Recurrence Formula for Bessel Function of the First Kind

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Theorem

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.


Then:

$\map {J_{n + 1} } x = \dfrac {2 n} x \map {J_n} x - \map {J_{n - 1} } x$

And:

$\map {J_{n + 1} } x = -2 \map {J_n'} x + \map {J_{n - 1} } x$


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Proof

From Generating Function for Bessel Function of the First Kind of Order n of x we have:

$\ds \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t} } = \sum_{n \mathop = - \infty}^\infty \map {J_n} x t^n$


Differentiating both sides of the equation with respect to $t$:

\(\ds \dfrac x 2 \paren {1 + \dfrac 1 {t^2} } \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t} }\) \(=\) \(\ds \sum_{m \mathop = - \infty}^\infty n \map {J_n} x t^{n - 1}\)
\(\ds \leadsto \ \ \) \(\ds \dfrac x 2 \paren {1 + \dfrac 1 {t^2} } \sum_{n \mathop = - \infty}^\infty \map {J_n} x t^n\) \(=\) \(\ds \sum_{n \mathop = - \infty}^\infty n \map {J_n} x t^{n - 1}\) Generating Function for Bessel Function of the First Kind of Order n of x
\(\ds \leadsto \ \ \) \(\ds \dfrac x 2 \paren {\sum_{n \mathop = - \infty}^\infty \map {J_n} x t^n + \sum_{n \mathop = - \infty}^\infty \map {J_n} x t^{n - 2} }\) \(=\) \(\ds \sum_{n \mathop = - \infty}^\infty n \map {J_n} x t^{n - 1}\)
\(\ds \leadsto \ \ \) \(\ds \dfrac x 2 \paren {\sum_{n \mathop = - \infty}^\infty \map {J_{n - 1} } x t^{n - 1} + \sum_{n \mathop = - \infty}^\infty \map {J_{n + 1} } x t^{n - 1} }\) \(=\) \(\ds \sum_{n \mathop = - \infty}^\infty n \map {J_n} x t^{n - 1}\) Translation of Index Variable of Summation
\(\ds \leadsto \ \ \) \(\ds \dfrac x 2 \paren {\map {J_{n - 1} } x + \map {J_{n + 1} } x}\) \(=\) \(\ds n \map {J_n} x\) by comparing coefficients
\(\ds \leadsto \ \ \) \(\ds \map {J_{n - 1} } x + \map {J_{n + 1} } x\) \(=\) \(\ds \dfrac {2n} x \map {J_n} x\)
\(\ds \leadsto \ \ \) \(\ds \map {J_{n + 1} } x\) \(=\) \(\ds \dfrac {2n} x \map {J_n} x - \map {J_{n - 1} } x\)

This is the first recurrence formula.

$\Box$


We prove the second recurrence formula by differentiating both sides of the original equation with respect to $x$:

\(\ds \dfrac 1 2 \paren {t - \dfrac 1 t} \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t} }\) \(=\) \(\ds \sum_{m \mathop = - \infty}^\infty \map {J_n'} x t^n\)
\(\ds \leadsto \ \ \) \(\ds \dfrac 1 2 \paren {t - \dfrac 1 t} \sum_{m \mathop = - \infty}^\infty \map {J_n} x t^n\) \(=\) \(\ds \sum_{m \mathop = - \infty}^\infty \map {J_n'} x t^n\)
\(\ds \leadsto \ \ \) \(\ds \dfrac 1 2 \paren {\sum_{m \mathop = - \infty}^\infty \map {J_n} x t^{n + 1} - \sum_{m \mathop = - \infty}^\infty \map {J_n} x t^{n - 1} }\) \(=\) \(\ds \sum_{m \mathop = - \infty}^\infty \map {J_n'} x t^n\)
\(\ds \leadsto \ \ \) \(\ds \dfrac 1 2 \paren {\sum_{m \mathop = - \infty}^\infty \map {J_{n - 1} } x t^n - \sum_{m \mathop = - \infty}^\infty \map {J_{n + 1} } x t^n}\) \(=\) \(\ds \sum_{m \mathop = - \infty}^\infty \map {J_n'} x t^n\) Translation of Index Variable of Summation
\(\ds \leadsto \ \ \) \(\ds \dfrac 1 2 \paren {\map {J_{n - 1} } x - \map {J_{n + 1} } x}\) \(=\) \(\ds \map {J_n'} x\) by comparing coefficients
\(\ds \leadsto \ \ \) \(\ds \map {J_{n - 1} } x - \map {J_{n + 1} } x\) \(=\) \(\ds 2 \map {J_n'} x\)
\(\ds \leadsto \ \ \) \(\ds \map {J_{n + 1} } x\) \(=\) \(\ds - 2 \map {J_n'} x + \map {J_{n - 1} } x\)

$\blacksquare$


Sources

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