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S o m e R e l a t i o n s b e t w e e n e s se l a n dL e g e n d r e F u n c t i o n sB y

J . C . C o o k e , S i n g a p o r e(Received M ay 3, 1956)

SummaryC e r t a i n i n f i n i t e i n t e g r a l s i n v o l v i n g B e s s e l F u n c t i o n s a r e e x p r e s s e d

i n t e r m s o f A s s o c ia te d L e g e n d r e F u n c t i o n s i n o b l a te a n d p r o l a t e s p h e -r o i d a l c o o r d i n a te s . S o m e e x p a n s i o n s o f B e s s el F u n c t i o n s i n s er ie s in v o l -v i n g L e g e n d r e F u n c t i o n s a r e a ls o g iv e n .

ntroduction and NotationT h e s t a r t i n g p o i n t i s t h e r e la t io n

( 2 ) ~ F ( ~ - m ~ - l )f e - ~ t - ~ g ,~ (e ) J n + ~ ( et) d t = ~ F ( , _ _ m P j ~ (/~ ) q j m ( O ,1 )

p r o v e d b y Cooke [ 2] i n a s l i g h t ly d i f f e re n t f o r m a n d n o t a t i o n , f r o m ar e l a t i o n o f Bailey. [ 1] . H e r e a n d ~ a r e o b l a t e s p h e r o i d a l c o o r d i n a t e sd e f in e d b y z = e ~ ,

@ = e % / { ( 1 - - 2 ) ( 1 + ~ 2)}.i s t a k e n t o b e r e a l a n d s a t is f ie s - - 1 < < 1 . @ a n d z a r e c y l i n d r i c a l

c o o d i n a t e s .W e e x t e n d t h i s r e s u l t to o t h e r i n t e g r a ls i n v o l v i n g t h e f u n c t i o n s Y ,K , I a n d H , a n d w e a ls o g iv e so m e s er ie s c o n n e c t i n g t h e s e f u n c t i o n s .

T h e n o t a t i o n i s Watson's t h r o u g h o u t . [ 7 ] .T h e m a i n p u r p o s e o f th i s p a p e r i s to p r o v e s o m e re l at io n s w h i ch ,

a s w e l l a s b e i n g i n t e r e s t i n g i n t h e m s e l v e s , w i l l b e o f uq e in s o m e w o r kw h i c h is p r o c ee d i n g in c o n n e c t io n w i t h c e r ta i n b o u n d a r y v a l u e p r o b l em s ,

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J . C. Cooke: S ome Re la t ions be tw een Besse l and Legendre Fun ct ion s 323i n v o l v i n g d u a l i n t e g r a l e q u a t i o n s , p a r t i c u l a r l y w h e n t h e m e d i u m i sn o t i n f i n i t e i n a l l d i r e c t i o n s , b u t b o u n d e d b y a c y l i n d e r o r b y t w op a r a l l e l p l a n e r

W e s h a l t f r e q u e n t l y r e f e r t o r e f e r e n c e [ 3 ] w h i c h w e s h a l l d e n o t eb y B .

W e s h a l l e x t e n d t h e n o t a t i o n d u e o r i g i n a l l y t o N i c h o l s o n [ 6 ] i nt h e f o r m

v ~ ( O = e - ~ ' ~ P ~ ( i O , (2)m mq~ ( 0 = e ~ ~ + ~ ) ~ - ~ Q ~ (~ ) 3)

I f ~ i s r e a l a n d p o s i t i v e q ~ (~ ) i s r e a l w h e n m a n d n a r e r e a l , b u tp ~ ( ~ ) i s n o t u n l e s s m ~ - n i s a p o s i t i v e i n t e g e r , s o t h a t h e r e t h e n o t a t i o ni s n o t s o u s e f u l . I t w i ll b e n o t i c e d t h a t t h e d e f i n i t i o n ( 3) i s d i f f e r e n t f r o mt h a t g i v e n b y Cooke. [ 2 ] .

