Mellin transforms for multiple Jacobi–Piñeiro polynomials and a q-analogue

Abstract

This work treats the Mellin transform of multiple Jacobi-Piñeiro polynomials. This allows us to put a number of irrationality and Q-linear independence proofs into the framework of Hermite-Padé approximation. A similar approach is presented for the q-analogue: the q-Mellin transform of multiple little q-Jacobi polynomials and its applications in irrationality proofs.

Multiple orthogonal polynomials associated with the exponential integral

“…There one uses a Mellin convolution of a (multiple) Jacobi-Piñeiro polynomial and a beta density, see Ref. [33]. Instead of a Jacobi-Piñeiro polynomial, we use a multiple Laguerre polynomial multiplied with a gamma density.…”

“…The type I and type II multiple orthogonal polynomials that were investigated here are related to the two‐parameter subfamily of the Jacobi–Piñeiro polynomials in, for example, Ref. [33] and to the multiple orthogonal polynomials associated with K ‐Bessel functions in Ref. [29], with confluent hypergeometric functions in Ref.…”

INTEGRAL REPRESENTATIONS FOR THE JACOBI–PINEIRO POLYNOMIALS AND THE FUNCTIONS OF THE SECOND KIND

“…E. Mukhin and A. Varchenko [7] show that the zeros of Q n along with the zeros of their Wronskian-type determinants are the unique solution of a certain Bethe Ansatz equation. C. Smet and W. Van Assche [12] show that their Mellin transforms can be applied to prove the Apéri theorem on the irrationality of ζ(3) and the Ball -Rivoal theorem on the infinite number of irrational points among {ζ(2m + 1)}. M. Adler, P. van Moerbeke and D. Wang [1] relate the Jacobi -Piñeiro polynomials to certain problems of random matrix minor processes in the percolation theory.…”