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arXiv · 1305.0228

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

Jorge L. deLyra
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Abstract

We provide a new, simple general proof of the formulas giving the infinite sums $σ(p,ν)$ of the inverse even powers $2p$ of the zeros $ξ_{νk}$ of the regular Bessel functions $J_ν(ξ)$, as functions of $ν$. We also give and prove a general formula for certain linear combinations of these sums, which can be used to derive the formulas for $σ(p,ν)$ 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 $ν$, 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.

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Jorge L. deLyra. 2014-02-13. On the Sums of Inverse Even Powers of Zeros of Regular Bessel Functions. https://arxiv.org/abs/1305.0228

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