Coffey, Mark W. and Lettington, Matthew C.  ORCID: https://orcid.org/0000-0001-9327-143X
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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Abstract
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: | Schools > Mathematics |
| Publisher: | Taylor & Francis |
| ISSN: | 1065-2469 |
| Date of First Compliant Deposit: | 15 April 2020 |
| Date of Acceptance: | 10 April 2020 |
| Last Modified: | 13 Jun 2025 15:30 |
| URI: | https://orca.cardiff.ac.uk/id/eprint/131014 |
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