)
叭ll/
十
k
nh之… k-O
(1.94) n?乙 (-OkfnJ 丁
k....O
(195)ZOLklxk'
( 1.96) 三。(斗)k乙k.+ 1) -
2n - 1 - 2
1:卜1(Cn :) .叫)
zc::) 2n - 2 '
[吁立]-= (-1)
n 2
n n n+.Ji)白- (1 - ...;王)
2-./言, (n ~ 1)
(1+1)n-(1·if sM)n sin 子,
21
n 、.. ,a唱
··圃
'-,2、
Enζ
'-a冒··
'
-12-
(1.99)
(1.100)
(1. 101 )
( 1.102)
叫?L
斤 2n 〕 =(-T22k+υ
n ‘‘,,圃
,唱,.
--,E
、-吨'』
幅-
A唱··· '
k ..,。
Z (;:::)3E 2n - 2 .. (2n - 1) 2 (n 注~)
"?/μ
)kc:::) 皿 (FJ , k 菌 o
ZN)k(::;)k;1 回(B(x忖 , symbolically ,
' where Bn(x)means B (x) 目 Bernou111 polynomia l.
n
n
Z C) k. k! -t-CJ W1}f ( 1.10') . ,
k +1 + 1) x" + 1 x k :oc O
主 C:)k
( 1.104) x E 飞(1- -E ( l x Iζ1 ) ' 2k
k.也 O (1 - 2k) 2
(1. 105) .z c:)•L E
, ( l x\.( 1, 81so va l1d for x =: -1 -J1万
0<1 k (1.106) Z (11(2:)
x 1+ 飞11+x , \ \x\< 1) - log 2k + 1 2
k == 1 k 2
c>ð
( 1. 107) L C:) arcs1n x 7r "'" dx = -::- log 2
(2k + 1)2 22k x 2 k=O
-1 ;5-
n r
2 2 C:J k" 2n + 1 f 2n\ ~ J I n \ ~ (1.108) ø 2:2: 1 (2:) L ( :)址
k-O 2 2' k=O
(1. 109) 5 - 211(子 (":1)o
2
n 旷宇 b ... ---5 ~ l t , t
当 O
(1.111) s -n(知 +2) s
2 当(5) 。'(1.112) s -
n( 15n2 十 18n + 2) S ,
当 ~(5)(7) o
n
(1. 11~) Z 趴1l/
千+
nn 向ζ
/ftE『tt飞
E 川-k2
飞飞
111/
』且
bA
nζ
,,,,,
.. EE--K
'''-呵ζ
门一十
4'-n -冉,
n- 42-吨4
,
k=O
、‘,,,
-H『
a, .. a咽··
-4., ,, .. 、.
、-EE
,/
…nz…
盯m
f( 对自 - k k _. k (1 - x) -- x~ f(τ- )
GENERALIZED LAGUERRE POLYNOMIALS 1
(111F)ILa)川
1 咱 x _ n ~ _ a T n -x ( = x e υ 哩 x e n' X l J ‘ (0)
Ordinary Laguerre Polynomiala: L (x) = L~V/(X) n u
k x
··飞EE
,/
kk .. pn f''EEZ、
nz… ‘
、,,,
,。
a唱a-
a咽,-
aE· ,,‘、
., a咽,
n /ι-
n· /』
nu AU aL· .q a-v o r p aE,., n· n
、‘.,,
a唱··
x ,,‘、
aE· + n‘ X E
-1 鸟,
飞、11/
nk ,,,, ••• EK
Lι
nz… 7t n'-k..k-1
(x 十 k)U ""(k+1)" -(x- 1)--,
(This 8nd the next two are speciel ceses of e general formula due to Abel.)
、
1l/
nOL川
(yk)n-k(x-yk)k 回n
x , n
(Abel) ) 付飞:县Z
n
(1.120)
三(:)旷 k(x-yk)
去 c :b~)Zk 幽
' (Abel) (1.119)
a 斗 1x x - 1
, I Zl <怦 1z= 一一--呵'b x
k-O 、
(Poly8 and Szeg巴)dz… 8 18 斗 bk' • I - . ---, zk =' X8 ,(88me conditions 8S in ebove.)
a + bk \. k J
(1 叫 Z (::::) , a discussion has been given by D~ Dick叫1[t7].
( 1. 12~) L (2:) (2k + 1)D 22k
Was studied by S. Ramanujan , in whose col1ected works one may find eveluations of this.
:1.124) ZC) 因
k n 、,,
z k y ,,‘、
k ‘‘,, MM LE X ,
,,‘、
n! L JXLY)k zn - k
k=O
(1.125) Z Bk(叩)BVKM)(P 十 qk) = ~h 飞了;:qnEE Bn(x 步 y ,z) ,
k=O x .., k-1
where Bk(x ,z) =毛τ(X + kz) (This 18 e Genera11zed Abel Convoluti句. Compere with (~.仙2)-(; .1韧) .)
-15-
n
( 1.126)
z… n
-= (1 十叶
飞
lII/
/'EEE、
jz… J
币、13
』/''
/,tzt r
L川
(1.127)
.D - L ('寸'〉 Ank '
and exp l1ci tly
A nk L(斗, 3 (刀 (k-j)n•
'
(Cf. Carlitz , Math. Mag. , Vol.~2(1959) , p.2的)
These numbers of Euler (also Worpitzky) area special case of numbers
due to Nielsen and which we might define 88 follows: 三、.- / q 飞 n
(1. 128) B~ _ ... /' (_1)K ( • l(r - k)口r ,q L..;、 1.. 1
k-=O 飞→ F
From this it follows that xn =(_1)叶 n :Ek=O
Tor all integersm> n·
-16-
Ix 十 k -
Bk ,m 十 1 t m
(1.129) Z(勺-j)
一十2
丁J
飞
rj
+ k 、EttE
,,f
+K J''EE飞
b且川。乙…
, k=O
(灿阳tt[72) , p.48)
(S伽ett(72J, p. 51)
Schwett gives v8rious other more compliceted exemples of this , such 88
formulas for
L>叫2? end 公l/
L民
/ftItt n7仨
.、d
n 飞,JX--K 咱
-4·-
r飞
-n
‘‘ •••• Ease'm
4tAE ++、
EEEa''
aa--m-B njkm ,,ESEE-
、,,,,
E·飞、
nh〉μ44TL咀
-3ak
、‘,,、‘,,
qsqL ZJZJ 4tAt •• dB·A·,,‘、,,‘、
k x
k 4d 、‘,,
X AE1 ,,.、
、EEElsF
aa·; --1-k -3K-/IIl飞
1tem
』且--、飞FFV-E
np〉
LZn飞』
k
··44d
、hEEEaa'
A』
as
---
nm
jJIlt
飞
x (Gould)
(Chung)
气/1-k
+n xhEt'' JFEE飞
fk
=/I
飞
飞lf
咱
kkk a-
、,,
xn1 /I
飞←
nFLM+LM 、‘,,、‘,,
ZJ'h
『
ZJ
『龟J
AB-aE-atdl ,,‘、,,‘、
= JC::)+ ' (SaalschUtz)
(1.135 ) LtC~2) r = (丁2) 立了:) h 一户
'
-17-
.、z,,,
tH
呼
.5)
:.6)
nz k-O
"艺k-O
k (-1)
-、11
』,/
xk /tl飞
-(丁?
TABLE 2
SU肌ATIONS OF THE FORM S:0/1
f号 [1 十 (:(:1:) }
x
x - 1
nO/μ叫
n
艺k
,n
三( :)
j - 1 , k 11: 1
币
_ s (x) n
1 ~ _ , D 十 2 _ # • n 十 1(1+~)S .(x) a:....:一一;- s (x) 十 X 十
n + 1 n 十 1 n
rO乙
k c: 1
-18-
-EH
'
j " 1
1 十 xk1 斗 x
1-x
(斗斗,
,
n k-1 n-1
Z ( -1) 十
(-1 ) ~2. 7) .:
2 C:) C:) 2{n 十 1 ) k = 1
2n - 1
z k-1 k n (2.8 ) (币1) ...
n + 1 ,
e:) k = 1
n 2k-1 2n
艺2 2
' (2.9 ) .,
k(2~) C:) k """ 1
cð
Z n , (r >1), \2.10) """ ( : ) r - 1 (:) k=n
C肖3
Z (n ~ 1) (乙 11) c=旬- , '
kC: 丁n
k = 1
n 。<:J 2
Z z rr (n~ 1) (2.12) = k2
,
们丁6
k = 1 k == 1
oa k+1 00
fxj+ Z x Z z + 1 , (2.1,) (:) =
X 千 1z 十 1 幽 k
k=O k 罩 O
R(x)< 告 z :/:. -1 , 0 , 2 , . . . 3
00 k+1 00回 k+-1 2 x z (-1) k z
( 1 : x ) (2.1h) =
(Z :k) z+k
k=O k-O R(x) <:号, z 手 0 , -1 , -2 , .
回才 9-
øo k 。。 k -k
lZ (xJ丁~- XL, k e
kr x+k k-O k! k _ 0 .
k
。。 -唱F
2 Z 川k
- ex. 一 X 飞f k ,
(丁丁 k' k' k 匾。 k 匾。
n
b .L, k-1
的艺k - 1
‘‘ .. ,, <u,,1
咱也正亨,
由世、p
n 呵'』
, "节'
k '_0 0
叫 2
句2k - 1
时 Zk = 1
:2 [ 1 - [为} -
a咽,-KL\j
1-nK HE吨'』
,飞
--rfEEL
kOL .,-4d
j - 1
7t》i
飞
J
-川I/
1-千k
--vh Ef'tt、
〈
J$iE、
17 r
勺Lh
2n n
s 冉+2(n 卡 1 )- 2n+2 Z
k - 1
飞飞t1J''
nk qι
kfiIL ‘‘,,因
a咽,由‘‘,,
,,‘、回.十k
.. (2川~Z • (Ljunggren)
2 有F-lC:) 18
、11BSJ
b皿mb且
呵'』
/Ftt飞
aE·· -、,,
b且
4,
22『+bh
同ζ,,a‘、
.K
x2LE+1
-K ‘、,,
41· ,,‘、 、
、,1,
Jf
kk 内4
/fttE飞
A唱··
bh
飞
J
nζ4E
2-+
k 吨ζ
,,‘、
bh
,‘-a'
'n 1
一+
-k .,,‘、
,
c X - -J 1 - x2 • Arcsin x ,
( lxl <1 , 8180 ve旧 for x_ 主 1 )
f2川 EXmJFT叶/1+1) ,
( l x , < 1, e 1s 0 va lid for x IIC + 1 )
-20-
k=O 飞EE
,
J
kk nζ
/J,, •• ‘、
k 、
JE2
4··"
、‘,,
--et (-十
b且
2 ,,墨、
-= 4 ~ k== 。
的 Z
圈子- ~ 1叫吁 1) ,
。。
n-1 z (1 - x) dx " (:"2萨 1 - x k...1
od
田 n! Z (kn 十1 )(kn 卡2) …… (kn + n)
k -0
n - 1
ftJk j z n-1 Z坦Z (- Ct/ .J (1 -但)- ) log
k' --- k ·但)k
k -= 1
臼JK 黯@
'
(E. H. Clerke) n n 千 1 k
(2.25) 艺 n 1- 1 z 2 -=' (Tor B. Stever)
(:) n 千 1 k 2 k=O k = 1 (Also sp也eciel C8. se
of (2.4) ')
(2.26) L 1
(X :丁(natural extension of (2.2); Gou工d)
- 2斗-
TABLE , SUM}.{ATIONS OF THE FORM S &2/0
‘‘,, n o --. &b u 噜A
o v n 。
nuv e AM n o m r e Ju n a v-,,‘、
飞
lIJf
咱.
+n x Jrtt\
鹏
、飞11J
b且
v·.
' //Il
、飞BEEF''
xk /,,IE飞
"γ己
,) _L_ C:j(Y:::k)_(叶飞o+j .
j)L (:)(0 :k) _乙::;1) ,
、
12/
y 牛'x
/fsk、
气/
由
b且
vw十
X
X J'Ett、EBEES-
n-K J'tIEK nz
5) L (:) ( :) -z (忏 n : j(+: -) .
6) L ( :)乙o :J 皿 Z(二 k)( n:k)-+[(::) 十(:)j.
7) L 仁兄斗、11
,
j'
XAM ,
/fl飞
41 吨J
J''E飞、
\飞11'
k
/fh 、
EI』,,F
/rttt飞
阳、三川 - 22-
" 一
8吟).2:ε~(工刀(乙仅2;儿:
10层]μ 儿:-1) 田→4÷4(2:) -」4;三P乒1卫正气2二纪:之巧飞℃;J川1飞飞V?γγ(←忖川叫.斗卅d1η♂ )户n
咐介J1民川J
---2n 二
2n乙
主/,
11飞
/l飞
"
、
l/hJ
xn
n 、电,,
n/l飞")斗
·『呵4-·十飞
l/
飞11/
飞
ljh
xnnn/FK 吨'h
呵'』!吨'』
ffUH1吨!'‘
叫罚飞fM
有飞
1-2
←
J2
自
勘飞
ll/
',
\1lIJ-1
4
、
i/.·p
kxah4
x2
-n
n·'
/IυhnhUH
川
\11//I
飞
.i
\1i/+
x+.:bAX
x2k
k2 //hv/l飞
/····L
川艺出?''已引艺…
「EK
171A) 艺 (:)(斗-干(:)(二:?缸子r:x.-:r) · (n~1 , O~rζÞ )
n e aM T AU n aA e r r a nr cu k a--r pnE
、、15''
k 唱,>
/Ftt
‘
、EEEEa''
xk /{利
n艺…
11).15) Z 叫::;)Eh-u(n ~ 2 • 。< r ~ n-1 )
(Erik Sperre Andersen)
-2予·
!CjZ 门)r:) =r~)r- 万=川) (丁) 2 •
1.17) .~. (:)(:) Zk - Z (:)c+: -J 怡- 1户,
jm)ZCX刀川
þ.19) L r:汇) l - ZCJ(X:J . 、
111/
xr ++ nn ,,/ttt\ 、飞J/r
x+ k
/t1'飞
、、1』』
J
nk /Il、
nz… nu
1721)Z (;rott 2(可n 二)l
Fb r e hu AU 唱A
O G r a w民
飞
i'/
L且
'bh xnζ 十四
ve Jr11飞
、、E,,
f
xk nζ
/FEE飞、o
n
勺Lh
l阳(臼μ阳叫?川川川22剖2约) Z三。 (:)C:工: :kJ护)k -t:~]
!(5.t!~> ~ (叫:1;XK+:?)-ifj;二;二
-24-
(, .24)
(, .25)
(, .26)
(, .27)
(, .28,)
(, .29)
(,.,0)
(,.,1)
(,.,2)
内1/
十/IIE\ Qll/ kk nn + x /
fi1、
、11tIJ
XK /iu n?/μ… 、
Il
』/
n1 内ζ
+ +
n
x2 /iE、
认ly/
kk nn 十x ,
Jftt飞
、飞KEEJ
X
十k
ffhv nz…
ZC:Y:::) -云rcγnE;二)f π(x242)'
Z C::)(::DZ;:::刀:引:(C:刀丁γγ2♂产2缸n~圄ι口:峦4峦峦[c扣(仅2川阳
L (:JC:j-t::刀,
z (:)(二 r) -(::刀,
2(:VDk zn( 叫: -) . 艺 (-叶叫叫1η)
L (-1 )k串斗卅j叫1ηJ)产叫k丁(:)(n二:二k)卜= (←串斗J1门)产2 (ι斗 )γ1叶川+剌忏川川(←忖川唰斗卅1η) '
= (:) ~:172;;!!fL)!
