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The fractional calculus and itsrole in the synthesis of specialfunctions: Part IIWilliam J. Grum a & Lokenath Debnath aa Department of Mathematics , University of CentralFlorida , Orlando, Florida, 32816, U.S.A.Published online: 09 Jul 2006.

To cite this article: William J. Grum & Lokenath Debnath (1988) The fractional calculus andits role in the synthesis of special functions: Part II, International Journal of MathematicalEducation in Science and Technology, 19:3, 347-362, DOI: 10.1080/0020739880190301

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INT. J. MATH. EDUC. SCI. TECHNOL., 1988, VOL. 19, NO. 3, 347-362

The fractional calculus and its role in the synthesis ofspecial functions: Part II

by WILLIAM J. GRUM and LOKENATH DEBNATH

Department of Mathematics, University of Central Florida,Orlando, Florida 32816, U.S.A.

(Received 24 December 1985)

This paper is concerned with applications of the arbitrary order derivintegralsof generalized hypergeometric functions to the synthesis of special functions. It isshown that the whole spectrum of special functions can be expressed in terms ofthree basic functions including the inverse binomial, the exponential and theBessel functions of zero order. Several formulae for the arbitrary orderderivintegral of the NFD and nGFD hypergeometric functions. A systematicmethod of evaluation of the derivintegrals of the important transcendentaland special functions is developed in this paper. Three tables are included in thepaper. Tables 1, 2 and 3 include Dx

x derivintegrals of NGFD hypergeometricfunctions for N=D, N=D-1, and N = D - 2 respectively. A listing of thereducible special functions includes exponential integrals and error functions,logarithms, inverse trigonometric functions, and their hyperbolic counterparts,incomplete γ- and β-functions, circular and hyperbolic sines, cosines, and sineintegrals, Bessel functions, generalized special functions such as Gauss, Kummerand other hypergeometric and elliptic integrals. It is shown that the derivintegraloperator forges a very powerful link among all the special functions.

1. IntroductionIn part I of this paper [1], the concepts and properties of derivintegral of arbitrary

(real or complex) order have been discussed. The Riemann-Liouville fractionalintegral R~"f and the Weyl fractional integral W~"J are discussed with theirextensions to real and complex values of a.

The purpose of the present paper is to illustrate specific applications of thegeneralized derivintegral in the realm of real analysis. It is shown that, by continuingthe two concepts of a generalized derivative-integral and a generalized hypergeo-metric function, all the transcendental and special functions can be synthesized fromthree fundamental functions: the binomial, the exponential, and the Bessel functionof the first kind. This discussion forges a powerful link between a myriad of specialfunctions studied as separate and unrelated entities.

2. Generalized hypergeometric functionsConsiderable work has been done in the past century or so by many mathema-

ticians generalizing the geometric series

into what has come to be known as the generalized hypergeometric function, denotedby NFD. This branch of mathematics was pioneered by Gauss and others who studiedthe properties of a function, the Gauss function, which today is recognized as a

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348 W. J. Grum and L. Debnath

subclass of this more general hypergeometric function. The Gauss function, denotedby F{ix1,oc2,P1;x) is defined by [2]:

l)k

where /?j 4=0, — 1, — 2.. . The a1( a2, /?! are real numbers representing the first andsecond numeratorial parameters and the first (and only) denominatorial parameters,respectively, and the Pockhammer symbol used in definition (2.2) is defined by

l), integer >(2.3 ab)

In view of the recurrence property of the gamma function, F(x+1) = xT(x), it is easyto prove the result

( a ) * = I r 9 r a + 0 ' - 1 ' - 2 - - - (2-4>The following result clarifies the value of the Pockhammer symbol when a is a zero ora negative integer,

( — l)kn\(-»)>=, ' , O^k^n (2.5 a)

(n-k)\(-w)* = 0, k>n (2.5b)

As a direct consequence of (2.5), the Gauss series (2.2) diverges if the denominatorialparameter /Jj has a value of zero or is a negative integer. This is true because the indexk will eventually exceed P1 in magnitude and force all the remaining terms in theseries to become infinite. Thus fit 4=0, —1, —2,.. . is a necessary condition for theconvergence of (2.2). Either of the numeratorial parameters may, however, assumezero or negative integral values. When they do (at = — n), series (2.2) will truncate atk = n and an nth degree polynomial will result.

