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Mathematics of Computation

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

Author: N. Liron
Journal: Math. Comp. 25 (1971), 769-781
MSC: Primary 33A40
MathSciNet review: 0304726
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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 $l > 2$.

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Keywords: Bessel functions of order one-half, roots of a transcendental equation, Hadamard’s factorization theorem, infinite sums of powers of roots, recurrence relations, multiplicity and splitting of roots at and from zero, Sturm-Liouville system
Article copyright: © Copyright 1971 American Mathematical Society