A structure of the Rayleigh polynomial

Abstract

The Rayleigh polynomial 2(v) has been defined [2], [3] in the following manher" Let J.(z) be the Bessel function of the first kind, and let j., m= 1, 2,..., be the zeros of z-J(z), IRe ( )l -< IRe (J,.+l)l, then 0)) 2 (,, " 2 ,.) n=l, and [x] is the greatest integer _< x. The symmetric function z,.(v) is called [1] the Rayleigh function of order 2n, and has been the subject of a number of investigations by Cayley, Watson, Forsyth and others [4; 502]. It is obvious from (1) that any structure of =() is closely related with that of 6(). However, no simple structure of the Rayleigh polynomial ,(v) is known so far. It has been shown [2] that 2.(v) is a polynomial with positive integral coefficients, that its degree is 1 2n

Determinantal Expressions, Identities, Concavity, Maclaurin Power Series Expansions for van der Pol Numbers, Bernoulli Numbers, and Cotangent

“…This is also called the Bessel zeta function in [3] and was originally introduced and studied in papers [4][5][6][7]. In paper [8], the authors derived closed-form expressions of the Bessel zeta function ζ ν (2k) for k ≥ 1 in terms of specific Hessenberg determinants.…”

On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian

“…Both Euler and Rayleigh analyzed eigenvalues of oscillations of physical systems (a hanging chain for Euler and a circular membrane for Rayleigh), which aroused their interest in computing zeros of the Bessel functions. By exploiting a differential equation of Riccati-type satisfied by the function z −ν J ν (z), Kishore [29,30,31] developed recursion formulas for σ 2 (ν), starting with the known expression, due to Euler and Rayleigh…”

Sums of powers of the zeros of the Bessel polynomials

“…where z(v, s) is the Bessel zeta function (also called the Rayleigh function [5], [6]). This enables us to give an explicit formula for cr^' (m = l,..., 2n) in terms of the Rayleigh polynomial, a polynomial that has been examined in [MATHEMATIKA, 37 (1990), [305][306][307][308][309][310][311][312][313][314][315] some detail [6], [7]. In fact we prove a more general formula (…”

“…Now if/(x) is any polynomial of degree n with distinct zeros r x ,r 2 ,... ,r n , with each r, 5* 0, it is easy to see that of a number of papers by Kishore [5], [6], [7] and many others. Kishore [5] proved that for m > 1 By properties of the Rayleigh function [5], [6], we have the following corollary.…”