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

DOI:
https://doi.org/10.1090/S0025-5718-1971-0304726-X

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 $l > 2$.

- N. Liron,
*Some infinite sums*, SIAM J. Math. Anal.**2**(1971), 105–112. MR**287054**, DOI https://doi.org/10.1137/0502010 - Ralph Philip Boas Jr.,
*Entire functions*, Academic Press Inc., New York, 1954. MR**0068627** - G. N. Watson,
*A Treatise on the Theory of Bessel Functions*, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR**0010746** - Nand Kishore,
*The Rayleigh function*, Proc. Amer. Math. Soc.**14**(1963), 527–533. MR**151649**, DOI https://doi.org/10.1090/S0002-9939-1963-0151649-2

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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