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15.3 Bessel Functions 549

Now we must collect all the coefficients of t , for each n. To illustrate the idea, look for the
n
4
4
4 4
4
coefficients of t in this product. We obtain t when x t /2 4! on the left is multiplied by 1 on the
5
6
5 5
6 6
right, and when x t /2 5! on the left is multiplied by −x/2t on the right, and when x t /2 6! on
2
2
2
the left is multiplied by x /2 2!t on the right, and so on. In this way, we find that the coefficient
4
of t in the above product is
1 1 1 1
6
7
5
4
x − x + x − x + ···
8
4
10
6
2 4! 2 5! 2 2!6! 2 3!7!
∞ n
(−1)
= x 2n+4 = J 4 (x).
2 2n+4 n!(n + 4)!
n=0
n
Similar analysis shows that the coefficient of t in equation (15.19) is J n (x) for each nonnegative
integer n. For negative integers, we can use the fact that, if n is a positive integer, then
n
J −n (x) = (−1) J n (x).
While this is not a formal proof, it is a plausibility argument in support of the generating function.
15.3.9 Recurrence Relations
We will state three recurrence relations involving Bessel functions of the first kind. In these, ν is
a real number.
THEOREM 15.10
d
(x J ν (x)) = x J ν−1 (x). (15.20)
ν
ν
dx
ν
Proof Begin with the case that ν is not a negative integer. Differentiate the series for x J ν (x)
to obtain

∞ n
d ν d ν (−1) 2n+ν
(x J ν (x)) = x x
dx dx 2 2n+ν n!
(n + ν + 1)
n=0

∞ n
d (−1) 2n+2ν
= x
dx 2 2n+ν n!
(n + ν + 1)
n=0
∞ n
(−1) 2(n + ν) 2n+2ν−1
= x
2 2n+ν n!(n + ν)
(n + ν)
n=0
∞ n
(−1) 2n+ν−1
ν
= x ν x = x J ν−1 (x).
2 2n+ν−1 n!
(n + ν)
n=0
Now extend this result to negative integers by using the fact that, if ν =−m, with m a positive
integer, then
J ν (x) = J −m (x) = (−1) J m (x).
m
THEOREM 15.11
d
(x −ν J ν (x)) =−x −ν J ν+1 (x). (15.21)
dx
The proof is like that of the preceding theorem.
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October 14, 2010 15:20 THM/NEIL Page-549 27410_15_ch15_p505-562