Page 22 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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Unit 1: Bessel s Functions

1.5 Generating Function for J (x) Notes
n
n
Prove that when n is positive integer J (x) is the Co-efficient of t in the expansion of
n
x t 1
e 2 t ...(A)
n
in ascending and descending powers of t. Also show that J (x) multiplied by ( 1) is the
n
n
co-efficient of t in the expansion of the above expression.
Proof:
x 1
t
Expanding e 2 t in powers of x i.e.
xt x
x t 1 e e
e 2 t = 2 2t

2 2
3 3
xt x t x t x x 2 1 1 x 3
= 1 ....... 1 , ....... (B)
2 2 3! 2t 2t 2 3 2t
n
In the above expansion, collecting the co-efficients of t , we have
n n 1 2 n 2
1 x 1 x x 1 1 x x
1 ...
n 2 (n 1)! 2 2 (n 2)! 2 2 2
1 x n 1 x n 2 1 x n 4
=
n 2 (n 1)! 2 (n 2)! 2 2

( 1) s x n 2s
x
= J n ( ) ...(C)
s n s 2
s 0
n
Similarly co-efficients of t in the above product is
( 1) n x n ( 1) n 1 x n 1 x x 2 ( 1) n 2 x n 2
= 1. ...
n 2 (n 1)! 2 2 2 (n 2)! 2
1 x n ( 1) x n 2 ( 1) 2 x n 4
n
= ( 1) ...
n 2 n 1 1 2 2 n 2 2
( 1) s x n 2s
n
= ( 1) s n s 2
s 0
n
= ( 1) J (x)
n
0
In the above product the co-efficient of t is
x 2 x 4 1 x 6 1 x 8 1
= 1 2 2 2 2 2 2 ....
2 2 2 2 2 3 2 2 3 4
x 2 x 4 x 6 x 8
= 1 2 2 2 2 2 2 2 2 2 2 ....
2 2 4 2 4 6 2 4 6 8
= J (x)
0

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