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Binomial polynomials mimicking Riemann's zeta function

Coffey, Mark W. and Lettington, Matthew C. 2020. Binomial polynomials mimicking Riemann's zeta function. Integral Transforms and Special Functions 31 (11) , pp. 856-872. 10.1080/10652469.2020.1755672
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The (generalised) Mellin transforms of Gegenbauer polynomials, have polynomial factors pλ n(s), whose zeros all lie on the ‘critical line’ ℜs = 1/2 (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould’s S:4/3, S:4/2 and S:3/1 binomial coefficient forms. Their ‘critical polynomial’ factors are then identified in terms of 3F2(1) hypergeometric functions. Furthermore, we extend these results to a one-parameter family of critical polynomials that possess the functional equation pn(s;β) = ± pn (1 − s;β). Normalisation yields the rational function qλ n(s) whose denominator has singularities on the negative real axis. Moreover as s → ∞ along the positive real axis, qλ n(s) → 1 from below. For the Chebyshev polynomials we obtain the simpler S:2/1 binomial form, and with Cn the nth Catalan number, we deduce that 4Cn−1p2n(s) and Cnp2n+1(s) yield odd integers. The results touch on analytic number theory, special function theory, and combinatorics.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Publisher: Taylor & Francis
ISSN: 1065-2469
Date of First Compliant Deposit: 15 April 2020
Date of Acceptance: 10 April 2020
Last Modified: 27 Nov 2020 21:31

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