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Bessel recurrence J(n+3)+J(n+5)

Short answer

Apply the standard recurrence J_{p−1}(x)+J_{p+1}(x)=(2p/x)J_p(x) with p=n+4 to obtain the identity.

Prove the following Bessel's function: J_(n+3)(x) + J_(n+5)(x) = (2/x)(n+4) J_(n+4)(x).

The standard recurrence relation of Bessel functions is

J_(p−1)(x) + J_(p+1)(x) = (2p/x) J_p(x).

Proof of the recurrence

From the two derivative results

x Jₚ′ = p Jₚ − x J_(p+1)       ... (a)   [from d/dx(x⁻ᵖ Jₚ) = −x⁻ᵖ J_(p+1)]
x Jₚ′ = −p Jₚ + x J_(p−1)      ... (b)   [from d/dx(xᵖ Jₚ) = xᵖ J_(p−1)]

Subtract (a) from (b): 0 = −2pJₚ + x(J_(p−1) + J_(p+1)), so

x[J_(p−1) + J_(p+1)] = 2p Jₚ.

Apply with p = n + 4

J_(n+3) + J_(n+5) = (2(n+4)/x) J_(n+4)

Hence J_(n+3)(x) + J_(n+5)(x) = (2/x)(n + 4) J_(n+4)(x).

✓ Answer checked by Claude Opus 5.5 (high effort)

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