Related papers: Mellin transforms with only critical zeros: Legend…

We consider the Mellin transforms of certain generalized Hermite functions based upon certain generalized Hermite polynomials, characterized by a parameter $\mu>-1/2$. We show that the transforms have polynomial factors whose zeros lie all…

We consider the Mellin transforms of certain Chebyshev functions based upon the Chebyshev polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line or on the real line. The polynomials with…

We present explicit expressions for the Mellin transforms of Laguerre and Hermite functions in terms of a variety of special functions. We show that many of the properties of the resulting functions, including functional equations and…

The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real…

In this paper a special class of local zeta functions is studied. The main theorem states that the functions have all zeros on the line Re (s)=1/2. This is a natural generalization of the result of Bump and Ng stating that the zeros of the…

We prove that all the zeros of certain meromorphic functions are on the critical line $\text{Re}(s)=1/2$, and are simple (except possibly when $s=1/2$). We prove this by relating the zeros to the discrete spectrum of an unbounded…

We use expansions with functions related to some special functions such as Hermite or Laguerre to get some conjectural expansions of the Riemann Zeta function in the critical strip involving a set of polynomials which have their zeros on…

We extend a result of Bump et al. to show that a large family of Sheffer sequences has their zeros - up to perhaps a finite number of exceptions - on a vertical line. We connect a particular such sequence to the Riemann zeta function via a…

We say that a function f defined on R or Qp has a well defined weak Mellin transform (or weak zeta integral) if there exists some function $M\_f(s)$ so that we have $Mell(\phi \star f,s) = Mell(\phi,s)M\_f(s)$ for all test functions $\phi$…

This paper deals with Mellin transforms of rational functions $g/f$ in several variables. We prove that the polar set of such a Mellin transform consists of finitely many families of parallel hyperplanes, with all planes in each such family…

Associated Legendre functions of fractional degree appear in the solution of boundary value problems in wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer,…

In this work we verify the sufficiency of a Jensen's necessary and sufficient condition for a class of genus 0 or 1 entire functions to have only real zeros. They are Fourier transforms of even, positive, indefinitely differentiable, and…

We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually…

A Mellin transform representation is derived for the associated Legendre functions of the first kind on the cut

In this work we establish some polynomials and entire functions have only real zeros. These polynomials generalize q-Laguerre polynomials $L_{n}^{(\alpha)}(x;q)$, while the entire functions are generalizations of Ramanujan's entire function…

We present a summation rule using the Mellin transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the…

A class of complex Fourier Transforms of exponential functions which have all their zeros on the real line is explored from a geometric perspective. These transforms belong to the Laguerre - Polya class, and it is proved that all the zeros…

We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0<\Re(s)<1.$ Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.

A systematic study is performed on the finite harmonic sums up to level four. These sums form the general basis for the Mellin transforms of all individual functions $f_i(x)$ of the momentum fraction $x$ emerging in the quantities of…

Every real Bank-Laine function of finite order, whose zeros are all real but neither bounded above nor bounded below, either has an explicit representation in terms of trigonometric functions or has zeros with exponent of convergence at…