The Gauss function may also be expressed in an equivalent integral form

F(ccl,cc2;P1;x) = r £ f f i ^ f 1<"*-1(l-<)''-'*-1(l-*O-"Idt (2.6)1 (<X2)l (Pi—CC2) J 0

where ^ x >a 2 >0 andOf course, there are other classes of hypergeometric series besides the Gauss

function. The simplest is the hypergeometric series with no numeratorial ordenominatorial parameters, which converges to the exponential function. That is,

Another is the hypergeometric series with one numeratorial parameter and nodenominatorial parameters. This hypergeometric series converges to the generalizedbinomial when the argument lies between — 1 and 1. Thus, we have

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Fractional calculus and the special function synthesis II 349

Another class of hypergeometric series is the Kummer function possessing onenumeratorial and one denominatorial parameter. The Kummer function is denotedby M^;/?,;.*) and defined by

M(ai;/*i;*)= E ~ T ^ 01 = 0 , - 1 , - 2 , . . . (2.9)

Unlike the generalized binomial and Gauss functions, the Kummer functionconverges for all x. Like the generalized binomial and Gauss functions, however, apolynomial of degree n results if at = — n. The Kummer function has also an integralrepresentation analogous to that of the Gauss function, and is given by

y f f (2.10)r(a1)r(/j1-a1) j 0

provided P1>a1>0.We next introduce the generalized hypergeometric function NFD, possessing N

numeratorial parameters and D denominatorial parameters. It is defined byiV

r . „ „ -i «, n(«i)k«*a 1 , a 2 , . . . q ) V v ^iND\T~R T' x \ = ^~D

where no denominatorial parameter is zero or a negative integer. As before, an nthdegree polynomial results if any of the numeratorial parameters is a negative integer.It is noted that the exponential, generalized bionomial, Kummer, and Gaussfunctions are all special cases of the generalized hypergeometric function.

Using the ratio test, we can determine the convergence of the hypergeometricseries (2.11), and conclude that the series (2.11) with no /?, = 0, — 1, —2,... has thefollowing properties:

(i) it converges for all x when N^D,(ii) it converges for \x\<l when N=D + 1,

(iii) it diverges for x^O when N>D+l.However, if any a,= — n, the resulting nth degree polynomial converges for all x

regardless of the number of numeratorial or denominatorial parameters. It is nowclear why the exponential and Kummer functions converge for all x whereas thegeneralized binomial and Gauss function converge only for |x| < 1.

The great advantage in studying the generalized hypergeometric function lies inthe fact that so many of the standard and special functions are special cases of the NFD

function. Thus any property of the NFD function is automatically a property of allthose other functions which fall in the class of generalized hypergeometric functions.In fact, any function which can be.expressed as an infinite series

and whose ratio is given by

M H 1 ( X ) _ (ai + k)(a2 + k).. .(aN + k)x

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350 W. J. Grum and L. Debnath

where the numerator is a polynomial in k of degree N and the denominator is apolynomial in k of degree D + \, is a generalized hypergeometric function withparameters <x1,a2, • • • <xN; Pi,... PD- Similarly, any function whose integral represent-ation is of the form (2.6) or (2.10) is a Gauss or Kummer function, respectively.

In subsequent sections, we will cast a number of transcendental and specialfunctions into hypergeometric form using one of the techniques:

1. perform the ratio test on its infinite series representation and ascertain a's andP's from the form (2.12)

2. determine a and fi by fitting its integral representation to either forms (2.6) or(2.10).

It is noted that technique 1 is frequently used since the infinite series representationis often readily available.

3. Derivintegration of the NFD functionAn attention is given to the development of a suitable formula for the arbitrary

order derivintegral of the NFD hypergeometric function, or even more general, aproduct of xp and NFD. Upon comparison of (2.11) with the form (6.1) of Part I of thispaper, it is apparent that the NFD hypergeometric function is really just aderivintegrable series. It should be pointed out that a = 0 and ck is nothing more thanthat horrible-looking ratio of products of Pockhammers out in front of the x* term. Inaddition, xk