A n e a r e r f o r m o f e q u a t i o n (1 ) i s( ~ c / 2 ) ~ I e - ~ ' t - u J~(@t) J ~ + ~ (c t) d t = p : m ( ) q , ~( r (4)o

w h i c h f o l l o w s f r o m e q u a t i o n (3 ) a n d B , 3 . 3 (2 ).S i m p l e r e s tr ic t io n s o n t h e v a r i a b l e s w i l l h a v e t o b e i m p o s e d f o r

c o n v e rg e n c e . T h e s e w i ll n o t u s u a l l y b e g i v en , b u t t h e y c a n b e w r i t t e nd o w n w i t h o u t d i f f i c u l ty s in c e t h e b e h a v i o u r o f t h e f u n c t i o n s a t t h eo r ig i n a n d a t i n f i n i ty i s w e l l -k n o w n . A s a l r e a d y m e n t i o n e d # i s a l w a y st o b e t a k e n t o b e re a l, w i t h - - 1 < / x < 1, a n d z m u s t b e s u c h t h a tR ( z ) > O .

I n t e g r a l s I n v o l v i n g J , Y , a n dI n e q u a t i o n (4 ) w e re p l a c e m b y - - m a n d u s e t h e f o r m u l a

Y,~(z) = ( s i n m 7 ~ : 1 { J m (z ) c o s m ~ - - J _ ~ ( z ) } . ( 5)W e t h u s o b t a in

(ze c /2 ) ~ ~ e - ~ t -Y~ Ym(@t)J ~ + ~ ( c t ) d t =( s in m ~ ) - 1 {p ~ -m ( ff) q . t ) c o s m z - - P ~ ( f f ) q ~-m ($ )}

= ( s i n m e l) - i [ p ~ m ( f f ) q ~ (r c o~ m ~ - - { c o s m :~ P ~ - ( f f) +--m mH - (2 /~ ) s i n m ~ Q . ( f f ) } q . ( ) ]

o n m a k i n g u s e o f B , 3 .4 ( 17 ) a n d B , 3 . 3 ( 2) .H e n c e w e h a v e

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324 J.C . Cooke:

(r e c l 2 ) ~ S e - ~ t - ~ Y , ~ ( e ) J , ~ + ~ ( c t) d t - = ~ (2/95) Q j m ( # ) q ~ ( ~ ) 9 (6)oOn co mb in ing toge the r t he r e su l ts (4) an d (6 ) me ob ta in

(r e c l 2 ) 8 9 ~ e - ~ t - ~ H ~ ) (~ )( qt) J ~ + ~ ( c t ) d t = -= q ~ ( r { P : ~ ~ ) ~ ~ i / r e ) ? Z ~ ~ ) } 7 )

w here th e - - an d + s igns go wi th H o) and H ('~) respec t ive ly 9A g a i n i n e q u a t io n (4 ) w e re p l a c e n + 1~ b y - - n - - 1~ a n d u s e th e

f o r m u l a H ~ ) (z ) = ( i s in m re ) -~ { J _ A z ) - - J A z ) e - ~ 'This wi ll g ive an in t eg ra l i nvo lv ing J~ n o ) on th e l e f t ha nd side,

a n d t h e r i g h t h a n d s id e isp m m{ si n ( . + ) 9 5} -~ { _ . _ ~ ( ~ ) ~ _ ~ _ ~ ( ~ ) - - e ' ( ~ + ~ ) ~ P Z ~ ( ~ ) q ~ ( ~ ) }

= ( i c o s n 9 5) -~ e ( ~ ' ~ + ' ~ ) ~ p zm ( ~ ) { Q ' 2_ ~ _,( ~ ~ ) - - Q T ( i ~)}on m ak in g use o f equ a t ion (3) and B , 3 .4 (7 ).

H enc e by B , 3 .3 (9) we havere e / 2 ) ~ ~ e - " t - ~ J . e t ) H(nl)+~(c t) a t -~

= - - i F ( n + m - k 1) /"( n - - m ) P~-m(~u)p~-m(~.)==9 P ( ~ + m + 1 ) ~ e - ~ ' ~

= - ~ F ( n - - m + 1) sin ( ~ - - m ) re P g ~ ( P ) P ~ ( i ~ ), (8)f rom equa t ion (2 ) and B , 1.2 (6).