、
1II/
'Ar n ,,IE
飞
r \i/ baH -r a ffttt飞
叭luv
fFE--、
b且…
勺L川
ZJ
!以5乒引号川; 艺 州 CJ比L汇(2nιn二n : k)斗k) -叫川(←归川-斗1
:~.~切!巧肪予另5引) 三;(斗卅d叫吨)户叫盯k气k(:仁:)(2'二斗!二:k) - <-i [(仙川:)+刑 '
川 Z 川了:γ::; 丁 f二丁 J;-1户,
('.~7) L (忏.: k)(斗
:'.;9) 去(叫
MO)Z (-rcynfb腼 2 乙; 1)( n 刁 2k 村,
川1)2(,1)KC)LfJE(叫
-ã>-
(~.42) Z (叫:γ:::)、
111/
xk 冉'』吨
ζ
nn 冉ζ/
ftt飞
、EEBEEtJ
XK If--飞
L且
-hJ3
叫~uZ川
回 (-1)nZ(-4(2:1)(勺 2k
1刀//It飞
十川π…
叭iv
呵ζ
冉'』
yn ,,,,,,
aE···
、
‘EI--
,,
xk //l飞
mr…
L且冉
'』
A1lF/ gk yn ffttat
‘
、飞111f
xk /fiI飞
bEh n「之…
ZJ L
且『
ZJ
b且
向4
9』
飞
1iJ/
h 内ζ
』且
xn 句ι
/ttl飞
、飞、11t2J
XK ,
JIl飞
bι
12… M ZJ -(叫
b晶
吨,』
2 λ叫ll/
少』
bA
kn //I1l飞
、-h1l/
xk //It-·
、
k nz… 民
JL
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n ,E
、-吨4
4'一 '
飞EE--
,/
xn /JtttL n 内
ζ内ζ
n 、川!j/
b且
hu-n n 冉£
fIII-飞
叫!'/
于
-K
X 肉'』
/fIIL -K nZ山
,。
L
斗ZJ
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z
x
,
/'Iiv 割... ,
f
、
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x ,
rll
飞
飞飞t
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nk /IIt'飞
b且、
,,,
aE, ,,‘、
nZω
、-z'
?『
L
斗ZJ ,
,.、
、
lJγ
xaf /fhv 叽1川V
+『+
'Ar MVI人
nk //t』EL
b且
nz… M咱
-27-
川,6(叫:)(::了)
9) L; (叫:yt)-C::) ,
币) L, (叫:)(气)~ c寸,
,k - z c){: ~~:) · -(叫马)
嚣队 :1::
K ZJ 、
气lj/
zk xn zJ VVAf xk ,,,,,
EEE、
bι
nz… 2
,
bh
nt』
、
1lI/
乱翩'n n n
dι
,
ftttt、
、飞11l
』
J
XK /ftlz、
bh
nu
nOLh
儿- 2才; +nD -) •
d(fU丁ftf?Lr z巾,·川ft/
十xn n rIlt-E
牛tn
It--/ 3z +f:
111la/
bEE
乱Me
2n2 ·-n nn 内ζnJ
』
/FEat--''1t飞
-EE
『『
tJ
、111IE/
fk
十
k
nn JFSE飞
/It--‘L
L且
b且
nz…moL
…
眠HM
川
, (Generelizes a formula proposed by B.C.Wong)
(B. C. Wo鸣)
-28-
n\ (气2X)(357)Z。(叫丁::Dz(-1〈
(;.59) 主(-1叫
n 、‘,,,
aT·· ,,‘‘、
十a咽'
、飞E『,,,
XA
口
,
//i飞
吨,』
JJ n
、‘,,
A唱··
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E
K 吨ζ
、
111''/
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,,,SEE-E飞
、、11
』/
x-K /''-1、
k 、,,,
aE-
nu
noρh
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4唱··
r。
ZJ
(, .62) .z= (叫:X2::D2kg1+;-1)yn二) ,
n 吨,』
飞
11/
'An ,ttSEEL 叭
1UJ
肉'』内
d』
xn nd ,,,E『LE飞
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J
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k=O
aEd n 呵'』呵
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4唱··
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酶
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,,『『lh飞
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n-K Jrl飞
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a咽,.
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n
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-29-
肉'』
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ll/
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x 叫
//n)
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hm+-/Il飞
1-
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nk /ftt飞nv
nzh
〉门
吨,』
4,F吨,』
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k x
ndnd
2\i/\l/
飞
lj
nknknk /i
飞/{飞!/{飞
nzγ二号。已
rlL
句"盯
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41 、
-tI'''
L崎冉
'h
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、、Ea--』
j
nk /I』tk
nz… 但
V69)zr:1)z÷(:::3 ,
。 70) 雪 C:Y →C:) 十二tiroJ n 1 十(斗)
2
1771)ZC)2 a 士(::)十二宁可:) .
口2) ! G: r -+C:) 十气气2:) 十 1 十~-1)" c:r · ~.7;) 艺 μ1) --ît) -土耳YJJt(-1)n
- 30-
(叫3川川川7协向A勾) Z (ιi二二k斗ι:1J ) E→÷封(:) _ J仨午干乎÷二严乒!卫卫咛)
叫 2 ι lr =→+(ω::D:D) - 」斗午平t=严!旦i气1'{2巳ω2飞节:)奸十 1 丁γ1)气n飞c:y ,
(~.76) L (: y (扣n卜.斗4叫2位叫旷k叫k)2户2 . n<(2 :二: :D) '
(~.77)
(~. 78)
(~.79)
"Ø.80)
(, .81)
~ cr kr
-
γ了:) ,
γ ♂ (2:1)'
~(叫
hz 吨'』
、1112/
nk 吨ζ,
/'』,、
L且、
.,
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rk 飞
ll/
b且
ftttlk nk /tilL r们/已
==- s r
,
k~O
飞飞is-J
吨ζ4
,
nn 呵'』
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n nt』
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FIt-L a-2 E
aERM
S5z n2(γ1)(2::3 ,
E(f '
-L (叫:rn : )2k ' k-O
., 飞
l/
nn 冉4f
''『aE飞
n ‘、,,,
A咽,.
,,t、
=
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- 31-
四町ζ
\1』/
nk rft『1、
k飞ll/
vvaA
、
nk m勺/已
k
" :~ ~/ ) 午J
k-O
k
z阳、
1112/
bak 吨'』
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飞
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nk //』t、
b且
nz… L
U吁民U
ZJ 、1l
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,唁k
vER产
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飞
nz…
k 飞
1ij
vt人
能
k
/ftt飞
nzh '
(~.85) Z(-1)k(;)(20 兰k -C:)-;纭,
Z(叫(; .86)
(~ .87川川8盯旷7
1;.88) 220 L (:)(2:) _ L (2: : :l2:) 庐,
n HP1(:}(? 叫 z ' (20 - 1) 2 2n
k=O
n Z(…)门 2n(; .90) / f \1 1 - 2 n -k It k k=O
-32-
..91 )
1.92)
j.9')
~.94)
:~.9乡)
nOL
(叫2::汀2:) _ 冉ζ
\llj/ nll
n
ftIl飞川k-O
Z (2::DfDJ-14::2;:
z (2:: 了)C:J阳)(斗1)'(-1附C:)-k-O
1, n .... 0 ,
。, nJ泣,
z (2: : :k)(2:) (2k-1 )(牛十 1 ) 4n
2 , (n 注 1) , - 山十1) ( 2:)
Z C: : :k)( 2:) 斗τ到/飞/
--n xn-4En +-x n7ti--1飞
/'''l飞- , (n ~ 0) ,
_2n _ í'
k-O
n .. 2
(2x +- 1)(2x 十~)…. (2x 千 2n-1)
(x t- 1) …. . (x +吟,(n ~ 1) ,
1二叫
飞
1I/
b且
b且
向J』IFtt1、
\飞lt/
ak 吨'』
r''''EEEE-
飞
nz… 叫1) ( 2:)
2n 2
- , c:, a唱E'
n LU寸吨
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nn nζ
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t』呵ζ由
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b且
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飞
B·E·--EEE.
,,
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2
-nn L
吨肉'』
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n。/μ川
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71 ny
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+'十
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b且
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LU『吨
ζ
nn h
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飞
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RVJ OJ 、‘,,
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、
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nn 呵'』
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-'3-
、11
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nn 内ζf''tEE飞
b且
向ζ
n 叫ζ
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kk 叫t』
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111JQ n-27、/-z
rL-ak
。)Ln (衷: 1)
叫丁)r:)士(-1)(x:才'
(,1)KCYJ:j2时,(叮
k-=O
nγ/μ
k-=r
辽 (叫:XTJ:7 皿 2吁:) . k-O
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na 飞
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8
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十
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叭ld,
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k n?/μ… A1I/ + kk
斗,
n ,,,,EEEE飞
飞飞EE
,
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n+ K
J't』『E飞
K NOL
… AW 时-2
‘、.,,
aE· ,,‘、
= 也r n
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nnT /fit-
川
lj/
+ kn 牛‘
JIlt、
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且
n /,ttt飞
k …?/μ
枫UY
(刀(":2)
c: r)
'
1 十 (-1)
1+(-1f-r
k-=O
、
EEEJJ
,
nn
、
E11FF
2r/ ,。
ιd
i---n
n nζ
K/Iil
飞
卜=
内ζRJ'k
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9』
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,,
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k,
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l‘飞
/Ittt、、飞、
1
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rr
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由
nk
fkJFI--
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平L…时-Z〕
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号鸟-
山) Z(k:j)C+l 了十) - .L: C:tk + j :叫j.k=O k-O
5109)ZC)「;丁
5110)Z (iyn
2n - 2k 2 -Cn2:j ·
k-O
k
(2n-k) 2k
飞1IF/
&4n /,l飞
n 肉,』咱
ι
n a币,
,,‘、
自
>/n
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h+
叭l/
nnrn
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飞
E
叭ij/E
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l/
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-12n. ·K-'
n
2nnr -nt
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hw1
人州)
飞U川
+k+k
nknn ,,,SEE-J''EEE、
I''SEE飞
-K-vh-K
3jill}
、川J}
陀ZZZ
二2
〕
rEL
4E
呵ζ
翼J
ZJZJZJ
川>2: (叫二nk)C叫 :71)E(仇 1) 三 ,
k- 。
~ .11日
\112/ nn zJ /ftt、
n 吨'』
吨,』
叫j/
+n b且
,,,ZBEEE、
\飞l/k
12 十nn L
件冉,』
/rtt飞
nz…
~.116) .、1l/
nn RJ /fft飞
n 呵,』
肉,』
2-5 、
11fJ
n +n -K ,,Il
飞
、、IJ/
k nJ』
nL
斗
" 呵'』
/Itt飞
nz…
-35-
12公)z (叫:)(nTm1)n艺1 十但412\
唔 (:)2 三卡 (2:) [J~l 才
n 呵ζ
叭l//
内J』-n n n
J』/FEEEE飞
、、1,,/
nk /Itt\ k
mz… 『
,t
n x 十
x 、
tte'/
n44 ,,,SIB-
k x
飞飞EES''
k44 VVA nk /'』E飞
nz… nu
帽
、、EEE
,,
b且
4J
/''BE飞
、
111/
nk frtIE飞
bι
、‘,,
,咱••
,,‘、
nZ川
‘‘.,, 内Y
A嘈•• a
唱··
o , j"n
(,1)n , j asn ,
川l//
n //,tt飞飞
吨ζ
n 吨,』
飞
11'/
kj
黯
/rttt
飞
Ett1/
t11t'''/
唱'唱'
k/th
十斗
、It--
,,
n-k
nkF2
吨'』,,,,
!li、
/ftl飞
mz川币2川
nud-叫4
叫ζ
hl/ /Il-·、
2
肉,』
1-k 飞
111J
b皿
+k n /JEEE飞
、EEEE/
nk /rl飞
k nZ川
冉ζ
内ζ
nz k - 1
kOLU ηl/
b且
L且
+ /il
飞
、111''
nk JFEE--』且
nZ川
句4
k 嚣 O
-36-
k
(斗 )n
- n
'
,
(R.R.Goldberg)
hZ j .. n + 1
斗门/
。。
(:r 00
(:) 2 126) Z -I ,仇杀 1) , 2n 十 2k 2n - 2k+ 1
k-O k=O
C咱
(飞) ~ (+lk 127) L 2 1 dx
τod1 - z2 91n2 x '
k=O
L C:) -Jfu1·zE J dx (+) , 1 - 2k
k-O o
BESSEL POLYNOMIALS.
9) Yn(x) = L C)C:, 1咐_) k
E F 入
?气。) 飞(x) = L k nd n
‘、.,,
x 叫ζ
,,‘、
•• ,k 、
li1/
乱圃,
-L
且
n fF』』』k、、
t111ts/ nk /,,EK
K 、‘,,,
aτ··
,,E‘、
,
m(-1)n nx D
呵,』
x e
2 -x e
JACOBI POLYNOMIAIβz
1)prb) 位)z Z (r】:丁:~:)c~lrk (干Jk ,
n 、‘,,
aτ··
,,,‘、
J?J f hu
飞E
,
x + n 十
a x
fEftL nx D
KU
x 十
a x ,
,‘、
= 2" n'
-;7-
LEGENDRE POLYNOMIALS
Definition by formula of Rodri恕les:
(3·132)PJN Jτn: (x2 - 1户,"2n!