NFD is also a derivintegrable series provided p > — 1.The formula for the derivintegral of a derivintegrable series has already been

developed in §6 of Part I. We need only to apply (6.5) with a = 0 to (2.11) to obtain

r a , . . . a 1) « U{a')k T{p + k+\) xp-*+k

' KFA —— ',X >= V ^-=r (3 1)N \Pi,...pD If *=o r r ,«x r(p-a+k + l) k\ K ' '> = i

or equivalently,

Dr NF\p^^D'x\rnp-a+x)x^F^ipu...pD,p-,+i'x\ (3-2)

provided p > — 1 and p — a. is not a negative integer. Finally, there is a special casewhen p = 0, then

U---PD J r ( l - «

f" CCU...<XN,1 Ift ft—i—~Z' \ *• '

provided a is not a positive integer.Remarks: Neither of these results are preferred formulae for the derivintegral of

a hypergeometric function. Both yield indeterminate results for certain choices of pand a. It is far more satisfying to find a formula that is valid at once for non-integraland integral values of a. This objective can easily be accomplished after a slightredefinition of the generalized hypergeometric function. The specific nature of thisredefinition is the topic of the next section.

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Fractional calculus and the special function synthesis II 351

4. The ^GFj, function and its derivintegralThe generalized hypergeometric function of the previous two sections can be

slightly redefined and renamed. One may define the generalized hypergeometricfunction, designated by nGFD to distinguish it from NFD, as [3]

r« • a 1 - nr(a i+l+/e)

This definition of NGFD is directly related to the definition of NFD of (2.2) in thefollowing manner, which can be easily verified by the reader,

Hence, any function expressible as a KGFD may be converted to an equivalentby utilizing (4.2). Similarly, by utilizing

K,...,«„. i_flr(/?i) rF r«i-i «w-i. j

D ft fl~;x " ^ N G F D + 1 —-^ ;a; (4.3)

any NFD may be converted into an equivalent NGFD+1.There are several distinct advantages to the NGFD representation over the

traditional NFD representation for the generalized hypergeometric function. First,the denominatorial parameters in the NGFD representation may take on any value,including zero or negative integral values restricted from the NFD representation. Adenominatorial parameter equal to zero is equivalent to a kl. Any /?,-= — n merelyforces the first (« — 1) terms to zero, creating an infinite series beginning with the nthterm. Conversely, if any numeratorial parameter in (4.1) is a negative integer, thefirst (« — 1) terms will become infinite. Consequently, this possibility is encounteredonly as the quotient

which represents a polynomial of degree (n —1).Secondly, upon applying the ratio test to (4.1) we obtain

from which it follows that the nGFD hypergeometric function

(a) converges for all x when N<Z),(b) converges for |x |< l when N=D, and(c) diverges for all x^O when N>D.

This is a somewhat neater or more aesthetically pleasing result than the analogousconvergence criteria of the NFD hypergeometric.

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352 W. J. Grum and L. Debnath

Thirdly, and most importantly, the formula for the derivintegral of the NGFD

hypergeometric does not yield an indeterminate form for the special case of integerorder derivintegration as does the nFD hypergeometric. One may obtain a formula forthe derivintegral of the product of xp and an NGFD hypergeometric function (p > — 1)by applying (2.11) directly to (4.1) with a = 0, which gives

L P C F P * 1 ' ' " " ' g j Y - x~\\ - x ' - " CF f / > » « ! , • • - , « i v 1 ,.,,\_PI,---,PD JJ [p-a,Pi,...,PD J

since the tjGFD hypergeometric is really just a derivintegrable series as defined by(6.1) in Part I. The special case where /> = 0 gives

rp [~«1» • • - , « * . I - . , -N{~rPD\~n W'X \~~X

\J>U---,PD Jn o

-<X,PU---,P

. . .> x

valid for all a.We note that the only restriction on either (4.6) or (4.7) is the requirement that

p > — 1. These results are far more appealing than the comparable expressions for thederivintegral of the NFD hypergeometric. Both formulae are at once simpler and lessrestrictive than their counterparts (3.1) and (3.2). Therefore, throughout the rest ofthis paper, the NGFD hypergeometric form will always be used when evaluating thederivintegral of any transcendental or special function that has been cast intohypergeometric form.

