Mellin transforms with only critical zeros: generalized Hermite functions

2013/08/30 by Mark W. Coffey, Coffey, Mark W.
Mathematics · #30D05 #33C20 #33C45 #42C05 #44A20 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #math.CA #math.CV #msc:30D05 #msc:33C20 #msc:33C45 #msc:42C05 #msc:44A20

paper · pdf · doi:10.48550/arxiv.1308.6821

17 pages, no figures

arxiv created 2013/08/30 · arxiv updated 2013/09/02

Abstract

We consider the Mellin transforms of certain generalized Hermite functions based upon certain generalized Hermite polynomials, characterized by a parameter μ>-1/2. We show that the transforms have polynomial factors whose zeros lie all on the critical line. The polynomials with zeros only on the critical line are identified in terms of certain ₂F₁(2) hypergeometric functions, being certain scaled and shifted Meixner-Pollaczek polynomials. Other results of special function theory are presented.

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