1n terms of generating functions , c:ao
三 tn fnh)E(1-23Et 十 t2 ). 去n=O
ρ
白?/μ
n .... 0
10 terms of the Hypergeometric fUnction ,
pn(x)z 附+ 1, -n , 1; 号~)
1-1eoy Bumme. tions of the form 5:2/0 1n the 1iterature involve
pn(仆,曰nd the f、011owing table Of equivalent hrmsof pnhj may be
found of use.
(~.1~3) P (x) '* 2-n n
n x
k、11//
nζ
-n 呵'』
/,tt飞\
、飞l/
n-K /It--k
mz… - 2k
(~.1~4) 飞(叫亏~r zcl (芒-}-)kk m:::: O
15)Pn忡 ZC)C:j C; Il
-~8-
n
(~. e6) 几位) - z (:)(2:) 2-k (~_ 1)k/2 ( x - (~ - 1忖k==O [才可
(}.1'7J p_(x) -= L ~:)C:) 产 rn
k-O
叫~ c:r: ::k) 2k x srxn嘀x + x-1) ' (R.P.Ke1isky)
/7(/2
窗户手 l 仇"气 +Jt)n"'0
叫z (-:X: ~J 产 c (-1川W , k-O
k qζ
x 钊Ij/
'n k f''EE、
\飞lj
nk ,
/t飞
k
o
nγLh
-L 十
万盯/syo
向1一节
wrp
()问 Pn (x) = + L, (X + Jx2 - 1ω 守主)n ,
k - 0 (valid for integral 乞> n )
-,9-
Integral form of L. SchlAf11:
p _(x) = ~-~~~ I u 2 Tr l I
(t2 - 1)n dt D ~. .n -r 1
2 U (t - x) ,
c ~here C 1s a circ1e counterc1ockwis6 about the point t ... x •
In genera1 one m8y define p_(x) for a11 rea1 va1ues of n. n
and relations such a8 the fo11o~ing are found:
‘ •. ,, a唱,
b
斗A『··. ,
、,,
( n + 1 (1 - x) Z (n: 1
。r if one defines T(z)E PZ 4
x 飞,
1/
x-x +--/tart-
k =-(1-x户 L c:r .k
z cr xk ,
k-O
then the above reads f(n) E 坷-n-1)
-件。-
A G即ERALIZATION OF THE VANDE~~ONDE CONVOLDTION
叫Z X+JEKzr:丁 y
y 十 (n-k)z (吃了>j 二;;二;fγ 才(contains (1.125) as special case)
MZCMKYCn寸 - Zc t: t- kZ)t-了
叫zc:丁(Yn-工) -ZC~:: , zk t (Jensen)
~.145; Z (丁才(PJ才 IC;
p + 1 I _..n-p-1 z t' • , (z - 1) ~ . t 0 三~p~n-1 ,
n+1 z - 1 • P-=D ,
z - 1 (Jensen)
(}.146) Z (X:γ;二 p rqk
(x t kZ )(y - kZ)
p(x+y-nz) 才- nxq
x(x "t- y)(y - nz) C:j · (归H恒问坤时阳叫8吨咐甲中g萨伊伊e盯enn - 1 n - 1
(:)(士刀 E 士C)Z川7} 艺nZ - n 十 k15(nz , kz) kz1
(Vnn der Corput) k=1 n (M-:7 (") .1 1时 Cj斗(d - 1) Z (dk - 2)! , (Cbu吨)
(k - 1) f (dk - k)l \ n-(cf. (7.18)) ,
k = 1
-41-
ADD工TIONS TO TABLE 3
》Fh,、
111,
J
.、1112JIVE
n1) -+Y-
、J
XmL
斗
竹叫川
/lh川
274
、11l
,
rrk
·+1
、
J
RJn+h1s r
飞
+mtd
'XUX+n
Ln/l
飞
blu
e'=-r
飞/,‘
x
adttt
飞
e
,hj
飞
II/)(
,,『
4EWU
,
.X+-4d'
田口飞
J
at4t41
啊M
州i/
〈
-nzVHH扣儿出
lon+++n
+泣)xmmk
xi'
叫
/ii((
/lt飞
O(
「
17LZ
fi<IILL--
n
土
23EPJ
2
旦
k
2=.-)
Y
LEEf-F+
=飞
1iJX
zrk kk
\l/+-3\l/ kkkix
-r-mrr xaitEFX-k /
Il飞,
nk/Il
飞
Y
叽ll'/tt
飞趴
1l/n
+'BK+m/l
吁飞
aa
、Jk
‘
11idr
x1xxk hJ』-'‘、
,SIZE飞
rktit-
飞
IIEEE飞
nyLMn?JLMn2Mm2ω
、‘.,、
1
,、‘,,、‘.,
QJD12 斗
555
4 ••
寸·叮
eAt
ZJZJIJ
(…3川川川…1巧叨叨5刃切3引) :趴C:川)C二: k) = 丁士子什钊τ斗(- xm寸丁Xm- 1)
ω叫5川乌):(Jj 儿二) = (x: 丁(b;? , m=Irl
飞
11j/
h)Z ιL4t
u·+ bhUta
hζ
fat-1+UX
-n
二
/fl飞飞
-ox -41hu/It--4
,
n--a+ -·工
nn
---r22 -ai22 s?V·
一--n
旦t
,""-++-+
飞
l/-x
王h
m
、-EII,
J--
+myn s++ /d川川
xa\11/;
、1i
俨回
484E1
』
sn
,,Et--v
··-LA
IrsEEE、=
..
itE1kk1-
=\1.h---a 、11
』,/Y-xnx
m-JIlt/SEE-+m/IIEU\il\l
呻..
illa--
,-
μkah+缸+
叫-uxjkk
/l
飞
/luv/ll飞
/l
飞
oa1c
-h\/二
nn\/μ=-YLPn?L
回
knkk
、.,,、‘,,,
F>
678
5FHJF
〉民
J
AB-4
,.a··A··
ZJZJZJ
可J
-乌2-
~. 159) :(JfjC::)=(J: (:)(γ) n: k 扎
., h 2
、EEE
,,/
n
n
+
向d
x JIt--飞
5.160) L (_1)k (叶)nω 丁号722a ,
川61);(-1)k(:〉(nLγz:: =(4)n(nL 丁户,
川62)i ←1)k(:)(n :丁产=川 (n 2:) 产幻,
3163)i(:}( 飞2)zffif: 〉 2n也 (Carlitz)
;;16ui 叫:)( ~2) =川
\lj/ , + kn + P
/IIlt 飞飞IIIJ
Km qq) /lil飞u
no
mYLMJ
n a
--14
\ll/
切
4Ea
pu
+ kmE + q在
/FEEt-飞
、EEEEt''
kn pp ,,,EEE-‘飞
ny
品RJ rb
冽··
'吨J
、‘,,
r e n n ,e T B
K 1L +' nad 飞
J0·K
2JK-
址
'W(dh
-n
飞
J
R
占JJJhhZY
飞
J
叫J』
Vuqr
飞
yxxk xr
、
r
、飞,
rk2 12kx +-11l/2( L且
AI
、EJ
Z+L22 、
Jn}y
、-EE
,,r
22nE yfIl
飞+23
,,IEEE
飞
+『
ζ
-x 2+florkntlo xn
『飞、/---2
『\、,/一-
rt2
,、
JK
,
dk
飞ll/=
队lj/=
takzk --'nn 「ζnn
r''』EE飞飞
D』
KIBE-rf''SEEt-
飞
-11·LLIll ++EK nk22 「4
『F』
'''ttt11
、
,,,st‘飞
n勺
LM
、‘,,
fo ro 冽l,
、J
k=O
(Brenner)
-斗3-
. (3.168 )
BFFL 2 、、lj
xa rFSE飞飞
= 飞ZEE-
,,,
kk ++ xn /'SEE飞EEE
,,,,,
k -K 4·--n x ,,SEE-、
2k + 1 n +k + 1 '
(3川川川16吻6句9 ) T (; : : )XY(「x ; k ) = ( 7) ' (3川川1η呐7mJO
(3川川川川1η171)叭川1忖) : (C; :二:)门(γ ) =(:) .
(3172)ic::) 仁++:) = (二) . …;(丁) (丁) =(丁) .
(Graham and Riordan)
(3.17鸟):(气了)(;1)=(:::) ,
(当丁 75) 二(斗(::7) 产 =(:) . n
(3.176)
k=O
n手P(3.177) 二
k=O
(3.178) 了k=O
(;;;)(;工) 2盐+1=(:::)
(JL:1)(P; 丁
( 2p : 2k)( p :丁= 2ntpr;丁户
= (γ)俨PtlMoriarty" , H.T.Davis ,et a工,
(3.177)-(3.178) are equiv.to (3.120)-(3. 121).
-乌鸟,
(n/2] :• 1)k (n - k\(k 飞 2山=川r{ n + 1 \
k~r \- I J \ k )~ r J 飞 2r + 1)
(Marcia Ascher)
。[乞 )k afK(n : k) ( :) 2n-2k-1 ♂( :r) (companion-piece to (3.179)
- these are inverse Moriarty formulas -
t> 2)
飞
E1tI''
4t
x+
a
/rttt飞
= 、、EE--EJ
k
x-
n
IIl-‘‘,E·E·-,,
kt ++ xk ,,SEEE、
kk 、.,,、‘,,
4 ••
AE
,,、,.、。咱』
n?JL=tyL=
krLK
、.,,
(Brill)
2b + 1 )(38 +
2b +
= (_1)b (2a + 2b + 1)!(2b)! (a + 2b + 1)!b!(a + b)! a
+ a ?」
//l飞
b 、.,,
At ,,‘、
= 2b + 1飞(勺/ c: 丁
(A. Brill , Math.Annalen , 36(1890) ,361-370)
ere 2t = 3a + 2b + 1
Discussions , proofs , extensions by Gould , Mathematica Mononga工iaeNo. 3 , August 1961)
1切:(叫 :ì 乙 :;1= 川
-鸟5-
TABLE .4
SU}.1l1.ATIONS OF THE FORM S 11/1
n (:) (~ ) (n: 1) ,1) 二 x+1 ...
C:) x - z + 1 e: 1) (:十~)k - j n + 1
n
Z川C)(b:k\-c
(b ~ c > 0) 1.2) ' n+c (nu) (b:k) (R. Fr1sch)
b - c
n n
L 川C) (xyuZC)(x-KF{1·HK) n-k y
I.~) , y 十 k
(Y:k) k=O
|月二(叫:) k g 1 ZCIB; (γ)(4;7kzo
E Sr (x),
rk 叭l//
+ /III\ rz… -
叭l/
1
一-Tn
-/'『tt、
1.5) -n x wh1ch 1s (1.42) again. s 1(n)E 2(2n , 1) , SO(x) =
x+n
n (:) 1.6) Z 2x+ n + 1
1:
(n+2x) 的k=O k+x (2x + 1) L -x
2n + 1
i.7) Z k(缸k tnH 十 2')k=O
k t- x
-乌6嗣
~.8 )
斗 .9)
4.10)
、、,,,
aEE' ,、.. -B
M句·
4. 12 )
k 1 当)
、‘.,,
LH『
a、.. . ,
""可
2: (-1 )k c:) 飞l
』
F/
E
KEES--
内ζ
、
-tf/-
n-+IMA 4'n-xaa
皿句ζ
x-It
』1飞
』
iI1·
xx 内ζ,,a··Et
~ (:) hofn;1)
÷仁(:)hO(2ni1)
-= 2
n =2
n + 1
民l/
-T+ 飞
IF/-mx
hn
可几
工'n
-x -JrtEE、
n 2k 20 n z _1{-1 ( :) 2 Z Z = 2
k (2~) k k
k a2 1 k =: 1 k -= 1
n
L (_1)k l :) 2k 2
=:
(2: ) 1 - 2n
kEO
L (:) (k: r)
-Z(n;7xk
.. C: r)
n+ r (x 千 1 )
n
(2k+1)j1+(-1)n+kj pk(X)
I n+k 手 1\(n-K+1)2k 十 1( 2
OLh
-乌7-
n ... x
'
Lt+2} (川)Z 川 2k n 牛 1
-…回- ' rEK 5 e:j k-O n 斗 k
n (工)寸「 4k +1 2k > 2 -刀
L.-J
C: 十 :k)2D 十 2k + 1
k-O n t- k
n
Z 川(:) n 千 1'回.. '
c+叶? 了 :tfk....O (2k 斗 1)( k
ITh1s may be rewritten as an infinite series actue l1y:
1 二二工 1 牛 (1吨)(2-n)1 + n 步'
. . . . . 1 (2-电… 2D 7 2 --‘万.4n 1 1 小· -(2D-1川'
(H_ F. Sandham)
'l 1 十 2号(:)h1(x:or; 与 ( 2:)
R(x) 主 o
1) 艺 (·1)YDK2\- 0 ,川ed 1 <以 2n - 1 , , o //k十叫 p being i
号(:)、 kx/n主 c + : -k) … +12(ijj3121;四)~------------~~-.
U ( ~) 名气 (71+1 ).1
- 乓(足协) (Jlt 't) … (~oI 2") ((j. 1 1:-王《例忖 1:; /J.• 1J1 廷4吁沙只
一川
饥艺阳
-斗8-
(件 .22)
(乌 .23)
(鸟 .2勾)
(乌.25)
(斗 .26)
(乌 .27)
(4.28 )
(乌.29)
ADDITIONS TO TABLE 4
:川k
2,(叶
I (-1
:川k=O
(:) (:r C)(:: y1 =
(:) (二 S1
C)Cn~ T1 =
(:)( :)自1
:)(:f =
J( :f 1) (:f
-- 2n +丁n + 1
., n cu =
1- (-1)n 1 2 n二7τ":"'" +
= u n - ?
n T
-宁
n 、‘,,
列E·
,,E、
n
ndq 『ζ'1-2
'+-+ -Ea rlJ11、
= v , n (-1 )nT
n
自- 丁 + (-1)n 1 z 斗- n -.n
1 + 1
-四-- • n
42 -e n
B =
= C = B - A n n n
nn llkA 『J』?』
n--+-n -nn-
fId--L
n B n
、、•. , 列i
,,.、
= n D =
(UO);(-1)k(:)(x;k)-2 X
x +卫
The first of the above eight sums arises natura工ly in a stat土sticalprob工em; it amounts to the evaluation of the moments of a certa土E
distr土bution. (Gould)
this is a 工so listed as (1. 鸟2) (because of intimate relation with (1.41)). This is a1so r = 0 主n (ι 的.