Finally, before closing this section, the effect of the appearance of a constant inthe argument of the fiGFD hypergeometric on its derivintegral is explored. Byapplying the scale change property (4.6), we obtain

1'-'*"; ex\ = d-'m\(exyKGFltt1'-••''"; cx\\ (4.8)P\y-yPD J . L \_PU--->PD J J

Or, equivalently, after using (4.6) with x—>cx,

|~ P>CC1TTJvP (IF rgl ' '">gjv-/-vll-r''UX<X N(jrr D — ^— ,CX \> — X

I \ _ P U - - - , P D J JJ>-CC,PI,---,PD

, CX

for all real c. Hence, the derivintegration properties of a hypergeometric function ofargument equal to a constant multiplied by x are not affected by the magnitude of theconstant.

5. The derivintegrals of 28 special functions

The most systematic way to evaluate the derivintegrals of the importanttranscendental and special functions is a three step process consisting of:

1. encode the special function into fiGFD hypergeometric form,2. use (4.9) to derivintegrate the result of the first step,3. decode the result of step 2 into familiar closed form, if possible.

The encoding process is most easily accomplished by using either technique 1 or 2discussed in § 2 to first obtain the tfFD hypergeometric representation, and then using(4.3) to convert this into the NGFD representation.

Decoding is not always possible since the particular nGFD hypergeometricresulting from the derivintegration may not be known by a special 'name'. Justbecause the derivintegral hypergeometric does not have a 'name', of course, does not

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Fractional calculus and the special function synthesis II 353

mean it is not a well-defined and perfectly valid function in its own right. The fact ofthe matter is that the number of functions which have been studied commonlyenough to merit special 'names' are so few compared with the total number that canbe generated from the t/GFD hypergeometric functions; it is no wonder that thedecoding is frequently not possible.

The results of performing the first two steps in the above process on 28 differentfunctions are tabulated below. No attempt has been made to perform the third stepon any of these 28 functions. This is because the denominatorial parameters of theresulting hypergeometric are always dependent on the value of a. Hence, differentclosed forms would result for different choices of a, and may be impossible for otherchoices of a.

The following symbolism has been used in tables 1, 2 or 3 to denote certainspecial functions [4]:

(5.1)~)0 0 ( l -x 2 s in 2 0)

and

fir/2E(x)= (l-x2sin20)1 / 2d0, 0 < x < l (5.2)

Jofor the elliptic integrals of the first and second kind, respectively;

-2\ j * ^ ^ (—1) X2 Cx

V^Joerf(,

for the error function:

f ̂ exp ( t) °° ( 1 )"̂ c"E1(x)=-Ei(-x)= —^—-dt=-y-\nx- Y - — ^ — . x>0 (5.4)

J x t n= i n\n

for the exponential integrals and y is the Euler constant.We write down the series representations of the Bessel functions [2]:

i2Hv

.W+.+1) <5'5>

and

- (5-6)

for the ordinary and modified Bessel functions of the first kind, respectively;

Si(x)= dt (5.7)

for the sine integral; and, finally,

f*sinh(0 ,ShiOc)= —dt (5.8)

Jo *for the hyperbolic sine integral.

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354 W. J. Grum and L. Debnath

6. Reduction of complex hypergeometricsWe recall that derivintegration of a NGFD hypergeometric function will, in

general, result in aN+1GFD+1 hypergeometric function. This is apparent from any offormulae (4.6), (4.7), or (4.8). Due to cancellation of like parameters; however, theactual result is frequently of the same order of complexity as the originalhypergeometric. This pertinent fact is obvious upon a brief scan of most of theresults in any of tables 1, 2, or 3.

1

2

3

A

5

6

7

8

9

in

11

1 9

13

Function

1

1-x

11+x

(l-x)-"-1

- o - l

In (1-x)

In (1+x)

/

arctanh (-Jx)

KQx)

arcsinh (-Jx)

N°FD

1

T(a +

XjC

W X

x " ,

r(i-6)2

V X f

representation

^ r _ -i

L - J

I)1 '|_0'*J

?FJ -; -x

.r-i-i l2[_ 0,0 'X\

r_i_i -j

1 — ot d — 1 1

f-2. -i. 1

*\-?-i' 1

DJ derivintegral

r0 -l

l G F l L - a > xix-' [" a 1

Ha+l)1 1-a'J

*-« r a i

v l ~° r^P 1 • v 1L J

xi/2-a r _ i I

-•'-' J r"-* l

s^T? • v 1

v l / 2 -a /->!? 2» 2 . . .

