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 ) \tan z = A(z)/B(z) , where A ( z ) , B ( z ) A(z),B(z) are polynomials and A ( z ) / B ( z ) A(z)/B(z) is odd. We prove that the function f ( z ) = B ( z ) sin ⁡ z − A ( z ) cos ⁡ z , f ( z ) 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 = ∑ k = 1 ∞ α k − 2 l , l = 1 , 2 , ⋯ {S_l} = \sum \nolimits _{k = 1}^\infty {\alpha _k^{ - 2l},l = 1,2, \cdots } , where the α k {\alpha _k} ’s are the nonzero roots of f ( z ) f(z) . We also prove under what conditions on A ( z ) A(z) and B ( z ) B(z) is S l = π − 2 l − 2 ζ ( 2 l + 2 ) {S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2) or S l = π − 2 l − 2 ζ ( 2 l + 2 ) ( 2 2 l + 2 − 1 ) {S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2)({2^{2l + 2}} - 1) , where ζ \zeta is the Riemann zeta function. We prove that, although Bessel functions of positive half-order, J l + 1 / 2 {J_{l + 1/2}} , have only real roots, perturbation of any one of its coefficients introduces nonreal roots for l > 2 l > 2 .