-件9-
TABLE 5
SU~~TIONS OF THE FORM S10/2
00 h
1.1 ) Z -~ 2 k2 (k: n)2 k - 1 k - 1
n 2 n+1
f 1-z4z·p. 飞1 z ~(D +.1) Z (:k)士'-(:Y 2n 牛 5fn+2) k 属 o
n +- 1 k.... 1
(Tor B. St&ver)
-50-
TABLE 6
SUMMATIC5S OF THE FORM S "/0
、-飞11'/
k •'
zn 吨,ι-z
/it飞
飞11t
』/
-m 斗
xn qLE
X
,,,EEEE
‘
飞11tt
,,r
nk nζ
/It--‘、
k
hz…
2) L (叫 20(2:::1(2::;Jj
.,, ,
z
.,,.-、,r
』,,,
z-x 句ζ-
,t、
-a,-
n • (Fjeldstad)
川llIJ
2n 牛'+
zz nt ft11·L 飞
IBI--J
n 2n ++ '鼻
X
句ιvvλ
nn 呵ζ
ff』''飞
n (什: t 'n)
C~γ: 2j ,
~'or 1n the symmetrical form
η 叫iv
牛'十
pp ,,EFt-
、、11
『
tI'
PK T+ nn ftI1t 、、11·l,
J
nk f
十
mm /Fsz-k rkm m
γρ←
...-'m+nT 'D )1 mf nf Pr
, (Th. Bang)
的li/
千
h
X UVEA户
nn nt rIt--k n 们
1j/
-n x ,
Is--
飞
、EE--J
xn fIII-L \飞lJF
-U圃'
xn
内FM
I''『t1
、、EEEESJ
XK ,,rtzl·飞
、飞llj
nk 呵'』
/It--、
b且
nu
hOLh
;) .L_ (_1)k C) (:)C : k) - ('~J -x.二}~川 )n ,
-z 叫:)(丁川仁).hz RJ
、飞1
』』/
nk 呵'』
/ft-l bι
、.,,
aE· ,,‘、
川
-RJ
nFafe RJ-n ,,‘、.
" 、.,,
,,‘、
田
\l'/ nn zJ fIlE1
、
、飞11lj
nn qζ
,,EEE-飞
n ‘,,, A哩,
' (A.C.D1xon)
k=O
-51-
b且
向J』ni!
、J
41m 十盹
hvd ky阳w
b且x 川11/
+k /tgEE‘飞
、11i/
b且』且
吨,』
uvnn尸
nk
内4
!l/flLl
吃σLN
』且vd' 』
且\1l』/
nk /Ittt飞
n?μ川
18 ) 兄9
c :)'叶2k …~ , ..- _ • (x ~ 11.刃. (Dougall) k) X 啊..X
XJ f/ 2
刊叮
<U 、,,,口
'x‘
m
nED
a'''
自灭J
引引
人飞?卜
"-2 ZJ 、1IJJ/
xk /Il1、
K JZ…
\10) 卖了/-xy ip. 但L)! (宅L)!/μ1 二土生
f70\k J X 1r X (-X;1 )尸吨,』
、
11ltf/
nn 内ζ
/ttlL 川ll/
吨ζL
民
nn LUT呵'』
/,,,,FEEt
k飞ltu,
叫ζV
人
n-K 吨'』
/,
IEEEL b且
hz…
川lj/
fr n
呵'』
飞
ll/MMH
气
2.-J
nJJnr
.,
/Iυ/(
n'r
一曲门
I/
rk-n
,电
d
--内,ι
ST-x
,
-z
•fn
.叫
l//y
?Z/IL
--BIttJ
A-r
,他,『,"
趴·l/~"·Ifl-
2kdFdM
咱飞
斗-LMF'E
---K.
同J』
nn
飞Etlt
,
f
2-nkj ,
Vuλh/lL
KK
们
i\11/
A
,,ESEE飞
bE同
A儿
U川八
y
飞
!/nk
nkn JIltLJ''tl飞
/lttL
K)
占J
飞i/
11xk r
、
r
、
/I11飞
nZ
川的2γz…
飞
Ij/
X4J /IItL 川li
川J
中''
/IIi飞
r \飞lIIJ
L且
xr /III-飞
、
1111ff
-r j4' 中tk
x /I'ttEEL \飞ll/
n-K ,
I'
』l
飞
.k
nz… RJ''
「ζ
民J
j16)Z (4CYEtiYC::;:)=(-1)rctD门'
17) L ( 气十γ:二川::) -(:)(:). . (Nanjundiah
:.18) L (X: 比 ::1)(:::)EC;:::)( 叶:丁,
,.19) .L:J:) ( :)( X r : : : -) - c: γ刀, (叫主)
i怡山削20划o叼) Z (-1护i
罩 Z<叫::13( 飞)C寸,
、叮/
叶n
一叶/
,但
l飞-4'n
\EE/-x
2nn-ry· 飞
ll/2
一』tt
n44JFik-/''ft飞.
(-K EXTT 、
112/-x
b凰
4J
肉,ι
/dM1·、飞
tlj/
\1|J/K k'n Bn
n
n2 句ζr-fJ'
』'''飞
J1
飞飞
12/
飞』
/nk
nk2 /lt飞
rFIt-
-KK nZ川nz…
4『句
ζ
吨ζ
呵,』 '
-53-
Z (-1 )k (2:)广丁 2n 十 1 (丁气?í.2~) ~ (-1)~ l kJl n - ' 2n + 1 + k (如: 1) k=O
Z (-1 )k C:)( 2川r 2n+ 1 - 1 1.24) 6 (-1 r.l k Jl n ,
2n 千 1-kk-O
叫l/
+n n 肉'』
''''EEEE
、
飞
l
,/
tm 'n
内,』
n L
崎旷VA
八
nk 吨,』
/IIt飞
b且
缸中/JU
-dk 、11EFF'
bA
n 吨'』'"
//tl、
冉Jh
\III/ nk 内ζ/
Ft』1、
k
nu
nO/μh
内ζ州l
/
一
,·丁,/--2 .n-
川l/hu--r
'Tn/it-h 12
叭lJ7'飞
22z-,
ftitra--
、-
叭lj/JJ
-n
ZJ'E
、
n h崎
m
r''tf1L 、‘EE--F2
nk11
,』,/
2ra /fi、
2k
占Jnn
128
a
rV
『
/i
飞
\lir
叫
『,,,,
F-Luh&-
nrJ-z2M ,‘,
ιkft
』1ts
罩、占
Jr
-LU
。可
口了一口n
F、
dkc
h
6-w , (Cerlitz)
k=-n (叫:)2(:127);三
川l/
+j n r'It『EK
、、-tt
』,,,
naJ J''『』飞
\Iυ
bι-
n
/jt』ttk
nζ 飞
、tlEFI
nk JI''BEE-
nz… au
")2(-1)KO2(2:::)s(-1严飞-1)nL;y:与丘L ,( ~X)
-5件-
内丘、
、,Eta
,r
xn /lt飞
、Ett
,,
I'
k
n
+2 x ftstE飞
吨ζ\
111』/
nk /'''1飞
n?L O ZJ ,
hu , (Thls ~nd the next tWD formulas ere essentiel1y the one g1ven 1n 汩的h. Rev. , V.17(1956) , pp.65,-4~ k=O
吨,』
、
11fF/
xn rIts-飞
、
11lfI/
-M mb
埠
n-n x nJι
/Iit飞
叫ζ、
、EE
』
1/
n "-vh x ,Itl飞、
nOL ZJ fhu
k-O
少』
、
1ltF/
n 十
n
x JItEE飞飞
、
1ltI/
b皿
nn 22 4I X //flt\ 内J』、飞11
』/
nK Ifttt飞飞
nz… 同
ζ灭J , (Meth.Rev. ,V.18(1957) ,p.4)
(Known to Le Jen-Shoo , 1867)
:6." )
飞
tit-/
k
注+
呵'』
b且
-//tl\ n
飞飞
lI/
呵'』·.,
i1
•lk
--Jn b且
4JI''tEE
飞
fftt1、
hB,,、
1112F/
飞
1/nk
kk/· 内J』
JISEE飞
fttlk 飞
111/η
nk·
吨'』,
t、
Jft』EE飞
、•... J--16 吁
勺乙hn?μ
叭叫I/
nζ
-n n 呵'』
It's-E、
、11t
』,/
nHaEM-FEEt飞
E
:6::列)2D - 2k
2
k o:: O
-EEB-』,/
川】2
//Il飞
川山2
f'',,I
EE飞
〈 -1) 步 (-1)j2
(Generelizetion of (6.~5) )
6.~5) ~ (-1
21 、
1iIJU
nAMA
nT
/fl飞
L
nE rt
、-呵ζ
土
li/
1-
主gk
阳
VHnf
k-K 呵Jh
n
n pvun
户
itnk ilt-//tL akk
/Il
飞斗
dlt/( tiinu
叫Jn?Lh
iu川
h
nk2 /IE
飞=k-O
内lIIJ/
2n fff』!
E
nζ
、
111/
轧匾
+k n ,,,『tEE1
、飞!/
nk 句ζ4
·Jt11、
k ,,‘、
hZ川
、‘,,
r。
ZJ ,。
-55-
吨ζ\
11』
1/
'如n
fI··』EE飞
飞
lI/
k 中'n
nJ』
n 只J/
Itt飞
向J』\111I n-k fJt11
nOL
… ?t z/
,
;, ;:8) L ( 斗 )k(: )-;LnxfL寸 』凰-内ζ
-一+
n-k nζ-
E 川
(2n - :n+ 2n)
,
\// 「JιJE飞
J
k ffL …
汀…
n (-1 )
2·n12
三C/ -coer 川 in (1 - x2)n Pn (叫(6 .~9)
(6.40) 认叮l/
内ζ-r VVAHh LK
叮/』
-n n nJι
川VEA
nK IF-SEE飞
k
飞hEJrt
、、,
恰?L」
J·
→♂
x n
p-rx
nu
(Cer1itz)
n - 2k - r x
(6 们);←d (:)(丁γ; 二 b丁=川(:) bn
(Gou工d; special case a 0 , b = 2 , x = 2n suggested by Samuel Karlin)
‘‘11t
』,
f
、-B
』,,,
ddd
+
++cd aca /,
FEt--/FEEt-K 飞
1jd\tl叫
a-ac
+
/lh
,ih
--= 、ZEE
,,,,、
EE--,,,
kckc ++++ abab J'SEtt飞/,
SEE-E飞
、11』,
1
、飞BEt-
dk
c-c-
kd
/,,EE--tSEEE-
飞飞EE--
,、飞
ZEE-F
bkbk ,,ll
飞
/''EE飞
?}Lkh\/UK 飞EJ
、.J
?」之
J
L
吁
L『
.. rbζU
Jt、,,.、
-56-
CM.T.L.Biz工ey)
(Bizley)
ADDITIONS TO TABLE 6:
(6 件的 tc)r:::k)(m+:_J=(:)( :).(R叫)
(6 句) ,t (:)(n:J(气: : k) = (:) ( :) .' (Ri 叫
(6 届):(:)(;)(X+:::-k) 平 ;γ: n) .
(6. 每7)
n?LW 、
11j/
Z +n y /''t11 、-EE
,,
J
Z +r r J''''1
、
= 飞飞EE
『,,
n -k z+r r ff』t11
、飞EElsJ
K zk +m yn + r JFEEt飞
、‘-EEB-
,
YK IFtst飞
(equiv. to formula posed by H. L. Kral工)
(6 崎)仨)吵了;二 1)(X: 丁了: k) = (:)(:) •
(extension of formula of David Zeit工in; equiv. to formula of Surányi.)
(如斗9县一口-2k
k .1 \ k (( 6.20)-~ '"芒,兰二eel (MacManõñ)
飞飞EEBEJ
na ff-、EEE,,
EE''
as--n a 「ζ,,,』EEE飞
嚣
\Itts/ ka J'tEEEtt 飞飞111jk
nnζ-
a
r'''EE--\EItl' 4t ak + k ,,,,EE飞
n?LM 飞·'
O FhJ fb ,,‘、
(C. Van Ebbenhorst Tengbergen , 1913)
(6.51 ) n?乙时
n
+ X JFEEl-
= 飞11
』,,
k + rr ++ aa + X JrEEEEE1 \EEESJ rk /
lat-飞
、-KEEF-
nk ''''EEE飞
n + γ+ : + r) (Contrast with (6.19) (Gou工d)
(652);(ufk)(二 k) C : k) = (丁 γ; 丁,
-57-
TABLE 7
SU1制ATIONS OF THE FORM S 12/1
(:) t: Z) 们 (n)k
k/ (:) (:)
(叶:+")气丁) ' 川斤。 k)\ kJ (寸z(寸7
o <_1)niX : j ( :) (刀
17.} ) Z ofO E
<_l)k (:) (X:) ( :X)
、‘.,,
L
斗. 哼,t
( n?户户k c: 0
2n (x _ n)! 、/-n:-
(X - n/2 月( -n/2- 如f
ft) ( 0 。~ j< n ·阳嗣.
n
Z (叫:) =:: (7.5 )
(。n n 2n x
j - n , (叶 r- 2 k 黯 O
(2:)
(7.6)
kvl/ n
们/主
.,
礼川
.. I/
缸
内J』
4J
甲吨'』
+→一\tj/
nnMFatu--内ζ-
叫ζ
」.
