Polynomials related to the Bessel functions

Abstract:

In this paper we examine the polynomials ${W_n}(a)$ defined by means of \[ - 4{e^{xa}}{[x({e^x} - 1) - 2({e^x} + 1)]^{ - 1}} = \sum \limits _{n = 0}^\infty {{W_n}(a){x^n}/n!} .\] These polynomials are closely related to the zeros of the Bessel function of the first kind of index —3/2, and they are in some ways analogous to the Bernoulli and Euler polynomials. This analogy is discussed, and the real and complex roots of ${W_n}(a)$ are investigated. We show that if $n$ is even then ${W_n}(a) > 0$ for all $a$, and if $n$ is odd then ${W_n}(a)$ has only the one real root $a = 1/2$. Also we find upper and lower bounds for all $b$ such that ${W_n}(a + bi) = 0$. The problem of multiple roots is discussed and we show that if $n \equiv 0,1,5,8$ or 9 $(\bmod \; 12)$, then ${W_n}(a)$ has no multiple roots. Finally, if $n \equiv 0,1,2,5,6$ or 8 $(\bmod \; 12)$, then ${W_n}(a)$ has no factor of the form ${a^2} + ca + d$ where $c$ and $(\bmod \; 12)$ are integers.