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

Jorge L. deLyra

Department of Mathematical Physics, Physics Institute, University of São Paulo

February 18, 2013

### Abstract:

We provide a new, simple general proof of the formulas giving the infinite sums of the inverse even powers of the zeros of the regular Bessel functions , as functions of . We also give and prove a general formula for certain linear combinations of these sums, which can be used to derive the formulas for 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 , with integer coefficients. We complete the set of known formulas for the smaller values of , extend it to , and point out a connection with the Riemann zeta function, which allows us to calculate some of its values.