/IIt飞-r'It--t
2k 2
(7ui(d(:)(;) 了丁丁a ••
、11EE
,,
J
4t yn +飞
J
xh /,『,』『E飞
+LW
u
.‘、EE
,,,,,
12 -nk yr
飞
J'ISBEE--、、EEEESE
,
F
AB--n x ,,,,EEt飞·
n 、、,,
at ,,.、
=
( ~:) -4n 2:)C:) 2 , -
(2:) , .7) L (-1 户 c:)c:
(n~) 产k c: 0
Z 川(才)( 2:) 2k+ 1 '.8 )
(n: k) (叫 k + 1) k=O
n -22n(…-j. z (-1)k(:f:J
2k 7..9) 2
(:k) (2叶1)k - 0 (节(叫1)n
Z • l)k C)(":) 2k n 7.10 ) 2 ( -1)
IC:
C:k) (2叶 1 ) 2n 卡 1k=O
n (22:) 三川k(:r :J 2k
7.11 ) 2 n - (-1)
(:k) 川) x( :J k=O
n 2k n
7.12) Z (-1)。(:k) (n+k)
〈可 n\ / -n - 1 \ 1 _ . n 寸、 n,\ 2 7 .1~)ζ-' l ,. JI. '- } f. , ., = ( -1 r" r /
T ~ \.k J\ n -k) fk 十 ι.J '-kl k 十 rk==O . J k=O k
QJ RJ
14) L 、11j/
」
-K-K 【ζ,
,,』a··.、
、11EEt-'
-k
h+ n
fFft11
Illl-/ 1, '2k ++ nn n
ζ
/fi--\ =(_1)n
产( -~/4)
(忡1) C:) k = 0 (2n 十 2k + 1)
k
) L(:)( 丁 ~k\ 2 jtx k) (γ) t-: 1
...
\1jJr-
一、111/
寸
-n
zn
一ι'n
千
-x
一/fi飞
XM /t『1、一 Z -21jJJ n
j =: 1
飞叫1l/
1
一十k
-f''』l飞、
}飞
1111/
nk fIIt--L bA
nu
nzh 一-
(rn)f 2
(rn - n)f (rn 世 n)I
n n m
117) L z+-z+ , (mJ注 n) , = 2
( :)k k -= 1 k =: 1 k = 1
n ((j'1)n) z (:) (j - 1)k j - 2 n~ 1 (Equiv. to 8 forn川a,18) =
1 - j of K. L. Chung)
(~: ) jk - 1 <3.1 斗8)k-O
:: -1 for n = O.
2n (可 n + x : y - jen : 2j z kC:X叩丁,19)
k 凶 (2;:3 (XDCO(2::3 J k=O
-60-
Meny closeà ex~ressions for sums of the form S&2/1 ere known from
the theory of the Hypergeometric 负lnction.
Definition: QQ
(-Dtuzk (7.20) F( e , b , c; z) - Z (-1 )k
C:) k... 0
where e ,b ,c ,z ert complex numbers.
(7 .21 )
'~7 .22)
(7.2,) n
.(7.24)
(7 .25 )
,川 (c - 1)! (c - a - b - 1)! F( a , b , c; 1) _
(c • e - 1)! (c - b - 1)!
兰cηn'k
nJ』、
.,,
x n .、
s 呵'』
,,,‘、
了。
IC才 1) ("气-j01
k == 0
C吨2
n LC.γyn/2k k :a:: 0
。。
艺(-n/2k-i)( 的: i) k=O
R(c)) R(e+ b) ,
IC cos nx (n reel , IX/4 衍'2 ) ,
-飞
111/
-U
且
b且
k2 2 1i/tl
飞
x-n-
飞
J1
·唱&-4B
,,-
hv←ι1
〈
mx --,‘阿
-za--、
kn-2 呵ζ』-,
1、
内/』-,
咽'-
a e r
sin nx 11:
cos x
1
一‘111/
千·一业
k
缸-Il‘-KA
X
』、
JT
圄
4··
-m一个
μ「
SE-x rLki-内
ζ
k-r
、
冉J』-,
内ζ-
、
Aa e r
- sin nx
bh 内Jι、
‘,,
x n .eu 内丘
,,,、、
cos nx "'" 11fJ/ kk
nζ
/tts2·L cos x
(n re e. l , I 斗 ι1 )
-61-
~6)
∞OLU
〈←川川E斗卅4衬1η甘咐)川户k门《 C. -丁: -!) - (a 十 b -告)! (-告)!
(8 -告)! (b- 主>! '
(e +- b +告'" 0 , -1 , -2 , . • . )
‘,,,, 『,1
肉''』 I
F--t
飞
、
1lJ/
2k -E-t』』,,
2·K JL
nu
∞勺Lh
2·k
叭llj/
kubuh
4干F(8 t- b).') 1
I (b -去)! (8 - 1)!
(巴专 b + 1 笋 0 , -1 ,-2 , • . .)
z:::…
8 - b
‘ •. ,, nRMF
内JFh艺 (-斗叫1
c:) (b - n! (-去)! 2' - b
(8/2 + b/2 - 1" (b/2 -8/2 ... 告)!
(b 1: 0 , -1 , -2 ,...)
7.49 )号 4C:JC~:)= 1 /叫号 r: 1) L K/n飞 x +丁~ nJ 乙 (y ,
) (:) _ ( :l ~-. l k)
= x - y
called Capell土 's relation by Harry Bateman , Notes on Binomial Coe~ficients.
Replacing x and y by x/z and y/z and multiplying through
by Zn , then let尘土ng z -• 0 , this gives the limiting n 寺、 k-丁 -k , n n
f。m yu 之, x~-'y-~" = (x&J._y~)/(x_y). Cf.(11. 的, (11.5).
k=1 -62-
.29)
.~O)
‘,,, 4·a RJ . 1
1.~2 )
Definitionl
可-= F( -川 n t j 川 'JZO(斗)k(2:)fD J ,
R; 配 F(- 2n - 1 ,去, 一川1l/
-4f 4,
-4Jb
且
一中
-,fJIlt、L
KJV ,I,,,,,、
川川l//
+k b且
川Z
…千吨
Then theee sums setisfy the recurrence reletions
,工+
吨'』-
-41 十一n
n-句4-
nt sn _ _ ~ 5~ + 'R~ j+1-7Wj , 'l\ j ,
2~n t- 2j 千 1
n f- j -t 1 自~ R~ 十 s~ 千 1
j t- 1 ".- j I - j - 1
Some apeci s. l values of thesesummations ere 88 follo 'W 81
r.~~) 5~1-~' (n~ 1)
、,,,
h
斗咒/ s;E1 , (n :;::二 0) ,
'.)5) 5~ _ n t- 1 1 2n 千 1
'.;6) 5~ _ ;(n 千 1)(n 十 2)2 - ,
2(2n 十 1)(2n +~)
'.,7) s;a(n+2)(n 千 ~)(11n + 10
4(2n千 1)(2n 手- ~)(2n 手 5),
'.~8) n (n i-2)(n 千~ ) (n .t 4)( 4~n -1 勿}54 -
8(2n 十 1)(2n + ~)(2D 't- 5)(20 千 7),
, .;9) Rn4 -= -.2n+L , (n 注 1 ) -, n
问) R~." -1 , (n ~ 0)
由63-
.41 ) n R1"- 0 ,
.42) n+2 , .‘,咽4 ‘'
'-;:2n +~)
.4~) 二(n 卜 2)(n + ~)
ι',"'" 4(2n' r~)(2D 十 5)
.44) R4 _ 21(n 十 2)(n+ , )(n 千 4)
8(2n 十~ )(20 + 5)(2n +- 7) ,
.与)
'.46)
'.47)
F(-n 告 j川 )EZ川
I .2j •. _ 2 C~) 人(… lì x) . (1 - 4z 川 ,
hιJ/
呵,』bk{
+·'+ /JIf--、飞11tl
nk JFFttt
o
n?Lh
-、E
,/
7''1lL -2k .. _ k .. .' _ n - k 2 ~~ (4z)- (1 阳也) 晶'
(These relations , (7.29) - (7.47) eppear in a paper by the author ~endi鸣 publ ication.)
t (~)+(;; ;) -n - k I X+ y + 1 -k=1 , (R.Lagrange) Ix + y - k\ X + Y + 1 - k
k=O ,-\ n I
l7. 斗9) - See page 62.
人乌8)
-6件自
TABLE 8
SUl.1J;:ATIONS OF THE FORM S 11/2
(8. 1) 2>叫2:)
'门广γ?
-65-
TABLE 10
SU阳ATIONS OF THE FORM S .11,/0
L
叶、11』
F/
XK JFSSE-飞
0
1Lh x 斗 2k s10 ~x (-2X)/
arx(-xyz , (7. <告)(Dougall) x
2) Z (:)\·÷ZOA , (Dougall / Staver)
k==O k-O n n k
?Z (;yzZ (-1)k(;ynjZM(:)2门'k-O k-O jeO
Z (_1)k (}:~ }n n- ky ~(::){ 2:) 14) / I (-1) k ̂ n J \ 2n J \ n k-=O
Z (_1)k Cn: 1) (川)~5) 2' ( -1 ) k l ~ -- ~ -J l ' n k-O
n 十 r k 1 飞 PWW
k 配 o j 1: 0
þ) z k ( -1) 、
11』
F/
r
nu·
n
i'ft1飞
9.
飞,,lj
r
k
,,ttzs't
mJ』、
、1'''
n4J /JIlt-、
KZ川
(0 , n ". r
!1 , n-r , zoy: 丁
',8 )二(_1)k 二C f(k: Y = (-1)了。)L217)n1千 2·1)r
、
1'
《川V''
' Z。 (;)A _ coef. of xn -;;(1-x巾 , (Car l1tz)
TABLE 11
SUJrnUTIONS OF THE FORJ.1 S I当/1
‘ •• , aE· . a
咽··
,电··
,,.、
ic) (:) (k: r) 。 c+: 丁
L (:r Cn +::丁( 11.2)
(11.3)
、飞EE
,,白
k-
z-
℃11
』
J
n-2
,,,E·E·-『-+
飞
11F-vuk
yk-
-+
,,SEE--
-EA
、EEEa,,--
xk-it1 ''''za、-
nyLM
(x+γ7 亿::)
(叫:寸'
(叫1) cn :, (飞?a::
' 20 + 1
--
、,,,,'
膺'-
-、飞,,/
+n-z y-+ /I1、
-ya
飞
EEl--+
z-
EX
+B
一,,,』
t‘、
X
』
,,tt
飞甲
(equiv. to a formula of H.L.Krall)
TKEJ/ 圄a1.
丁
-+k
『n
-/JSE飞\
飞飞BEar
yk /ftl飞
、EElsF
XLAi--rrtt'111 仇}/y+
-a k
注/寸、
--、飞
EBB
at
yn
x+
+
n
xI''Et
飞
''''t1、、
(11 .的 L (_1)k k
at-at
+ n
x - y ' F -
--Harry Bateman (Notes on B土nomial Coefficients)
The limiting case of this , when we_r穿place x by x/z and y by y/z.and multiply through with zn+l~ 工et~ing z 一亏~Ois precisely (1.60). Bateman does a sim土工ar thing wi th a series of form S:2/1. The result suggests ana工。gies between power relations and binomial sums.
(11.5)
、EEE
,,
f
a·--4t
k-K Y+n x r'SEE--、
、EE·E·''
yk ,,,,『EE飞
、11EE,,
XK J''It1、
k 飞,J
AB-J·、
n?yLM EEEEE
,J
ya-Its-飞-
'-飞、‘E』,
1-
xnJFF『K飞一
n -T112/ -A' --K -n 』,,,EEE、、
1
x - y
Ba teman , Notes on Binomia工 Coefficients.
-67-
TABLE 12
SUMHATIONS OF 四E FORM SI2/2
C:>O (气 27)( 12 .1) 1 十 2
二 C) (:) - ' (丁γ
valid for R(x t y) > -告
(12.2) 1 十 2 L Ht (:丁( :) (X:k)e!j
lIIl: , C:y) va 1id for R(叫 y)美·告
(12.~) Z -lll/
飞飞lF/一1
xk
一十
k
/t
一户ll
-、飞飞''』,,,
、
11/-k
nk
•+k
fri飞
-n
E 告气 1 十
At/-2 2
一叭11/
fh
一十n
x-x 27Ilk frit-飞… ,
(12.5)
(:) (:) (n ~γ :j
Z (n ~) (, j z 二二jin 仁 :D~o C- n:X-':j
nOL k
、‘,,,
a、,,
,,.、、
气
(Y!J
-叭lJ
1
一十n
-/'11飞
牛J
C}f//、
'言(12.4)
k... 0
-68-
的 Z (:Y 2k + 1
CT: +) 2 2k. 2 (n-k+1
k
.. 2D + 1
、飞-tsj
n
勺/r
4k + 1 11:
cγγ24k( This moy be rcwritten in the form
n (J: 扩.5) Z
(件k + 1) .2 4k
(…) k-O n+k
(2n 千 2k 斗1)2
n (:) C: 丁ε.9)
(X : k)(X + Y + : + n + 1) k=O
=
•
nrk 1 宁 (-1 )
2
4n T 1
注-k
y-wJ
Harry Bateman (notes on bin~mia工 coefficients) ; see Gould , Duke Math.J. , 32(1965) , p.706.
1
了;丁
1-uh
,
-69-
TABLE 16
SU1<1MATIONS 'OF THE FORl-1 S :4/1
-::.. ~j
(-1 )k C :)( x : ) (: : k)(: ~ :) (16. 1) 2 (:) k :0: 0
22n (γ)(E丁~)n - (-1) ,
(Equiv. to e result of Beiley.)
;二 k (可川丁t:::n)(飞丁(16.2) /, (-1) e. +kH b 斗 kJ\ c-{ (2: 千 :jk .., 0
d 千 k
n (2e 千 2n)! (2b 十 2n)! (2c 十 2n)! (n 手 d)! (2n).' d/ a:: (-1γ
低, (8 t- n)! (b +- n)1 (e 千 n)! (2d t 2n)! (8 + 2n)!
(a 十 b 专 ~n)! (e 千 c +- ~n)l (b +- c 千 ~n)/
(b 斗 2n)! (c + 2n)! (巴千 b 手 2n)! (e + c 手 2n)/ (b 昔 c+ 2n)!
~ith e , b , c reel end ~here d IC e 斗 b+ c .沪,n
(Equiv81~nt to 8 formule of W. A. Al -S 81e.时
(1602(:)(; 〕(njJf:11(:Y=「 ;7(y;7(H.L. Krall)
-70-
TABLE 17
SU阳,~TIONS OF THE FORM S 1~/2
、‘,,
aE· . 咱,•
... 三(叫-z)(:) (:) kl e+:可 e 十:+k)
= {X + z)! (y +z)! (z/2)! (x + y + z/2)!
(x ,+ z/2)! (y+z/2)! (x 中 y 斗 z)! z!
