Zeros of Regular Bessel Functions

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.