IGNOU MPH 1 2026 - Using the generating function for Bessel functions of the first kind a

Question

Using the generating function for Bessel functions of the first kind and integral order

$$g(x, t) = \exp \left[ \frac{x}{2} \left( t - \frac{1}{t} \right) \right] = \sum_{n=-\infty}^{\infty} J_n(x) t^n$

Obtain the recurrence relation

$$J_{n-1}(x) + J_{n+1}(x) = 2 \frac{n}{x} J_n(x)$

Also using the generating function show that

$$J_0(x) + 2J_2(x) + 2J_4(x) + 2J_6(x) + \dots + 2J_{2k}(x) + \dots = 1$

19 Jan 2026

Answer :

Word Count : 132

Starting from the generating function [ g(x,t)=\exp!\left[\frac{x}{2}\left(t-\frac{1}{t}\right)\right]=\sum_{n=-\infty}^{\infty}J_n(x)t^n . ] Differentiate both sides with respect to (t): [ \frac{\partial g}{\partial t} =\frac{x}{2}\left(1+\frac{1}{t^2}\right)\exp!\left[\frac{x}{2}\left(t-\frac{1}{t}\right)\right]. ] Using the series form, [ \frac{\partial g}{\partial t}

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