I f we r ep lace ~ by e - i~ ~ and e by e~" c we ob ta in( re v /2 ) ~ I e - e t -~ '~ Jm ( f f t ) Hf )+g~(c t) e lt =

9 F ( n + m + 1) ~ e~ni~= ~ ~ F ( n ~ m + 1) sin ( n ~ m ) 95 P ~ m ( / u ) P ~ ' ~ ( - - i ~ ) . (9)

Su b t r ac t ing equa t ion (9 ) f rom equa t ion (8 ) an d d iv id ing by 2 ig iv e s o n t h e l e f t h a n d s id e a n i n t e g r a l i n v o lv i n g J m Y . + 5 a n d t h e r i g h than d s ide is equa l t o~ ( n ; - - m t 11

9= PZ ~(# ) {e- ~ ' ~ / ~ , (ir + e 5 ~ P ~ ( - - I r 2 s in ( n - - m ) 95

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Some Rela tions be tween Besse l and Legendre Fu nc t ion s 325o n m a k i n g u s e o f B , 3 .3 ( 6), a n d s o w e h a v e

( . , / 2 ) ~ . f e - ~ t - ~ J m e t ) Y . + ~ ~ O d t = ( / 2 ) p ; ~ ( ~ ) R ~ (~ ), (1 0 )0w h e r e/ ~ ( r = { sin (n - - m ) a } -~ { e - ~ ' ~ P ~ .( i r + e ~ ' = P'~ ( - - i r (11)w h i c h i s r e a l i f ~ i s r e a l .

W e n o t e t h a tF ( - - ~ m )

R : m ( ~ ) - ~ F ( - - n + , m ) / ~ (~ ) (1 2)b y B , 3 . 3 ( 6 ) a n d B , 1 . 2 ( 6 ) .

C h a n g i n g t h e s ig n o f m i n e q u a t i o n ( 10 ) a n d u s i n g e q u a t i o n (5 ) w eh a v e o n t h e le f~ h a n d s id e a n i n t e g r a l i n v o l v i n g Y ,~ Y n + ~ a n d o n t h er i g h t h a n d s id e

~ / 2 ) s in m ~ ) - ~ { P Z ~ ~ ) R .~ ~ ) c o s ~ ~ - - P ~ ~ ) R : ~ ~ ) } =

= ( = 1 2 ) ( s i n m z ) - 1 R ~ ( r { P _ - .~ 1 (/~ ) c o s m z / ( - - n - -m )F ( - - n + m ) p m ,_ ,(~ ,) }b y B , 3 . 4 ( 7 ) a n d e q u a t i o n ( 1 2 ) .

H e n c e w e h a v e o n u s i n g B , 3 . 4 (1 3)~o(:~ c / 2 ) ~ f e - * t t - ~ Y m (Q t ) Y = + 8 9 d t = Q - ~ _ i ( / z ) R ~ ( ~ ) .

0

F r o m e q u a t i o n s ( 2), (4 ), (8 ) a n d (1 0) w e d e d u c e t h a t

0= P : ~ ( ~ ) q ': (~ ) - - (2 i /N ) Q Z m ( ~ ) q ': (~ ) + ( i : 2 ) P Z ' ( ~ ) R ~ ( r - -

- m R~ ~ ) .

(13)

14)ntegrals nvolving and K

I n e q u a t i o n s (7 ) w e w r i t e c e i n s t e a d o f c a n d w e o b b a in2 c l ~ ) ~ ~ d - ~ t - ~ K , ~ et ) I . + ~ c t ) d t =

0= e T 89 q ~ ( ~ ) { P n m ( f g ) ~ ( 2 'i/;7 10Q ~ - m ( ~ ) } .

H e n c e w e h a v e o n a d d i n g a n d s u b t r a c t i n g e q u a t i o n s (1 5)(15)

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326 J . C . C ooke:

2 e / ~ ) ~ f cos z t t - ~ K A e t ) I . + ~ ( e t ) d t == q m ( r { c o s % ( n - - - m ) ~ P ~ m ( / z ) - - ( 2 / z t ) s i n % ( n - - - m ) ~ Q z m ( /z )} , ( 1 6 )a n d a s i m i l a r e x p r e s s io n i n v o l v i n g s in z t .