2>^ 2Crlo,i-«' J

Table 1. Hypergeometrics with N=D.

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Fractional calculus and the special function synthesis II 355

Function NGTD representation DJ derivintegral

14 exp(x)

15 exp(—x)

16

17

18

19

20

0GFl - 5 - ; - * aT'cGF, — ; - x

0GF \—;x\ x~" 1GF2\ ;X\L-w J L-a. ~n J

^-^GFJJ-^— ; J

26

2 7

Table 2. Hypergeometrics with N=D — 1

Function NGFD representation DJ derivintegral

(v>-2) ° 2[o,v;

0,v,v/2-a

22 7 ^ ( v^ *» /2 G F r jL ,( v>2 ) 2L0,v'

23

24 sinh^v*) ( ^ ) / o G FL°.2

y/2 1,v,v/2-«' J

25 cos(2V*)

T;x\ y/nX-" 0GF2\~—» —2 J L—2. ~

Table 3. Hypergeometrics with N=D — 2.

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356 IV. J. Grum and L. Debnath

By taking advantage of this cancellation property, the sequence of operations

1. multiplication by x^l~p

2. derivintegration to order P1—ot.13. multiplication by x"~Xl

may accomplish the transformation

(6.1)

XGF^HGFI"1'-••'""; ex]LPi PD J

] (6.2)

and

N_1GFD_^N^GFD_\^^-,cx1\ (6.3)

This transformation converts an NGFD hypergeometric to an Ar_1GFD_1 hypergeo-metric. The only restriction on this important transformation is that p\ > — 1 topermit the derivintegration. Examples of some results of this transformation include

which, from entries 9 and 3 of table 1 with a=j, is recognized as

x-1/2£>1/22X(Vx) = V7i( l -*r 1 / 2 (6-4)

Also, we have

which, from entries 12 and 8 of table 1, is recognized as

y/nxDl12^-1/2 arcsin (Jx)] = arctanh (Jx) (6.5)

The above three step transformation may be applied to an NGFD hypergeometric Ntimes, reducing it to a 0GFD_N hypergeometric. Such a hypergeometric has thegeneral form

xp0GFD_J- ^ — ; ex] (6.6)

LPN+I>--->PD J

The sequence of operations

1. multiplication by xfitf*'~p

2. derivintegration to order PN+1

will replace a denominatorial parameter with a zero. Thus

r;ex] (6.7)

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Fractional calculus and the special function synthesis II 357

provided fiN +1 > — 1. If the above two step transformation is repeated D — N times,all the denominatorial parameters may be replaced by zeros. An illustration of thisreduction procedure

! ^ JJJ ^ J (6.8)leads to the relation

JnDlJ2 {xDl12 [x~112 erf Qx)]} = exp ( - *) (6.9)

from entries 17 and 15 of table 1.We have just seen that by a sufficient number of operations (each of which is

either multiplication by a power of a; or derivintegration with respect to x) any f/GFD

hypergeometric (with all denominatorial parameters exceeding — 1) is reducible to a0GFD_N hypergeometric in which all the denominatorial parameters are zero.Hence, all the transcendental and special functions discussed in this paper mayultimately be reduced via the above-mentioned operations to one of the 'basis'hypergeometrics:

l-x

v<\_->x]>= 0GF0\—;x\ \x\<l (6.10)

—;*1 all x (6.11)

(6.12)

or their complements:

(6.13)

(6.14)

(6.15)

Finally, by reversing the above reduction process, any hypergeometric (whosenumeratorial parameters all exceed — 1) may be synthesized from one of these threebasis hypergeometrics of their compliments. This idea of using the generalizedderivintegral to synthesize the various special functions from elementary functionshas been discussed by Oldham and Spanier [3], and others. Sections 7-9 will bedevoted to the details of such function synthesis.

7. Synthesis of N=D hypergeometricsThe functions representable by a hypergeometric having equal numbers of

numeratorial and denominatorial parameters may be synthesized from the basishypergeometric (1 — x)~* or its compliment (1+x)^1. We begin with the 1GF1

hypergeometrics. Any of entries 3-8 of table 1 can be synthesized from the basis intwo steps.

