Singularities of bessel‐zeta functions and Hawkins' polynomials

Abstract

The asymptotic behaviour of a sequence of polynomials cm = cm(v) satisfying cm+2=12[(m+3)cm+1−∑k=0mcm−kck] 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.

Sums of powers of the zeros of the Bessel polynomials

“…We note that Grosswald Using the method of Section 1, we can find a generating function for the Hawkins polynomial. We can get the same result by replacing x by 1/x in formula (2.9) of [8]. We have Throughout the rest of the paper we use the following notation -l), (3.6) ) After multiplying both sides of (3.4) by /"(*), we compare coefficients of which appears to be somewhat simpler than a similar formula in [8].…”

“…Stolarsky [8] proved that <) 1 (m=0,l,2,...), (1.3) where c m (v) is the Hawkins polynomial [4], [8]. Grosswald [3, p. 93] stated that "it would be very desirable to have simple explicit formulae" for a^ for arbitrary m. Stolarsky [8] pointed out that equation (1.3) "is perhaps an answer to this question, but the c m (v) are themselves a bit mysterious".…”

“…(3.2)These polynomials occur in a formula of Hawkins[4] for the residues of z(v, s) at its possible poles in the left half plane. Stolarsky[8] established the asymptotic behaviour, and other properties, of c m (v). The relationship of c m (v) and the Bessel polynomial is given by(1.3).…”

On the non‐homogeneous quadratic Bessel zeta function

“…More precisely, and for completeness, we can state the following results. Notice that the contribution of the non-homogeneity term q~ is shared equally among all the poles; in other words, the homogeneous Bessel zeta function has the same poles, but with (possibly) different residua (see [21]).…”

Vacuum expectation value of twist fields

“…It was thus dubbed the Bessel zeta function in Ref. [21,22], it is also known as the Raleigh function [23], for even integers s = 2, 4, 6, . .…”