D e a l i n g s i m i l a r l y w i t h e q u a t i o n s (8 ) a n d (9 ) l e a d s to2 c/n) ~ S em~ t - ~ Im(et) K n + ~ c t ) d t =

_ r ( n + m + 1) n em~,~*~= - - F ( n - - m + l ) s in (n ---= m ) et P : ' ~ ( t z ) P : m ( - 4 - i $ ) (17)

H e n c e w e h a v e( 2 c / z ) ~ i e o s z t I , ~ ( q t ) K n + ~ ( c t ) d t =

F n + m + 1 ) :~= - - P ( n - - m + I ) 2 s i n ( n - - m ) 7r P ~ - ~ ( # ) { e - ~ r P Z ' ~ ( i r +

+ e ~ p j m ( _ _ i r (1 8)a n d a s i m i l a r e x p r e s s io n i n v o l v i n g s i n z t .

W e n o t e p a r t i c u l a r l y f r o m e q u a t i o n s (4) a n d (1 6) t h a t i f w e w r i t en = m + 2 s , w h e r e s i s a p o s i t i v e i n t e g e r o r z e r o( n / 2 ) f e - * t t - ~ J m ( ~ t ) J m + ~ , + ~ / , ( c t) d t

= ( - - 1 ) s ~ c o s z t t - ~ K m ( p t ) I ,~ + ~ s + ~ ( c t) d r , (19)a n d s e v e r a l o t h e r s i m i la r d e d u c t i o n s m a y b e m a d e . T h e s e r e s u lt s m a ya ls o b e o b t a i n e d b y a d o u b le i n c r e a s i n g a n d d e c r e a s in g t h e a r g u m e n tb y 1//9~ i n t h e s t y le o f D i x o n a n d F e r r a r . [4 ] .

Ser ies for an d JW e s h al l u s e t h e f o r m u l a f o r t h e G e g e n b a u e r p o l y n o m i a l

C , , ( z ) = 2 - F ( n + 2 v ) F ( r + t//2) ( z - - 1 ) - P = + 5 - ~ ( z ) / F ( 2 v ) F ( n + l ) ,20)

a s g i v e n i n B , 3 . 1 5 ( 4). I f z i s r e a l a n d - - 1 < z < 1 w e r e p l a c e z ~ - - 1 b y1 - - z ~ .

W a t s o n [ 7] g i v es t h e f o r m u l a

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Some Relat ions betwe en Bessel and Legendre Fu nction s 327J ~ _ ~ ( z s i n ( b s i n ~ ' )(z s in ~5 s in ~ ' ) ' - ~ exp ( i z cos ~b cos r =

2~P(v) | i " s ( ~ , + s ) J ~ + , ( z )- V 2 ~ ) ~ r 2 ~ + s ) ~ C ~ c o ~ ) ~ ; c o s r

P u t v = m q- 1~, cos 9 = # , cos ~ ' = i $ , z ----- 2 c an d w e hav e ,u s i n g eq u a t i o n (2 0),J~(e 2) c-Z~= 7g~ / 2 z 2 c ) X ( -- 1 ) s (2 m + 1 -}- 2 s ) / ' ( 2 m -{- 1 -}- s) F s q - 1 )

x J m + y ~ + , ( 2 c ) P~5(t, ) p7~5(~). (21)I n a s i m i l a r w ay b y w r i t i n g z = if= i 2 c i n s t e a d o f z = 2 c w e h a v e

2 m + 1 -+- 2 s) / '( 2 m -+- 1 ~ sI r a ( 2 ~ e ~ /( 2 ~2c) 2 i ~- P s + 1 ) I m + 8 9 P S ~ 8 ( # ) P ~ 8 ( r (22)

H e n c e w e h a v e o n a d d i n gz ~ ~ ( 2 m ~ - l ~ - 4 s ) / ' ( 2 m - + - l - ~ 2 s )

Im(2q) cos 2 z - - A / 1 2 -z e 2 c ~ X ( - -1 ) I ' ( 2 s + 1) V~ s=Op ~/i't, - m Im+~A+2, 2 c) m+2,(tt) p~ +: ,(r (23)

t o g e t h e r w i t h a s i m i la r f o r m u l a f o r I m ( 2 ~ ) s in 2 z b u t w i t h 2 s a n d 4 so n t h e r i g h t h a n d s id e r e p l a c e d b y 2 s ~ 1 a n d 4 s q - 2 w h e r e v e r t h e yo c c u r .

