Infinite sums of roots for a class of transcendental equations and Bessel functions of order one-half

Abstract:

The roots of Bessel functions of order one-half are special cases of roots of transcendental equations of the form $\tan z = A(z)/B(z)$, where $A(z),B(z)$ are polynomials and $A(z)/B(z)$ is odd. We prove that the function $f(z) = B(z)\sin z - A(z)\cos z,f(z)$ even or odd, satisfies the conditions of Hadamard’s factorization theorem, and derive recurrences for sums of the form ${S_l} = \sum \nolimits _{k = 1}^\infty {\alpha _k^{ - 2l},l = 1,2, \cdots }$, where the ${\alpha _k}$’s are the nonzero roots of $f(z)$. We also prove under what conditions on $A(z)$ and $B(z)$ is ${S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2)$ or ${S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2)({2^{2l + 2}} - 1)$, where $\zeta$ is the Riemann zeta function. We prove that, although Bessel functions of positive half-order, ${J_{l + 1/2}}$, have only real roots, perturbation of any one of its coefficients introduces nonreal roots for $l > 2$.