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 2 F 1 (2) hypergeometric functions, being certain scaled and shifted Meixner-Pollaczek polynomials. Other results of special function theory are presented.
It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for square matrices give rise to similar superalgebra structures, pointing to novel symmetry types in their complementary parts. In particular, this provides a unifying framework for the composite 'most perfect square' symmetry and the related class of 'reversible squares'; moreover, the semi-magic square algebra is identified as part of a 2-gradation of the general square matrix algebra. We derive explicit representation formulae for matrices of all symmetry types considered, which can be used to construct all such matrices.
We study the interplay between recurrences for zeta related functions at integer values, 'Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudocharacteristic polynomials for zeta related functions. We begin with a recent result for ζ(2s) and some seemingly new Bernoulli relations, which we use to obtain a generalised Ramanujan polynomial and properties thereof.I would like to thank Professor M N Huxley for all his invaluable support and guidance in this problem and Professor M W Coffey for his perceptive appraisal.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.