(17.2) 三( : ) fx 4-~:) ( :)
(x+7+n)(": )
for R(x 斗 y +功)> (~心a11 / D1xon)
(叫: t n) (y +: + n) .:;
c: 丁(叫 y:叫 n)(Saa 1schðtz )
~17.~)
w勺乙k-O
Let
(-告)!(z- 告)!(x+;-1)f(z-x +;+1),frw&1
(弓J.)!(与1)1 (z- 斗斗! (z -号1)1 、 ,
仰,y) - (1 - z产叶子 (_1)k (1 ~ 2X)( - X ~告)(yγ号 zkι~ 0 (* ~ x) (- Xγ- i)
-
(17.4 ) k=O
and then it follows that f(x ,y) _ f(y ,x) (Bailey.)
)夺(_1)k _(飞)(-2:) (_x:) f=i t27、 (-x叼
zk ~ f (J: )(丁)?了?
2
z '
(Th. Clausen)
-71-
(21.1) Z (_1)k
TABI且 21
SUMMATIONS OF THE FORM s.6/0
L
崎EIBEE
,/
nn ZJnJ
』
/IJIt飞
、-BEES,
f
n
n
冉'』
,,tltE
飞
n 、、,,
A咽··
,,a‘、
E
ZJ
、、11'/
b且
-n n ZJ /It--\ ZJ
、11i/
nK ZJ ,
ISEEEE飞
2』
内,,
'
TABLE22
SU阳~TIONS OF THE FORM S 15/1
(22.1 ) Z 时 一叭l/
-呵'』
4·-+K EA-
7f飞
k/ +k au /it--L \飞iyb
ι
+ en 冉
ζ.
/Iluv \飞i/
ban + bn
呵,』.
i同ω
/JI飞
飞飞11
』
J
LM + an
肉,』.
E
rrftlE
飞
\11/ hk /'』EE、
(2a) f (2b)! (2c)! (n + d)! (2的! (n + a 手 b)!- (-1)
州 (n +8 )t (n 十 b)! (n +c)! (d 手- 2n)! a{ b! c!
(n 寺 a -f c)! (n+-b 手 c)!
(a 令 b>! (a + c)f (b +c)! • '
where d _ a + b + c
By use of the 1dentity
ck) (γ丁
(γ)(2:::丁
and the reøult 1s tabulated under that headin,. Special C8ses
可iUW
RJ fIiL 、飞EEF/
bh +k n EJ flttL J\j hk /
'E·E·、
bEm
hz… 呵
ζ冉ζ
ZJ 、飞EE
,/
nn zJ /ttttk
n 、,,
aτ··
,,‘、
E (知)! (2n)! (刑, nf '
-73-
(2~.1) )~
TABLE2~
SUMlt.ATIONS . OF TIÆ FORM S ,4/2
飞川一刻!/
vd··--U
且
n
由y
fttE飞-中'n
、-EE--x
-K-'-
x--||i
n-·、11itJ'
IIl
飞
Z2
\飞--J
一
YLEvek vvn一-T
xk-x /I飞一/tlh
飞
(2:)( 2:) ( x: 7)
Cx : 才(气 -j(Follows from(17δ) .)
-7乌-
TABLE 21占
SUMMATIONS OF THE FORM S :~I:当
国 kC) ( :) ( :) 十 27/7,1(·1) (γ)C门口7
(x+;才
C:Y) c:) ==
,,,.-、,,
飞,
-x
z-
一斗
4'一-z
y-rt zy x-z (-+ ''一
y
z-rk vw-vw f·
一十
x-x
,
ve lid for R( x 十 y+z)> -1 (Doúge. ll)
(The 8uthor shows in e pener ewaiti吨 publication , that this
reletion implies FjeldEtad's relation - (6.1). )
T
-75-
TABLE ~1
SUMY.ATIONS OF THE FORM S .4;'
(~1 .1 ) 何?/]
( _1)k c:可 +k- 〉(;)(;)(;)k J C+:才cγ丁C+:+jk-O
R"
(x 十 c>! (y -卡 c)! (z + c)! (x 和 y +z 手 c)!
(y 十 Z 十 c)! (z 啡-x +c)! (x+-y 千 C)f cf ' (DougeU)
ve11d for R(x -+- y +z 千 c) > -1 ,
:~1.2) jfcx- 1 : '/2)( :)(- :) -
1· 飞,,
z-a'' ,‘‘.,
--z x--,,E‘、,-vd·
.,,·-‘‘ .. ,.四
ve--x
--Ft
、
X
二,,,
r
飞
-x
B
(Beiley)
-76-
TABLE 71
su阳ATIONS OF THE FORM sI6乃
One of the m08t gener81 identit1es known 1s th8t of Dougall , which
m8y be expressed 88 8n 516/5 or 88 B 7乌 "ith unit 盯&Um8n也·
(71.1) 艺(衬「「门盯+丁丁::-丁刀咒)川川川(:ο:)(:)( :)(:) ( 叶y γ +刊仰山2c肌e川手
("t口+丁;7+丁γ丁叹飞γ7川忖("叶+γO叽丁(X+ Y+飞?;7+忖0+)y
(97.1)
C+γJC寸+j(叫γ丁 C:j了:+ n) (Y+:+丁了?丁(叫:+c寸
TABLE 97
,
Expreesed 88 e 7气。ne hBS the equivalent form of Dougall's formulB
「息, 1 -1- 8/2 , b. C t d t e , - n ;
7'6 I 1 a/2 , 1 +8斗, 1;-息 -c , 1+a-d , 1 +8吨,
(1+ak(1+a-b-e)n (1+8·bd>n(1 • aHC4)n.. (1+a-bh(1+a , ck ( 1中amdhn十a-b-c-d)n
'
where 1 + 28 .. b 乎 e千 d + e - n , n... 0 ,1 ,2 ,当,..
8nd (x +n - 1)! 户-t- n - 1)
(x)n-r,-唱、~- n!l. J -飞 n i
-77-
TABLE X
SU肌ATIONS OF THE FORM S.p/O
An Bsymptotlc formulal
n p -1
2 (:r pn
侈;-)2 2
'‘~ a8 n -;.. 00.
-VP k-=O for p _ lnteger 二~ 1 •
(Polyp Bnd Szeg~.)
n
kf?P?Lxjp Z C:) (n+1)!P x - kr - x.- ~ - - '
(k+1)p ( ) ln+ (n+1)p k-O x x
(Gould)
~) t. L~:-l) L' -l)j Cp;:) C+r:jγ - C) '
VB l1d for integral r ~ 1, P~ 1.
4) Z (斗)kLγ)(…tyfr· 〉 -(1;γ ,
,5) a咽,
E
P 叭l/
-r n· r Its-aEEL ---EES
,J
+ b晶
nr Fa /Ittt飞
b且,
,‘、
问Z…
r
'
(The threereletlon8 immedietely ebove ere connected "ith very
generel eXpansions of Worpitzky , e8 shown by Cerlitz , end were
'rediscovered' by veriωs modern writers , in perti叫er by Shen汩的.)
-78-
(X .6)
(X .7)
Defini tion: 飞(q) - Z (:) (Stever)
nzvhzoq KP (5tever)
Theo the following formulas are knownl
(X.8)
(X .9)
(X.10)
, (X .11 )
(X .12)
(X .1~)
(~) -= ~ 5_(~) n ,1'.....' - 2
一25n ,2(~) ...,一卜 50(~) 十
‘‘,,, ZJ ,,,‘、
4·· n qu
., 、、.,,
ZJ ,,E‘、
'1·· n qu
zJ n E 、
‘,,,
RJ ,,,、‘
ZJ ., n
qu
5 _ J, ( )) = --.!!当 (n + 1L s~ '~(~) 口,"" 2 n-I
民 A
2 S ~(~) = ~二- 5_(~) -二~ (n - ~) s ~(刃,n ,~-" 6 n-' . 6 n-1
- (staver)
n2 5 n(~) .... (7n2 - 7n + 2) Sn_1(~) + 8(n - 1)2 Sn-2(当)n'" - ,... -, -n-1
(Franel)
,
This relation just giveo Wes f~r8t proved by J. Frane1 , 1n answer to a
que8tion posed by C. A. Leieant in the first volume (189均 of the
/.. . .. .. / journe1 .!'Intermédiaire 但旦 Mathematicien8. Freoel a180 found the
r
followiog formula es well a8 other interesting resultsI
(X .14)
(X .15)
J SJA)s2(2n-1)(5n2-5n+1)s JA)
十 (40 - ~)( 4n -问(An-5)S JA)
主 (:r 小言:c:1;(Franel / St~ver)
(寸(叫-79-
where: C( q. + 1) n ,k
、飞、EEE
,
J
-K4d ,,,aEE飞、J
KTU归
、飞ht1/'
"K IF-飞
c(q: (Nenjundiah)
'his genorel result includcs several speciel ceses in the prcceding
Ibles.) Thus:
k ‘ •. ,, a
τ··
,,,‘、
E
飞飞,,
♂
5·K
(n ;, 1)
( :J (0)
- (-1)
飞11t』
ff
nub
且
fft1
、
回
‘、,,,
,E·· ,,‘、.
户V
11if/ nK If-1、、
z 、‘,,,
向ζ,,,、
FM C( 当) =C) 了:丁
阳c喇M蜘圳胁灿8时盹巾h加10n
owever this 1s too εxtc盯nsive to list in the present tables.
x.16) 主川( :) (1 :1 b 1k) (2 :2b2k
) nu --llif k
r b
r
+m r a
JIt--、
(X 丁 7) I (叫:)k11Et--/
‘飞'he''
k22 RJ』-bc
a ,,,,EE飞
,
rttIFfitt
、
、
1lij/
、114
kt4l alLUFUW
a ,,I1、
/Il--\
•
飞tBIll-
,
J
、飞'1
』,
krr rbc
a rtt『t、
//tIll--
、
= 0
when b1 c 1 + b 2c 2 +… +b r c r <:n
(again a trivial consequence of (z.8))
n
C:汁:1(af) ( a(r) (x.18 ) • • • == 0 ,
k=O
hen bAc A + b~c~ +…+ b c L n 1-1 . -2-2 . -r-r -.
(another trivial consequence of (z.8))
-80-
TABtE Z
In this tab1e we 1ist some genere1 expansions , operations on
series , and a few ‘ use负11 binorniel coeffic1ent identit1ee.
In formu1as (Z.1) through (Z.1l) f(x) 1s understood to be any
polynomie1 of degree less than or cquel to n in x , thet 18 , a
T川 z a X1 i
i= 0
(Z .1) (Legrange series) n
Let g(x) _汀 (x - X i ) , g'(x) , Dxdx) ,
and let distinct realnumbers xo , x1 ,...,xn be given •
Then n
订出
41 x··xi 叫y)- L x - x
k i
n
事 g(x) Z k=O
k x
veEF十一J
k四
X
X-P··E
x
(Z .2) (Taylor series)
你十 y) 81:' γ 二巳 r(k) (y) ι..J' k' k=O
r(x T y) -二。三(-1)J (:) r(k - l十 y)
(Z .4) (Worp1tzky - Nielsen series)
叫X十川-1)也 I(叫:'让(叫m :j r(j _ k -t- y) '
prov1ded m ~ n.
(Z. 乡) (Melzak, prob. nr. 44饵, Amer. Math. 以onthly , v. 58(1951) , p.6~6; a speciel cese 0 1' (Z.1). )
叫.J
/'l nO/乙
趴lj/
十
n
y ve + T(x - k)
y+k '
(Z .6) 川y) z 1
y十 kHDYTh+y)
1111/ nk /i飞
b且
nz…
、叫lj/
+n /,Itt飞
ve f(x - k)
(y+k)2 '
(Zo7) Z (-1俨 C) T(X - k)
k c f川 Z
1-k - 1" (x)
r
(Z .8) L(叫 \ 0 , 11' nL r. 1'(k) ~ < 蝇飞,
{ (-1"" n' a , 11' n e: r. 、 n
旦[j j X J X 、气, ._ f .,、
e ~ 一一 / I (_1)A ( - J f( j - k) '-/ j rι.J ' "' l k) jzo k z O J
Relation (z.8) i8 very useful; we have numerous interesting cases by choosing f; e.g. (1.13) , (6. 斗1) , (X.16) , (X.17) , (x.18) , etc.
(Z .9)
WZ k
一二:- t'(k) _
k' k 匾。-
,
-82-
(Z .10) 、11lt
,
rb且
缸'?
/Iiw nz… f'(y 手 k2 )
2 .2 x -.K
_ (_1)n f(二十 y)十 士 C") f:~) '
2.(. - n) C::J n
(Z .11 ) Z 付(:) f(y + k2)
C:J k-O (x2 - k2) l k
n f(x2 + y) f(y) - (-1)
(:n)c:丁十 2x2 2x(x - n) t n)( 2n
~h1ch 18 e restatement of (Z.10).
(Z .12) f(x)
rπHVK
(Z.1~) 立(-1)k (:)
-- r?乙
k =< 1
rz f(Uk )
• r
订x - U
k k .... 1
1 - 1 1 .fo k
provlded r..> n.
m k rvu
c ~Uj 古川 J for r > m.
-83-
Z.14) Let f(x) be. e polynomial of de~ree n 十 1 10 x. Then
2 (-1 )k C) f(y - k)
(k 斗 x)(k+ z)
E(Jx)jc::j Z.15) Let f(x) be a polynomiel of degree nt 2 1n x. Then
n
~(叫 :J f(y - k)
(k + u)(k + v)(k t- w) k=O
u - 1 f、(y +u) -(n + 1)(n + 2)(u - v)(u - w)
(叶:) (仆:)n+2 n 千 2
v - 1 f、(y+ v) f(w 千~ y)
(n 十 1)(n+ 2)(u - v)(v - w) (川:)0+2 n + 2
~ing partial fract100s one may evaluate ~ (-1)k(~) f(y-k) 1n general. r k=。
有 (k -r Ui ' 1=1
-8件-
(Z .16) Let f( x) 二艺 -1aixi
Then L(叫:)f(y-k)c: r
) -…
r?JU 的一叭l/
,问-4·k
fLEK f-Ji\ 、
1l/-
rk ,
J'''EE飞
k
k = 1
(Z.17) 1n the preced1ng relati0n , let r ~ n. 7hen if f(x) i8 of degree 2n-1
in x ,
艺 (_1)k (:) f(y - k) 卡 f(y千 k)
= f(y)
k=O
(Z.18)
c: n)
If f(x) ~ f(-x) , f(O) = 0 , end f(x) 18 of degree 2n-2 in x , then
n
Z ←1可 = O.
kl tt丁k-=O k
(Z .19) A useful ooeretionel formule:
(♂吵 y =: (n!) r L:
e Ak = Z
k 斗 n-1x ve a唱··
n ·俨
-KX D
-K AHH
(k + n-1 )_,
(叫叫: -) l寸寸 r
(Cf. Carlitz , Amer. Meth. Monthly ,
Vo1.37( 1930) , pp. 的2-9)
-85-
A very usefu1 genere1 technique in summing series 18 to meke U8e of
the identity (x D扩 ιkr Xk This i8 whet 18 used in e11
the series in t he Be tables Where a series conteine the term Kr xk
Bnci . 1.丁 f :rs 1n no other connection 1n the 8eries. Schwett gives a
吃、 rr b~D~d discuseion of its uses.
