Theoretical analysis of Fourier series of Gegenbauer function - methods for integration of Gegenbauer function and Fourier coefficient

Carlson, B. C.

Greiman, W. H.

English

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The Four ie r  S e r i e s  o f  Gegenbauer's Function 
Hard copy (HC) 
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R. C. Carlson and W. H. Greiman 
Iowa S t a t e  Univers i ty ,  b e s ,  Iowa 
ff 653 July 65 
1.  In t roduct ion .  If 
Cv is  known t o  have t h e  N 
C ~ ( C O S  V e )  = 
(1 .I 1 
N is  a p o s i t i v e  i n t e g e r ,  t h e  Gegenbauer polynomial 
r ep resen ta t ion  [ 3 ,  vol .  2, p. 1751 
m=O m! (N - m ) !  
where (v), = T(v  + m ) /  r ( v ) .  The Four ie r  s e r i e s  of  Gegenbauer's P i e t i o n  
C"(cos e)  with  genera l  (poss ib ly  complex) cy does no t  appear t o  have been 
1 
given previous ly ,  even i n  t h e  s p e c i a l  case  o f  Legendre's  func t ion  P = 
We s h a l l  f i n d  t h a t  
o! 
cy % *  
If Re v >, 1 t h e  Four ie r  c o e f f i c i e n t s  do n o t  exist f o r  genera l  cy because 
C (cos e)  i s  no t  i n t e g r a b l e  over an i n t s r v a l  conta in ing  t h e  po in t  8 = n. 
If Re v c 1 and CY i s  a p>t ive  i n t e g e r  N, t h e  first f a c t o r  of A vanishe 
if N + n i s  odd; s i n c e  A = An , (1.2) t hen  reduces t o  (1.1). 
V 
01 
n 
-n 
We remark t h a t  Cegenbauerls func t ion  (mul t ip l i ed  by a cons tan t  t o  
This work w s s  supported bjj t he  Ames Laboratory of t h e  U. S. Atomic 1 
Energy Commission and by t h e  National Aeron'autics and Space Administra- 
t i o n  under Grant KsG-293 t o  Iowa S t a t e  Universi ty .  
/ 
I 
https://ntrs.nasa.gov/search.jsp?R=19650019891 2020-03-17T01:59:29+00:00Z
4 
2. 
give it t h e  value u n i t y  a t  8 = 0)  has a n i c e l y  symmetrical expression i n  
t h e  no ta t ion  of t h e  hne rgeomet r i c  R funct ion [ 11 : 
where P is an as soc ia t ed  Legendre funct ion.  An important s p e c i a l  case is  
2. The Fourier  c o e f f i c i e n t s .  Gegenbauerls func t ion  is  defined [ 3 ,  vol .  1 ,  
p .  1781 by 
% 
We assume f u r  t h e  p re sen t  t h a t  Re v < 8 , so  t h a t  t h e  hypergeometric series 
converges abso lu te ly  f o r  0 = x [5, p. 251 and hence uniformly over t h e  
*- 
i n t e r v a l  (0, T). The Four i e r  c o e f f i c i e n t  
(2.2) 
can then  be found by i n t e g r a t i c g  term by term. Fror.1 t h e  elenentaqr formula 
( 2 . 2 )  
. 
3. 
w e  g e t  
(2.4) 
k.I-0 (v  t -$ t n )  (1 t 2n)k k! r(24 r(0 t 1 )  ( V  t +), n! k 
The las t  series, obtaiced from t h e  preceding one by p u t t i n g  m = n t k, is 
a F series wi th  u n i t  argument. If Fie w < 1 it converges and can be summed 
by Watson's theorem [ 3 ,  vo i .  1, p. 1891: 
3 2  
(2.5) F (n  - c, 2v t a t n ,  -$ t n ;  v t 4 t n ,  1 t 2n; 1 )  3 2  
S u b s t i t u t i o n  i n  (2.4) gives  an expressicn f o r  An t h a t  
