Mellin transforms with only critical zeros: Legendre functions

Coffey, Mark W. and Lettington, Matthew C. ORCID: https://orcid.org/0000-0001-9327-143X 2015. Mellin transforms with only critical zeros: Legendre functions. Journal of Number Theory 148 , pp. 507-536. 10.1016/j.jnt.2014.07.021

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Abstract

We consider the Mellin transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line View the MathML sourceRes=1/2. The polynomials with zeros only on the critical line are identified in terms of certain View the MathML sourceF23(1) hypergeometric functions. These polynomials possess the functional equation pn(s)=(−1)⌊n/2⌋pn(1−s)pn(s)=(−1)⌊n/2⌋pn(1−s). Other hypergeometric representations are presented, as well as certain Mellin transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	Mellin transformation, Legendre polynomial, associated Legendre polynomial, hypergeometric.
Additional Information:	Pdf uploaded in accordance with the publisher’s policy at http://www.sherpa.ac.uk/romeo/issn/0022-314X/(accessed 04/11/2014)
Publisher:	Elsevier
ISSN:	0022-314X
Date of First Compliant Deposit:	30 March 2016
Date of Acceptance:	14 July 2014
Last Modified:	18 Nov 2024 05:30
URI:	https://orca.cardiff.ac.uk/id/eprint/66831

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