Special functions and the Mellin transforms of Laguerre and Hermite functions

We establish a new connection between moments of n × n random matrices Xn and hypergeometric orthogonal polynomials. Specifically, we consider moments E Tr X −s n as a function of the complex variable s ∈ C, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden et al. [F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Math. Phys. 57 (2016)] on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order n → ∞ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials. Contents 48References 50 1 arXiv:1805.08760v5 [math-ph]

show abstract

“…The following analogue of Proposition 6.1 for the Laguerre wavefunction is essentially due to Bump et al [20] and Coffey [22]. Proposition 6.3.…”

mentioning

confidence: 90%

Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

Cunden

Mezzadri

O’Connell

et al. 2019

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…), δ = 0 or 1, the generalization to Mellin transforms of Laguerre functions has been made [3,7] and now the polynomial factors are a family of other 2 F 1 (2) functions. The Laguerre functions are L α n (x) = x α/2 e −x/2 L α n (x), for α > −1, and their Mellin transform is of the form M α n (s) = 2 s+α/2 Γ(s + α/2)P α n (s).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Mellin transforms with only critical zeros: Legendre functions

Coffey

Lettington

2015

Journal of Number Theory

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…The polynomial factors of the Mellin transforms of Bump and Ng are realized as certain 2 F 1 (2) Gauss hypergeometric functions [10]. In a different setting, the polynomials p n (x) = 2 F 1 (−n, −x; 1; 2) = (−1) n 2 F 1 (−n, x + 1; 1; 2) and q n (x) = i n n!p n (−1/2 − ix/2) were studied [23], and they directly correspond to the Bump and Ng polynomials with s = −x.…”

Section: Introductionmentioning

confidence: 99%

“…The polynomial factors of the Mellin transforms of Bump and Ng are realized as certain 2 F 1 (2) Gauss hypergeometric functions [10]. In a different setting, the polynomials…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Binomial polynomials mimicking Riemann's zeta function

Coffey

Lettington

2020

Integral Transforms and Special Functions

View full text Add to dashboard Cite

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' ℜ s = 1/2 or on the real axis (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2 and S:3/1 binomial coefficient forms. Their 'critical polynomial' factors are then identified as variants of the S:4/1 form, and more compactly in terms of certain 3 F 2 (1) hypergeometric functions. Furthermore, we extend these results to a 1-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation p n (s; β) = ±p n (1 − s; β), similar to that for the Riemann xi function.It is shown that via manipulation of the binomial factors, these 'critical polynomials' can be simplified to an S:3/2 form, which after normalisation yields the rational function q n (s). The denominator of the rational form has singularities on the negative real axis, and so q n (s) has the same 'critical zeros' as the 'critical polynomial' p n (s). Moreover as s → ∞ along the real axis, q n (s) → 1 from below, mimicking 1/ζ(s) on the real axis.In the case of the Chebyshev parameters we deduce the simpler S:2/1 binomial form, and with C n the nth Catalan number, s an integer, we show that polynomials 4C n−1 p 2n (s) and C n p 2n+1 (s) yield integers with only odd prime factors. The results touch on analytic number theory, special function theory, and combinatorics.The authors would like to thank Dr J. L. Hindmarsh and Prof. M N Huxley for their helpful comments and suggestions. 2010 Mathematics Subject Classification: 11B65 (primary), 05A10, 33C20, 33C45, 42C05, 44A20, 30D05 (secondary).

show abstract

Special functions and the Mellin transforms of Laguerre and Hermite functions

Cited by 4 publications

References 19 publications

Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

Mellin transforms with only critical zeros: Legendre functions

Binomial polynomials mimicking Riemann's zeta function

Contact Info

Product

Resources

About