Recurrence relation of Bessel Function

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aryajur

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recurrence relation for bessel equation

How can I prove the recurrance relation of the Bessel Function of the 1st kind? i.e.

J(p-1, x) + J(p+1, x) = 2p/x J(p, x)

Any hints would be appreciated.
 

bessel function recurrence relation

Use generating function
\[g(x,t) = e^{\frac{x}{2}( t-\frac{1}{t})}\]
From
\[ e^{\frac{x}{2}( t-\frac{1}{t})} = \sum_{=-\infty}^{\infty} J_n (x) t^n\]

\[ \frac{\partial}{\partial t} g(x,t) = \frac{1}{2} x (1+ \frac{1}{t^2})e^{\frac{x}{2}( t-\frac{1}{t})} \]
\[=\sum_{n= - \infty}^{\infty} n J_n(x) t^{n-1}\]

Plug back generating function:
\[\frac{x}{2} ( \sum_{n= - \infty}^{\infty} J_n(x) t^n + \sum_{n= - \infty}^{\infty} J_n(x) t^{n-2} ) = \sum_{n= - \infty}^{\infty} n J_n(x) t^{n-1} \]

compare the coef. of the polynomial with the same power
 
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    aryajur

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