S o m e o t h e r s i m p l e r s er ie s d e d u c e d f r o m a n o t h e r f o r m u l a o f W a t s o n ,n a m e l y / 0 ) ~ z ~ + ~ - ~c . . . ~ s in O ) - / ' ( 8 9 (2 s in 0 ) ~ - ~ X ~ I ( 2 v + n ) C ~(eos 0 ) ,d o n o t ap p ea r t o h av e b een n o t i c ed ex cep t i n s p ec i a l c a s e~ . [ 5 ] .

I n t h i s f o r m u l a w e w r i te v = m q - 1 /2 z = r t , a n d t h e n p u ~ r t cos 0 == t z , r t s in 0 ----- t~ ; w e th u s o bt ai n

& J m ( ~ t ) = ( t r ) " X ( t r ) " P ~ n ( e o s O ) / n ! . (24)n~OR e p l a c i n g t b y i t w e h a v e

e i*~ I , ~ ( Q t ) = ( t r ) m i ~ ( t r ) " P ~ ( c o s O ) / n !It Oa n d s o

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328 J . C . Cooke: Some Rela t ions betwee n Bessel and Legendre Fun ct ions~

c o s t z Im(Q t ) = ( t r)m 2 ( 1) s ( t r )~ P ~ 2 , ( c o s 0 ) / ( 2 s ) , (2 5)w i th a co r r e spond ing s er ie s fo r s i n t z Im Q t) .

T h e s e e x p r e s s i o n s m a y b e r e g a r d e d a s g e n e r a t i n g f u n c t i o n s f o ra s s o ci a te d L e g e n d r e F u n c t i o n s o f t h e f i r s t k i n d .

Prolate Sp heroida l oordinatesThese coo rd ina t e s a r e -defined b y t he equa t i ons

Q e~v/{( I __ 2)(}~ - - I)},a n d s o al l o u r r e s u lt s m a y b e e x p r es s e d i n t h e s e c o o r d i n a t es b y w r i t i n ge~ i~ = } a n d r ep l ac ing c b y c e ~ l~ . These l e ave z and ~ unchanged .

W e q u o t e t ~ o e x a m p l es o nl y. W e o b t a in i m m e d i a t e ly f r o m t h e sesubs t i t u t i ons a nd equa t i ons (4 ) and (3 )

o(zw /2) ~4 I e - ~ t t - ~ J m ( e t ) I n+~ /~ (c t) d t = e - ~ m P ; m ( t ~ ) Q ' ~ ( } ) . (26)Simi l a r l y from th e s e ri e s (23 ) we de duco

I,,~(2e) cos 2z - - - -/ ~o) ~Zo -1)~ 2 m+ i + 4 ~) ~ 2 ~+i+2 ,), F 2 s + 1 ) xX J m + v ~ + ~ , ( A c ) P ~ , ( #) P ~ 2 , ( } ) . (2 7)

ReTerences[1] B a i l e y , W . N . , Pro e. L ond . Math. Soc., 40 (1936), 37.[2] Cooke , J . C . , Proe. Camb. Phil. Sot., 49 (1953), 162.[3] E r d e l y i , A . (ed) , Higher Transcendental Functions. (New York, 1953).Ba tem an M anuscr ip t P ro jec t .[4] D i x o n , A . L . a n d F e r r a r , W . L . , Quart . Jour. Maths. (Oxford Series) ,

l 1930) , 122.[5] K n i g h t , R . C . , Quart. Jour. Maths. Oxford Series, 7 (1936), 126.[6] Nieholson, J . W., Phil . Tran s. R oy . Soc. A, 224 (1924).[7] Watson, G. N., Th eo ry of Bessel Func tions (Cambridge, 1944).