The Rayleigh polynomial

Kishore, Nand

doi:10.1090/s0002-9939-1964-0168823-2

Cited by 26 publications

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“…In a previous paper [4] a structure of Φ n (v) has been given. This in turn leads, through (2), to a corresponding structure of σ n (v). And since for v = 1/2, σ n (y) is expressible in terms of the Bernoulli number B n it is natural to enquire about a structure of B n corresponding to that of σ n (v).…”

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confidence: 93%

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A representation of the Bernoulli numberB_n

Kishore¹

1964

Pacific J. Math.

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Section: (5) (!) = ("I) Lmentioning

confidence: 93%

“…The function σ n {v) and the polynomial Φ n {v) have been defined in [2] and [3] respectively. Let J y (z) be the Bessel function of the first kind, and j v , m be the zeros of z~*J v (z), then (1) <?M = E (Jv, m )-2n , n = 1, 2, 3, , where [#] is the greatest integer ^x.…”

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confidence: 99%

A representation of the Bernoulli numberB_n

Kishore¹

1964

Pacific J. Math.

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“…Howeer, it is our conjecture that the polynomial () itself does not hve ny negative iategml zeros [2].…”

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confidence: 83%

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) 2 (,, " 2 ,.)

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confidence: 99%

A structure of the Rayleigh polynomial

Kishore¹

1964

Duke Math. J.

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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

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“…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 (…”

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confidence: 99%