Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-11T01:47:28.113Z Has data issue: false hasContentIssue false
  • English
  • Français

Singularities of bessel-zeta functions and Hawkins' polynomials

Published online by Cambridge University Press:  26 February 2010

Kenneth B. Stolarsky
Kenneth B. Stolarsky
Affiliation:
Department of Mathematics, 1409 West Green Street, University of Illinois, Urbana, Illinois, 61801, U.S.A.
Get access

Abstract

The asymptotic behaviour of a sequence of polynomials cm = cm(v) satisfying

is established. These polynomials occur in Hawkins' formula for the residues of a Bessel-zeta function at its possible poles in the left half plane. The results imply that cm(v)/cm(0) converges uniformly to cos πV on compact sets. This in turn implies that, for v not a half odd integer, all but finitely many of the possible poles are actual poles.

Type
Research Article
Information
Mathematika , Volume 32 , Issue 1 , June 1985 , pp. 96 - 103
Copyright
Copyright © University College London 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abramovitz, M. and Stegun, I. E.. Handbook of Mathematical Functions (Dover, New York, 1968).Google Scholar
2.Burchnall, J.. The Bessel polynomials. Canad. J. Math., 3 (1951), 62–68.CrossRefGoogle Scholar
3.DeBruijn, N. G. and Erdős, P.. Some linear and some quadratic recursion formulas. Kon. Ned. Akad. v. Wetensch. (A), 54 (1951), 374–382; 55 (1952), 152-163.CrossRefGoogle Scholar
4.deBruin, M. G., Staff, E. B. and Varga, R. S.. On the zeros of generalized Bessel polynomials, I. Proc. Kon. Neder. Akad. A, 84 (1981), 1–13.Google Scholar
5.Feller, W.. An Introduction to Probability Theory and its Applications, 2nd ed. (Wiley, New York, 1960).Google Scholar
6.Grosswald, E.. Bessel Polynomials, Lecture Notes in Mathematics, 698 (Springer, Berlin, 1978).CrossRefGoogle Scholar
7.Hawkins, J.. On a Zeta Function Associated with Bessel's Equation, Doctoral Thesis (University of Illinois, 1983).Google Scholar
8.Ismail, M. E. H.. Bessel functions and the infinite divisibility of the Student t distribution. Ann. Probability, 5 (1977), 582–585.CrossRefGoogle Scholar
9.Ismail, M. E. H. and Kelker, D. H.. The Bessel polynomials and the Student t distribution. SIAM J. Math. Analysis, 7 (1976), 82–91.CrossRefGoogle Scholar
10.Ismail, M. E. H. and Kelker, D. H.. Special functions, Stieltjes transforms, and infinite divisibility. SIAM J. Math. Analysis, 10 (1979), 884–901.CrossRefGoogle Scholar
11.Ismail, M. E. H. and Miller, K. S.. An infinitely divisible distribution involving modified Bessel functions. Proc. Amer. Math. Soc, 85 (1982), 233–238.CrossRefGoogle Scholar
12.van der Poorten, A. J. and Tijdeman, R.. On common zeros of exponential polynomials. L'Enseignement Math., 21 (1975), 57–67.Google Scholar
13.Watson, G. N.. A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge, 1966).Google Scholar