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358 W. J. Grum and L. Debnath

First, the generalized binomial may be synthesized from (1— x)~x by firstmultiplying it by xf and then derivintegrating the result to order a. We clearly seethat

a>-\ (7.1)

along with entries 1 and 3 of table 1, leads to

xy-\ a>-\ (7.2)

the appropriate synthesis formula. Secondly, since

^ ! (7.3)

along with entries 1 and 5 of table 1, yields

D-1(l-xy1=-ln(l-x) (7.4)

a synthesis formula for the logarithm function. Of course, this result agrees with thewell-known relation of classical calculus [5].

Yet a third synthesis, this time for the hyperbolic arctangent,

D; J [*~ 1/2(1 -X)~ *] = 2 arctanh Qx) (7.5)

arises from entries 1 and 8 of table 1, along with

^ (7.6)

Finally, the three syntheses

(7.7)

x (7.8)

and

^" 1 [*" 1 / 2 ( l+^)" 1 ] = 2arctan(Vx) (7.9)

arise from the same procedures applied to the complementary basis hypergeometric

Each of the 2GF2 hypergeometrics listed in table 1 of §5 may also be synthesizedfrom the N=D basis hypergeometrics. The elliptic integral of the first kind may besynthesized from the basis (1 —x)'1 by first forming its product with x~llz, thenderivintegrating the result to order —\, followed by a repetition of the above twosteps. Mathematically, this is expressed

^ ] (7.10)

which, based on entries 1 and 9 of table 1, is equivalent to

(7.11)

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Fractional calculus and the special function synthesis II 359

A synthesis formula for the incomplete beta integral follows from

x° 2GF2^f^; xj (7.12)

and representations 1 and 11 from table 1. It is

D;1[x°-1D;b[x-b(l-xr1]] = r(l-b)Px(a,b) (7.13)

Also, since

^ ^ ^ J j J ^ ^ ^ ( 7 . 1 4 )and

^ JJJ [ | j i ; -*] (7.15)the synthesis formulae

^ J (7.16)and

(7.17)

for the inverse sine function and its hyperbolic counterpart follow immediately from1 and 12 and 2 and 13 of table 1, respectively.

The final 2^F2 hypergeometric, the elliptic integral of the second kind, providesus with our first difficulty due to the —§ numeratorial parameter. To synthesize E(x),one makes use of the identity

J (718)Then, since

^ ^ ^ ^ ^ J ^ ^ ^ i ] (7.19)the elliptic integral of the second kind can be synthesized according to

(7.20)

Prior to closing this section, the synthesis formula for the Gauss function isdeveloped. Recalling that the Gauss function is expressible as a general 2GF2hypergeometric with one zero denominatorial parameter, the synthesis may beperformed with the four steps

j^JJ^ [̂ ] (7.21)producing the desired 2^F2 hypergeometric. Combining the above with 1 and 2 oftable 1 yields

r i - T £ ) 2 ) f X 1 +otl)l+CCz; 1+Px'x) {122)

for a 1 )a 2> — 1.

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360 W. J. Grum and L. Debnath

8. Synthesis of N=D— 1 hypergeometricsAll the iGF2 hypergeometrics listed in table 2 may be synthesized from the

N=D—1 basis hypergeometric

(8.1)L u J

or its complement

r- iexp( * ) - o & ^ o , *j (8.2)

Specifically, we have

which, after comparison with 15 and 17 of table 2, gives the synthesis formula for theerror function of argument y/x.

D;1[x-ll2exp(-x)]=yJnerfy/x (8.4)

The incomplete gamma function is synthesized by the two step process

D~1[xa~1exp(-x)]=-Y(a,x) a>0 (8.5)

a

while the exponential integrals require three steps in their syntheses1 ) ] = .E1(*) + y + ln;e (8.6)

= Ei(x)-y-\nx (8.7)as is evident from 18, 19 and 20 of table 2.

A general synthesis formula may be developed for the Kummer function, since itis merely a ]GF2 hypergeometric with one zero denominatorial parameter. In threesteps we have

[ ( ^ ] ] { ^ ] (8.8)which translates into

f ^ L + n J ' | i ± l J (8.9)

upon comparison with 14 of table 2 and use of (4.2). The only restriction on the aboveformula is (Xj > — 1.

