On the Sums of Inverse Even Powers of Zeros of Regular Bessel Functions

TL;DR

This paper presents a new proof for formulas related to sums of inverse even powers of zeros of Bessel functions, extends known results, and explores connections with the Riemann zeta function.

Contribution

It introduces a simple, general proof for these sums, derives a formula for linear combinations, and extends the results to higher powers up to p=9.

Findings

Sums are ratios of polynomials with integer coefficients.

Extended formulas for p=1 to p=9.

Connected sums to Riemann zeta function values.

Abstract

We provide a new, simple general proof of the formulas giving the infinite sums $\sigma(p,\nu)$ of the inverse even powers $2p$ of the zeros $\xi_{\nu k}$ of the regular Bessel functions $J_{\nu}(\xi)$, as functions of $\nu$. We also give and prove a general formula for certain linear combinations of these sums, which can be used to derive the formulas for $\sigma(p,\nu)$ by purely linear-algebraic means, in principle for arbitrarily large powers. We prove that these sums are always given by a ratio of two polynomials on $\nu$, with integer coefficients. We complete the set of known formulas for the smaller values of $p$, extend it to $p=9$, and point out a connection with the Riemann zeta function, which allows us to calculate some of its values.