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

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.

arXiv comment15 pages, 1 figure. Fixed a calculation error that had rendered the explicit expression for p=9 incorrect. Completed the references and acknowledgements