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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 SI = *- 1 aT 2t, 1 = 1, 2, * * *, where the ak's are the nonzero roots of f(z). We also prove under what conditions on A(z) and B(z) is S1 = ir212r(2l + 2) or S1 = Wv-212r(21 + 2)(221+2 - 1), where r is the Riemann zeta function. We prove that, although Bessel functions of positive half-order, JI+1/2, have only real roots, perturbation of

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