[Paper Review] Mellin transforms with only critical zeros: generalized Hermite functions

Mark W. Coffey|arXiv (Cornell University)|Aug 30, 2013

Mathematical functions and polynomials11 references3 citations

TL;DR

This paper investigates Mellin transforms of generalized Hermite functions parameterized by μ > -1/2, showing that the resulting polynomial factors are proportional to scaled and shifted Meixner-Pollaczek polynomials. These polynomials have all zeros on the critical line Re(s) = 1/2, establishing a direct link between special function theory and the Riemann hypothesis via hypergeometric functions and functional equations.

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 $_2F_1(2)$ hypergeometric functions, being certain scaled and shifted Meixner-Pollaczek polynomials. Other results of special function theory are presented.

Motivation & Objective

• To analyze Mellin transforms of generalized Hermite functions defined via parameter μ > -1/2.
• To identify the polynomial factors arising in these transforms and determine their zero distribution.
• To establish a connection between these polynomials and known orthogonal polynomials with zeros on the critical line.
• To explore functional equations and symmetry properties of the resulting polynomials using hypergeometric identities.
• To provide a framework for constructing polynomials with only critical zeros, relevant to analytic number theory and the Riemann hypothesis.

Proposed method

• Derives Mellin transforms of generalized Hermite functions using their representation in terms of Laguerre polynomials and confluent hypergeometric functions.
• Expresses the polynomial factors as terminating hypergeometric functions $_2F_1(2)$, specifically $_2F_1(-n, (μ+s)/2; μ+1/2; 2)$.
• Identifies the polynomial factors as scaled and shifted Meixner-Pollaczek polynomials via hypergeometric transformation identities.
• Applies the Plancherel relation for Mellin transforms to show that the weight function $|\Gamma((\mu+s+\varepsilon)/2)|^2$ corresponds to the orthogonality measure.
• Uses known orthogonality and zero-separation properties of Meixner-Pollaczek polynomials to infer that zeros lie on the critical line and are simple.
• Establishes functional equations such as $P_n^\alpha(s) = (-1)^n P_n^\alpha(1-s)$, linking symmetry to critical line zeros.

Experimental results

Research questions

• RQ1 Do the Mellin transforms of generalized Hermite functions yield polynomial factors with zeros only on the critical line Re(s) = 1/2?
• RQ2 Can the resulting polynomial factors be expressed in terms of classical orthogonal polynomials with known zero distributions?
• RQ3 What is the functional equation structure of the Mellin transform polynomials, and how does it relate to symmetry on the critical line?
• RQ4 How do the hypergeometric representations of the polynomial factors connect to Meixner-Pollaczek polynomials?
• RQ5 Are the zeros of the polynomial factors simple, and do they interlace as expected from orthogonal polynomial theory?

Key findings

• The Mellin transforms of generalized Hermite functions produce polynomial factors that are proportional to Meixner-Pollaczek polynomials $P_n^{(\mu+\varepsilon)/2+1/4}\left[i(1/4 - s/2); \pi/2\right]$.
• All zeros of the polynomial factors lie on the critical line Re(s) = 1/2, as confirmed by orthogonality and known properties of Meixner-Pollaczek polynomials.
• The polynomial factors satisfy the functional equation $P_n^\alpha(s) = (-1)^n P_n^\alpha(1-s)$, indicating symmetry about the critical line.
• The zeros of consecutive polynomial factors $p_n^\mu(1/2+it)$ and $p_{n+1}^\mu(1/2+it)$ interlace on the critical line, a hallmark of orthogonal polynomials.
• The polynomial factors are expressed as $_2F_1(2)$ hypergeometric functions, and their transformation identities confirm the critical line location of zeros.
• The Plancherel relation for Mellin transforms confirms that the weight $|\Gamma((\mu+s+\varepsilon)/2)|^2$ corresponds to the orthogonality measure, validating the zero distribution.