Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Infinite sums of roots for a class of transcendental equations and Bessel functions of order one-half
HTML articles powered by AMS MathViewer

by N. Liron PDF
Math. Comp. 25 (1971), 769-781 Request permission


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 $l > 2$.
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 33A40
  • Retrieve articles in all journals with MSC: 33A40
Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Math. Comp. 25 (1971), 769-781
  • MSC: Primary 33A40
  • DOI:
  • MathSciNet review: 0304726