1n terms of the general number8 of Nie 1sen type (see (1.128)ebove)
we heve
(Z .20) 川Z…
n 4' E E Ve
r 、、ls/
x nu x /
ttE、
n
Bktm+ 1 ve nu 让
川μUO\叫
一\J/-B
In term8 of the Stir1ing numbers of second kind (differences of zero) ,
two other use fU l generel formules ere
:z.21) 毛才 y 川Z zk . B~ . .d x ,z
‘ •. , x ,,.‘、
,ι
bA
俨: -J n
k X n 1r
一-~- . B1c • D二 r(x) • kr •
z .22) (x Dx ) n f(x) = L
-86-
.
,
(Z.2~) Let F(n) - Z ( -1尸( :) ~(吟,
and suppose thet f(k) does not depend on n.
Then the series may be inverted to give n
'J '. ..k - 1 I n 飞f( n) a: / ( -1 ) ~ . , } F(时,
ιJ 飞 kl
‘ k JC 1
and e180
Z T(k)E z (-1)lu1(;::)F(k)
k = 1 k ... 1
(Z .24) ... 、•. ,, ,、EW
,,.‘、
伊A
\111j/ b且
4d
/IIf飞
,、J、
,,
A咽,,
,,‘、
nv
k
勺乙μ
E ‘、,,,
k ,,.‘、
曹r
&L e L
t )ten
n ‘‘,,, a唱··
,,‘、
E ‘‘ •• ,
bA ,,‘、
霄ι
\i/ xk //iIEE
飞
b且‘‘.,, 4·· ,
,E‘、
n。jL…
bk叫咱b且
,··-圃
'A-x 飞
llj/
nk ,,,,,EtL
K
nu
n?Lh \111/ -n x /IlIL
(Gould , ~4-J )
(Z .25) z ( :) ~(k) -LH厅: X) z{ n ~于(j 们)k=O j 信 O
(Z.26) Suppose that DxUn(x) 皿 n Un_1(x).
Then the 8ummetion 气飞川咐l(ρ(仅川xk..。、 ι ,
~es the property thet It is 1ndependent of x , 1.e" DxSn(x) = o. Hence the series
~ey be eva1ueted by letting x take eny convenient va1ue. (Math. Tripo8 , 1896,
or see Hardy , Course of Pu re Mathemetics , 9th Ed. , 19l山 , pege 27队)-87-
(Z .27)
nz 阳)=÷ z 艺 ttjrjk f(时,
k=O j 嚣 1 k-=O
where 也J
2γi
r =e
(Z .28) Z (r:) f(rk) 斗 Z Z{:J f(k)ω 气jkk=O
(Z .29) Let 小雪 (_1)k C: 丁子去k=O
and suppose that. f(k) is independent of n.
Then
川一川
叭i//
/,,,,、
问z…
Ei巴L - F(n 泸 2)n 十 2
(Z.~O) 号。(-1)kC) f(k) … e川
independent of n. Then
艺 (-1)吁:) f(k T 1)
k+1
.. T( 0) - F( n 1- 1 )
n +-1 k=O
(Z .~1) -K
和-e
叽lM7
牛1-
nn fIt--L 我lll'
?k /'Il
、
n。/UH
川jJ
、川j
2·x-2 于十-··于b
nn-n 同£-叫,』
,,tt21、
-/Itt\
k 伊·&
叭i/
2x 千fs
寸
nk /I』lL
nk /II』1
2…
-88-
n (:) n
Z n
Zt~l二) r(时(Z.;2) r(k) a: (-1 )
(叶:X) e1γ7 k=O k-O k 千 x
n [+J [旦~1J
(z.;;) Z 忡Z ~(2k) + Z r(2k千- 1)
k = 8 k =[8 t 1] k-[+J
[n ; 1J n (z.纠)
L(泊问 ~Z Z r(2k 1- 1) 1'(2k) - . k :o: 8
k ...(咛] k-[+J
(Z.;5) [n ;1 ]
r - 1 n
艺 2 r(rk 十 j) - Z r(k)
j - 0 E8 + r;17 j J k...a k-
r
‘‘E,, a
唱··
>/n a唱,
>/r ,,‘、
., ‘、,,
4d f k r ,,‘、
T 川艺
、,,,‘,,
-a··-
,,
、‘,,,卢、
4''
圃
'3"ζ
,二
实J
霄,
ι
-r q白
,,,‘、
'E O 俨'飞
J
K
erk gT E c ····aE-a
气/
r-F}e
丐/
nn
,、/
erdvu e hu φu
aH aE‘‘、,,
回
6
4.‘
ZJ
』u
m··qu ,,.‘、,,
a‘、
k=O j=O k=O
n r
(zmzpk)-mrrgap(川(k -t- n )}
(Z.;8) 2~ k l'川 (0 + 1) Z r(k)
k - 1 k == 1
KZ川
nz… '
皿89恒
(Z.~9) 千F(h)!a22n n!(囚,告)! ,
(z.4o) n
d』内J』
飞11
』
IJ
hn JIBE飞
刷刷
‘ •. ,, a电··
,,‘‘、
E
飞EEtt''''
Agn 儿Il\
、‘a'
A叩··
FJt
町
,、,
fi
‘、
(帽 n -去)! (n- 去)! .. (_1) n 汁,
(Z .42) 且 z 叫 1(~) x - z 十 1
「,/飞/飞j
飞1
』
f-飞飞,,/
Z
牛'一千+
,
n-xn 』,
E1.
,,,··、
-、-t
,
J'
叫,叫一们n
/tt飞
-x
-,,''''飞
{)JIjI飞
(Z.句) n LU丁
冉'』
飞
11I/
r 2n j''』1、
n
‘i,Efs
,,F
+-2+ n-n 内Jι-
内t
J''Et--飞
(z.44) 了了) ~(4D2:) 产
C: 于 (."X:) <n-OPCfDBZ::(3(;)2
(刀(z.46)
(Z .47) n n
ζ
k qι
nt b且
n 、,,
a咽,.
,,‘、
事
\11lts/ τ吉
k
-nn fii--
飞
川lυ-
/MH川-1i/
-MK-AOK J-T/i···
、
nnE L
且Tqι
/ftt1、一
(Z.48) (;)1月 ,
-90-
(z.49)
(Z.50)
(Z.51)
(Z.52)
(Z.5~)
(:)---; 二节ω
了了) _l4n:j(沙 J
( : ) ( n ~ k)(:k) (η
kf川
、飞ttIJ'
4tk 』且
,
JFI『atL
、们叮tj/
-b
且
JII『'1、
ba 川lJJ
4『
KK V儿
'中t
bhb
民
/Iltt\
1..0
1 " j
(2:::1f::DE(-1叫:KYL-72知,
川lJ叮
叮ζ
吨'』
ν1λ
n 2n /IItt 也llIF
nk ,,,,E·E·-、
n nt·内ζ·
叭丁/、叶
ll/
14·izx -n/itl
飞
bhitt
『!
vu儿
nKJ
nk-/IIIt飞
XX同严』
k
,
UU』A
X 内J』
n
nrtSEtt
ηll/
-nk ,f
、
J''FEE-EE
、
\叭11,
/\11
』/
gkk --x-hn/Ilh
川•• 、
汕大叽
l
叮
akfIll-f''E,ltth飞
民JKU
RJRJ q缸,"
-91-
(Z .57)
(Z.59)
(Z.6o)
(ι~" : 1)(二:1)(川)(汇f仁:
(γ)(门刀升〉Eι/" ( ":) (" +: -~ c: 丁2:::丁-(':γ~2) .
'
n
叼/乞
Aj/ /lil飞 k+x
(It is by this difterentietion thet the finite hermonic series cen be
introèuced into a series of binorriel coefficients; e.g. , see reletions
of this type such as that obteined from (' .44) by u se of (Z .2~) , or
Goldberg's reletion ('.124) , etc.)
(Z.61 ) (:)厂: 1) 坦时( :n)(丁)
蝇'
《咽,
BIBLIOGRAPHY
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-93-
12. Chrystal , A Textbook of Algebra , Reprint by Chelsea Publ.Co. ,N.Y.1959.
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-9每-
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;1.
;2.
;;.
~4.
~5.
;;6.
;7.
;8.
Vol. 61(1954) , 251-25~.
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-95-
55. T. Narayana Moorty , Some summationformuleefor binomiel coefficientB,
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des Meth{maticiens ,Vo1.1 ,p.47.
57. Leo l-'loser , prob1em Nr. E799 , Amer. Y.eth. .t-1onthly , 1948 , p 、 30.
58. T. S. Nenjundieh , On e formu1e of A. C. Dixon , Proe. Amer. Meth. Soc. ,
V01. 9(1 9,58) , 308-311.
59. / , Remerk on e note of P. Turén , Amer. l.:eth. Monthly ,
Vol. 65(1958) , p.354 •
60. Eugen Netto , Lehrbuch der Combioetorik , Leinzig , 1901 , ~ec.Ed.1927. Reprinted by Chelsea Publ.Co. , New York , N.Y~ , 1958.
61. N. Nielsen , Hendbuch der Theorie der Gemmef、unktion , Leipzig , 1906.
62. _, Treit: éí6mcnteire des nOlrbres de Bernoulli , Peris , 1924 , note p.26.
63. N. E. N巴r1und , Yor 1e sung ,: n_ ð b~r Dif.fere.~ze,n reçþQ.ung ,.. >>~r.l1n , 1924. Reprin~ed-by ChelseaPUbl. Co. , New York , N.Y.; 丁 95件.
64. Q. P61ye end G. $zeg巴, Aufeeben undLehrs8tze , Berlin , 1925.
Note Vol. 1, pert 2 , problems 206-218.
65. John Riorden , An introduction tocombinetoriel enelysis , N.Y. , 1958.
fr6•• R咿 1 , Eine ber口的enswerthe Ide川t l! t , 问pe'e Archiv , (2) Vo1. 10 , 110-111.
See review io Fortschritte , Vol.2; , p.262.
67. Heinrich August Rothe , Formulee de eerierum reversione , etc. , Leipzig , 179~.
白. H. F. Sandhem , verious prob1ems , Nr. 4519 ,etc. , Amer. Math. Monthly.
69. L. Schlßfli , ErgAnzung der Abh. ðber die Entwickl. des Produkts...etc. ,
Jour. r6ine u. engew. l-1eth. , Vol. 67(1867), pp. 179-182.
70. O. X. Schl&milch. Compendium der h色heren Anelysis , Breunschweig , 1866 ,
,Vol. II , pp. 26-31.
-a『.•
7 , Recherches sur les eo~fficients des fecult6s enalytiques ,
Jour. reine u. angew. Jv:eth. ,飞TOl.44(1852) , ;44-'55.
'2. 1. J. Schwatt , Introduction to the operetions with series , Univ. of Pe.
Press , 1924. Reprinted 1962 by Chelsea Pub工. Co. , New York , N.Y.
-97-
7~. B. Sollertinsky , e summ8tion equiv81ent to Chur唱's problem , J pro !,osed 8S Question Nr. 12 , l' Intermédiaire des J.~athematiciens ,
VoL 1( 189句, page1.
74. Tor B. Stever , om summasjon ev poteneer ev binomielkoeffieienten ,
Norek Matematisk Tidsskrift , 29.Krgeng , 19饵, 97-10~.
75. B. A. SWinden , Note 2299 , on notes 202; end 209句, lv:ath. Gazctte ,
Vol. ~6(1952) , p. 277.
76. K. v. SZily , see review in Fortschritte , VoL 25(1 ,S9}) , 405-6.
77. P. Terdy , Sopra alcune formole reletiva ai coefficienti binomiali ,
Journel of Betteglini , Vol. ~(1~6刃, 1-:~.
78. C. ven Ebbcnhorst Tengbergen , ðber die IdentitAten...etc. ,
Nieuw Arch. Wiskde. , Vol. Hl( 19~4) , 1-7.
79. G. Torelli , Qualche formola relativa all'interpret a. zione fattoriale
delle potenze , Batt. Jour. Vol.;,; 178-182.
80. D. C. Wong , various problems in Amer. Math. Monthlj.
81. J. Worpitzky , Studien 咀ber die Bernoullischen und Eulerscher> Zahlen ,
Jour. reino u. angew. l-lath. , VoL 94(18ô~) , 20~-2~2.
82. prob. ~~77 , Amer. ~eth. Monthly , 1929.
8~. prob. 句51 , ibid. , 19另, p.432. see solution , ibid.Vol.61(19功),65~-4.
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• ERRATA SHEETS FOR THEORIGINAL PRINT工NG OF THE TABLES -page numbers refer to original pagination
ERRATA ,.;5 of' ma1ntableø ,如d 11ne up f'r om bottom , read [72J 1n阳ad of 72
'-5, 1n{136) , on right read t+( , 1)n instead oT1·(-1)n.
•
). 8 , 10 (1δ8) , read n 1T 1nstead 01' n T τ- •
,. 9 , in (1.71) , read -x instead of'
_x2 z .膨 z_ .
(1 + x)2 (1 + x)2
, 1n (1川….d C": 1) …叮~. 14.1 , 1n (1. 118) , addcondit10n that "注 2.,.17 , 1n (2.16) , on r1ght read 1nstead of' • 一k' k . ~. 18 , 1n (2.24) , read (:")… or(:n) •
p. 21 , 1n ('.2') , on r1ght hand s1de read n~ . _. n
p. 27 , 1n ('.72) , read ...l旦L 1nstead of
2 1nstead of n
二4in.2.