applying s e v e r a l  t iees  t h e  dup l i ca t ion  formuls. f o r  t h e  
t h e  r e l a t i o n  r ( z )  r(l - z) = SI csc m. The resu l t  is  
can be s i m p l i f i e d  by 
gamma func t ion  and 
1 C Y - n  1 r ( v  t 2 1 T b J  4- 2- a t n  C Y - n  (-1 )" sir ,io! 
-1 sic TV sic n ( 2 )  [ r ( v ) ]  r(1 t -) r(1 t 
sir, TF(V + 
0 - n  2 a t n  C Y - n  
(2.6) An = - . 
2 2 
Elementary rearrangement o f  t h e  sire func t i cns  now l e a d s  t o  (1.2).  
If R e  w >, 3 t h e  ser ies  (2.1) no longer  converges uniformly over (0, n )  
if it does not  termir-ate. I-iowever, t h e  a n a l y t i c  coc t inun t ion  of Gauss t hyper- 
eeometric funct ion '5, p. 2911 shcws t h a t  c (cos 8) i s  then of t h e  o rde r  of  
(cos &I 
W 
CY 
1 -3w 
as 8 - 'li (except t h a t  t h e  s ingui  a r j t y  i s  loga r i thmic  i f '  w = ) . 
4 .  
3rovided t h a t  ?e L < 1, t h e  function i s  ir,tegrab;e over  ( C ,  n ) ;  furthermore,  
it s a t i s f i e s  ccmciiticns [5, p. 1641 s u f f i c i e n t  t o  ensure t h a t  i t s  Fourier  
s e r i e s  converges (except when 8 i s  an odd m G l t i p l e  of n )  and r ep resen t s  t h e  
functior,.  
To shcw t h a t  t t e  Fourier  c o e f f i c i e n t s  are s t i l l  given by (1.2) i f  
3 5 Re v < 1, one can e i t h e r  use a n a l y t i c  cont inuat ion i n  v o r  j u s t i f y  
d i r e c t l y  t h e  term-by-term i n t e g r a t i o n  of (2.1). The second method i s  t h e  
easier i f  one uses  t h e  f o l l o w h g  themem [ 4 ,  D. 451: If C urn(@) converges 
1 ~ r L f n d y  over ( C ,  r - E )  f o r  every (small) p o s i t i v e  E , and i f  
Z J'" lu,(0) 1 d e  converges, t hen  C u (e) may be i n t e g r a t e d  term by term 
over  ( 0 ,  TI. 
m 0 
The f irst  assunpt icn cf t h e  theorem i s  ? la i r - ly  s a t i s f i e d  by 
urn@) = 
Moreover, w e  have 
(v + 31, m! 
The l a s t  series converges i f  Re v < 1 ,  and t h e  proof of  (1.2) i s  now complete. 
Eq. (1.3) follows from (2.1) by use of  t h e  r e l a t i o n  [I, Eq. (2.5)] 
F (a,  b; c ;  x) = R(a; b, c - 0 ;  1 - x, 1 )  2 1  (2.7) 
and t?!e quadra t i c  t ransformation [2, Eq. (5.1)] 
(2.2) 2 2  2 R!a; b, b; x , y ) = R(a; 2b - a ,  a - b + 3; (x t y) / 4 ,  xy) . 
- , ,  . 
5 .  
References 
1. B. C. Carlson, L z u r i c e l l a l s  hypergeometric funct ion F , Journal  of Math- D 
ematical  Analysis and Applications,  vol. 7 (1963), pp. 452-47C. 
2. E. C. Carlson, Normal e l l i p t i c  i n t e g r z l s  o f  t h e  first and second k inds ,  
Duke Mathematical Journal,  September, 1964. 
3.  A .  Erdglyi ,  W. Magnus, F. Oberhett inger,  and F. G. T r i c o c i ,  Higher 
Trmszendental  F x c t i c n s  New York 1953. 
4 .  E. C. T i t c h a r s h ,  The Theory of Funct ions,  2nd ed., Oxford, 1939. 
5. E. T. Whittaker and G. N. Watson, Modern Analysis,  4 t h  ed., Cambridge, 1?27. 


The fourier series of gegenbauer's function

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