9. Synthesis of N=D— 2 hypergeometricsAll the Bessel functions and modified Bessel functions of the first kind, sine and

cosine of argument 2 y/x and their hyperbolic counterparts, as well as the sine andhyperbolic sine integrals, may be synthesized from the N=D — 2 basishypergeometric

t ; * l (9.1)

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Fractional calculus and the special function synthesis II 361

or its complement

J0(2V*) = 0 G F 2 r ^ ; - x l (9.2)

First, all the Bessel functions are synthesized from J0(2^Jx) in two steps

/ j (9.3)as is the cos (2y/x)

Cos(2s/x) (9.4)

The sin(2^Jx), however, requires only one step

/ (9.5)

for its synthesis. Analogous expressions for the modified Bessel functions and thehyperbolic cosine and sine functions of argument 2^Jx are

(9.6)

^ ^ y / (9.7)

andD;1/2I0(2s/x)=^rsinh(2jx) (9.8)

Secondly, all the ±GF3 hypergeometrics from table 3 may be synthesized in threesteps from the basis o^F2 hypergeometric or its complement. For example, since

(9.9)

then entry 28 of table 3 immediately gives

D; '[X-'D; 1!2I0(2y/x)] = - ? - Shi (2V*) (9.10)

the synthesis formula for the hyperbolic sine integral. Finally the complementrelation is

D;\X-ID: 1 / 2 J 0 ( 2 V * ) ] = 4 - S i (V*) (9-1 !>

for the sine integral.Of course, the synthesis formulae of the last three sections are not the only

conceivable ones. In many cases, several other synthesis routes could be followedfrom the basis to the synthesized function. For example, there are four routes from a0GF0 hypergeometric to any given 2G-F2 hypergeometric (unless it possessesredundant parameters). Hence, the arcsine function of argument y/x could also havebeen synthesized by

x i /*Dji/2[*-i2)-1[*-1 / 2(7-*)-1]] = 2>/7tarcsin>/* (9.12)

in addition to (7.16). Generally, the synthesis given in the preceding three sectionshas been the one with fewest steps.

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362 Fractional calculus and the special function synthesis II

10. Closing remarksIt is clear from the above discussion of derivintegrals of generalized hypergeo-

metric functions that the whole spectrum of special functions can be expressed interms of three basic functions including the inverse binomial, the exponential andthe Bessel functions of zero order. The use of derivintegration combined with onlysimple algebraic operations (multiplication by constants and powers) reveals thateach of the reducible special functions may be synthesized from one or the other ofthe three basic functions. A partial listing of the reducible special functions includeexponential integrals and error functions, logarithms, inverse trigonometric func-tions and their hyperbolic counterparts, incomplete gamma and beta functions,circular and hyperbolic sines, cosines, and sine integrals, Bessel functions, gen-eralized special functions such as Gauss, Kummer and other hypergeometricfunctions and elliptic integrals. It seems that the various derivintegral synthesisformulas for the above-metnioned special functions developed in §§ 6—9 constitutemore appealing definitions than those customary in terms of definite integrals,differential equations and the like. Indeed, it is possible to assert that derivin-tegration removes the need to recognize many special functions as functions in theirown right. Thus, if an elliptic integral is nothing more than a derivintegral of aninverse binomial (7.11), is any purpose served in regarding it as anything but thederivintegral of the inverse binomial? Need it be graced with a special name?Similarly, since the hyperbolic sine integral is a derivintegral of the zero-order Besselfunction (9.10), is not the symbol

^ - \ (lo.i)

more informative than the symbol Shi (2y/x)l Hence, the derivintegral operatorforges a very powerful link among all the special functions. Each may now be viewedas a member of one of three families: the inverse binomial family, the exponentialfamily, or the zero-order Bessel family.

References[1] DKBNATH, L., and GRUM, W. J., 1988, Int. J. Math. Ed. Sci. Technol., 19, 215.[2| ANDREWS, L. C , 1985, Special Functions for Engineers and Applied Mathematicians (New

York: Macmillan Publishing Company).[3] OLDHAM, K. B., and SPANIER, J., 1974, The Fractional Calculus (New York: Academic

Press).[4] DUTTA, M., and DEBNATH, L., 1965, Elements of the theory of Elliptic and Associated

Functions with Applications (Calcutta: World Press).[5] DEBNATH, L., 1978, Int. J. Math. Ed. Sci. Technol., 9, 399.

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