2
p. 40, (4.10) , tbe sum shou1d be
n
宁Lh
(:) r" : 1)
x(x 午 1). • .(x牛 n)p. '0, 1n (;5 .95) , read (x + 1)
.
p. 40,如 (4.1') ,read n+ r 1nstead 01' n r.
p. 50, 1n (7.11) , l'ead( I 1nstead o1'{ 1 飞 2n I 、 x/
p. 7咱, 11ne ;5, read (Z.12) 1natead of' (Z.1').
p. 79, change statement a1'ter (Z.'5) to read ITh1s 1s the genera1
cas8 f'or (Z.,,).I
,.80 , 1n (Z.'9) , on left read .J1f (2n>! /2",
). 80, 1n (z.48) , on r1ght read γ( J . n .
1. 81 , 1n (Z .52) , read k 1nstead 01'
-99-
1nstead of'
(了) . 1nstead 01'
kk .
ERRATA - Combinatorial Identities by H. W. Gould
page 6 formu叫1.46)
气~ 7) 叮~4)
{z(J+3}+h d for 1
24(n+3)
page 18 formula (2.24儿 line 2
α2 α3
n ~ L read n ~ I
page 21 :formula (3.22) in le:ft-most sum
-2k for 2-L.- read
k nJι
p nJ』n
J』 •
page 35 , in third d土splayed formula from top , .n .n t.. t
for ---- read .-=0 n n 品
-100-
Errata f'or: G∞1d, "Combinatoria1 工dentities, "
page 4,件 (1.31). Rep1ace si♂(言) by si♂(~)
k-1 page 13, eq.(1.106). Rep1ace (_l)A by (-1)
page 26, eq.(3.58). Multip1y the right member by
n-nf』
间可♂
page 26, eq.(3.57). Multip1y the right member by
严吕e 31, eq. (3.工03). Eepkce 2nfby ♂-k
page 43, eq.(6.5). 卫讪ce 叫叫2) by 主[1 + (-l)nJ
page 44, e吐·(6.8).Dehte(4)k
page 44, eq.(6. 工0). 工nsert (_l)k i的。 the s urom.ar也
page 47, eq.(6.35). Mu工tip1y the second sum by 22n
page 51, eq.(7. 工9) • Rep1ace (且-x:y-1) by (n+x:y-1) n
page 66,叫. (31. 工). Rep1ace (x + y)! by (x + c)!
---compi1ed by E1don R. Hansen , August , 1969.
-10丁-
FURTHER ERRATA
2τri/r p.7 , in (1.切, addzwhere taL z e f&-
k-1 p.14.1 , in (1.117) , for (x + 1)~-' read (k
1 2n+2x飞p.26 , in (3.57) , for l ~2n"""^") read
A 、 k-1+ 1)
for
-BJO --、飞〉/r---
n,
dk
(2n~竹. Cf. Han5en'5 form 。 f this correction.
p.38 , in (3.1 斗7) , read
k=1
p.51 , in (7. 丁 8) , the sum = -1 for n = O. 工ndicated sum correct
for n ~ 1.
p. 5丁 in (7.17) , in left-hand sum , insert the factor 1 k .
A工50 , for n~ m read m 共 n.
p. 65 , for (2,.阳 )iead (24.1).
p.86 , item #51 , for 'Todskrift' read 'Tidskrift'.
p. 87 , item #60 , add ref. to Second ed土tion ,. 1927.
p.3 , introduction ,工ine 丁 0 , for 'indes' read 'index'.
p. 69 , in (x.13) , for (n_1)3 read (n_1)2.
p. 14.2 , in def. stated with (丁 .127) , deletethe factor (_1)k.
H. W. Gou工d
1 May , 1972
-102-
Research and Pub1ications. selected 1ist: H. W. Gou1d
1. Note on a paper of Grosswa1d , Amer. Math. Month1y , 61(195斗) , 251-253.
2. Some genera1土zations of Vandermonde's convo工ution , Amer. Math. Month1y , 63(1956) , 8鸟-91.
3. The Stir1ing numbers , and genera1ized d土fference expansions , University of Virginia Master's Thesis , June 1956. iii+66pp.
4. Fina1 ana1ysis of Vandermonde's convo1ution , Amer..Math. Nonth1y , 6斗(1957) ,鸟09-415.
5. A theorem concerning the Bernstein po1ynomia1s , Math. Magazine , 31(1958) , 259-264.
~. Note on a paper of Sparre Andersen , Math. Scand. , 6(1958) , 226-230.
7. D土xon's series expressed as a convo1ution , Nordisk Mat. Tidsk. , 7(1959) , 73-76.
8. Note on a paper of Steinberg , Math. Magazine , 33(1959) ,斗6-件8.
9. Combinator土a1 Identities , a standardi~ed set of taö1es ~isting 500 binomia1 coefficient summations , Morgantown , W.Va. 1959-60. 100pp.
10. Genera1ization of a theorem of Jensen concerning conv。工utions ,
DukeMath. J. , 27(1960) , 71-76.
11. Stir1ing number representat土。n prob1ems , Proc. Amer. Math. Soc. , 丁 1(1960) , 斗斗7- 鸟51.
12. The Lagrange 土nterpo1ation formu工a and Stir1ing numbers , Proc. Amer. Math. Soc. , 11(1960) ,也21-斗25.
13. A q-binomia工 coeff土cient series transform , Bu11. Amer.' Math. Soc. , 66(1960) , 383-387.
1 鸟. Fje工dstad's series and the q-ana1og of a series of Douga1工,Math. Scand. , 8(1960) , 198-200.
气 5. The q-series genera1iaation of a formu1a of Sparre Andersen , Math. Scand. , 9(1961 ) , 90-9乌.
才 6. Note on a paper of K1amkin concerning Stir1ing numbers , Amer. Þ1ath. Month1y , 68( 丁 961 ),鸟77-斗79.
17. A series identity for find土ng conv。工ution ident土ties , Duke Math. J. , 28(1961) , 193-202.
18. The q-Stir1ing numbers of first and second kinds , Duke Math. J. , 28(1961) , 281-289.
19. Some re1ations invo1ving the finite harmonic series , Math. Mag. , 3乌(1961) , 317- 321 •
20. A genera工ization of a prob1em of L. Lorch and L. Moser , Canadian Math. Bu11. ,斗(1961) , 303-305.
21. A binomia1 identity of Greenwood and G1eason , Math. Student , 29(1961) , 53-57.
-103-
22. (with A.T.Hopper) Operational formulas connected with two generalizations of Hermite p。工ynomials , Duke Math. J. , 29(1962) , 5丁 -63.
23. Congruences involving sums of binomial coeffic立ents and a formula of Jensen , Amer. Math. Month工y , 69(1962) ,鸟00-402.
24. A new conv。工ut立。n formula and some new orthogonal relations for 主nversion of series , Duke Math. J. , 29(1962) , 393-斗04.
25. Generalization of an integral formula of Bateman , Duke Math. J. , 29( 才 962) ,鸟75-斗79.
26. Theory of binomial sums , Proc. W. Va. Acad. Sci. , 3乌(1962) ,158-162 (196~).
27. Note on two binomial coefficient sums found by R土。rdan , Ann. Math. Stat. , 3斗 (1963) , 333-335.
28. The q-analogue of a formula of Szily , Scripta Mathematica , 26(1961) , 155-157 (1963).
29. Generating functions for products of powers of F土bonacc主 and
Lucas numbers ,丁 (1963) , No.2 , 1-16.
30. Solution to Prob工em 470 , Math. Mag. , 36(1963) , 206.
31. Solut土。n to Problem 505 , Math. Mag. , 36(1963) , 267-268.
32. Sums of logarithms of binomial coeffic土ents , Amer. Math. Monthly , 71( 196件), 55-58.
33. The functional operator Tf(x) = f(x+a)f(x+b) - f(x)f(x+a+b) , Math. Mag. , 37(196斗), 38蛐斗6.
3斗. Solution to Problem 521 , Math. Mag. , 37(196件), 61.
35. (with L. Carlitz) Bracket function congruences for b土nomialcoefficients , Math. Mag. , 37(196斗), 91-93.
36. A new series transform with applications to Bessel , Legendre , and Tchebycheff polynomials , Duke Math. J. , 31(196乌), 325-33乌-
37. The operator (ax~)n and Stirling numbers of the f让st kind , Amer. Math. Monthly , 71(196鸟), 850-858.
38. Problem 5231 , Amer. Math. Monthly , 71(196鸟儿 923. See SG工utions ,ibid. , 72(1965) , 921-923.
39. Binomial coeffic土ents , the bracket function , and cornpositions with relatively prime summands , Fibonacc土 Quarterly , 2(196件) , 2斗1-260.
40. The construction of" orthogona工 and quasi-orthogona工 number sets , 72(1965) , 591-602.
41. An identity involv土ng Stirling numbers , Ann. 工nst. Stat. Math. , 17( 丁 965) , 265-269.
斗2. A variant of Pascal's triangle , Fibonacc主 Quarterly , 3( 丁 965) , 257-271. Errata ,鸟 (1966) , 62.
-101•
43. Inverse series re1ations and other expansions invo1ving Humbert po1ynomia1s , Duke Ma th. J. , 32(1965) , 697-711.
44. Note on recurrence re1ations for Stir工ing numbers , Pub工.工nst.Math. (Beograd) , N.S. , 6(20)(1966) , 115-119.
句. Cwi th J. Ka summatioJ丑lS , J. Combinator土a1 Theory , 1(1966) , 233-2乌7.Errata , ibid. , 12( 丁 972) , 309-310.
乌6. Note on a q-series transform , J. Combinatoria1 Theory , 1(1966) , 397-399.
斗7. (with C. A. Church , Jr.) Lattice point solution of the genera1ised prob工em of Terquem and an extension of Fibonacci numbers , Fibonacci Quarter1y , 5(1967) , 59-68.
鸟8. Comment on Prob1em 602 , Math. Mag. ,斗。 (1967) , 107皿108.
斗9. (with Maurice Nachover) Prob1em E 1975 , Amer. Math. Month1y , 7件(1967) ,斗37-件38. See solut土ons , ibid. , 75(1968) , 682-683.
50. Note on a combinator土a1 identity in the theory of bi-c。工。redgraphs , Fibonacci Quart. , 5( 丁 967) , 2乌7-250.
51. The bracket function , q-binomia1 coeff土cients , and some new St土r1ing number formu1as , Fibonacci Quart. , 5(1967) , No.5 ,斗01-423. Errata , ibid. , 6(1968) , 59.
~2. (with Frank1in C. Sm土th) Prob1em 5612 , Amer. Ma th. Month工y ,75(1968) , 791. See solution , ib土d. , 76(1969) , 705-707.
53. Prob1em H-1 乌2 , Fibonacci Quart. , 6(1968) , No. 斗, 252. See solution , ibid. , 8(1970) , 275-276.
54. Note on a paper of Pau工 Byrd , and a s。工ution of prob1em P-3 , Fibonacci Quart. , 6(1968) , 318-321.
55. Note on two binomia1 coefficient ident立 ties of Rosenbaw且 J.
Math. Physics , 10(1969) ,斗9-50.
56. The bracket function and Fonten{-Ward generalized binomia1 coefficients with app1ication to Fibonomia1 coefficients , Fibonacci Quart. , 7(1969) , No.1 , 23-乌0 , 55.
57. Power sum identities for arbitrary symmetric arrays , SIAM J. App1ied Math. , 17(19~9) , 307-31 6.
58. Invariance and other simp1e techniques as a genesis of binomia1 土dent土 ties , Nieuw Archief voor W土skunde , (3) 丁 7( 1969) , 21 乌-217.
59. Remarks on a paper of Srivastava , J. Math. Physics , 12(1971) , 1378-丁 379.
60. Comment on a paper of Chiang , J. Math. Physics , 12(1971) , 17也3.
61. Reviews #2033 - 203件, Math. Reviews , 38(1969).
62. Noch einma1 die Stir1ingschen Zah1en , Bemerkungen zu Noten von Johannes Thomas und F. Klein-Barmen , Jber. Deutsch. Math.-Verein. , 73(1971) , 1 件9-152.
-105-
63. Solut土。n of Problem 783 , Math. Mag. , 鸟斗 (1971) , 289-291.
64. Equal products of generalized binomia工 coefficients , Fibonacc土Quart. , 9(1971) , No. 鸟, 337-3斗6.
。二. Explicit formulas for Bernoulli numbers , Amer. Math. Monthly , 79(1972) , 斗斗-51.
5.6. Solution of Problem 70-21 , SIAM Rev土ew , 1 乌 (1972) , 166-169.
67. A new symmetr土 ca1 combinator土a工土dent土ty , J. Comb. Theory ,
68. The case of the strange binomia1 identities of Professor Moriarty , Fibonacci Quart. ,
69. Some combinatoria1 identit土es of Bruckman - a systematic treatment with re1ation to the older 1iterature , Fibonacci Quart. ,
70. Two remarkab1e extensions of the Leibniz differentiation formu1a ,
_4 --A new primality criterion of Mann and Shanks and its re工ation to a theorem of Hermite , Fib。且acci Quart. ,
72. A remarkab1e combinatoria1 formula of Heselden , Proc. W.Va. Acad. of Sci. ,
73. Eva1uat土on of the finite hypergeometric series F(-n , 1/2;j+1; 的,
7斗.
'-~<::.
76.
The moments of a distr土bution as a genesis of comb土natoria1identities , Fibonomia工 Cata1an numbers , ar土thmet土c properties and a tab1e of the f土rst fifty numbers ,
A new greatest common divisor property of the binom土a1 coeff土cients ,
77. P。工ynom土a1s associated w土th an 土ntegra1 of Eu1er ,
~吧zR, Types of orthogona1土 ty for arrays of numbers ,
79. Speci豆工 funct土ons associated w土th the higher derivatives of y f X J-
80. Brill's series and some re1ated 吃xpansions obtainab工 e in closed form , Mathematica Mononga1iae , #3 , August , 1961. 14pp.
81. Exponential b土nomial coeffic 土ent ser土 es , Mathematica Monongal土ae ,#件, Sept., 1961. 36pp.
82. Note些 on a calculus of Turtn operators , Mathematica Mononga1iae , #6 , May , 1962. 25pp.
8二. The Bateman notes on binomia1 coeff土cients , edited by H.W.Gould , August , 1961. 150pp. to appear. Based on 560 page manuscript compr土sing three versions of the origina工 work.
-106-