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We investigate the connection problem for the Jackson integral of type $A_n$. Our connection formula implies a Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific multiple series. Introducing certain elliptic Lagrange interpolation functions, we determine the explicit form of the connection coefficients. We also use basic properties of the interpolation functions to establish an explicit determinant formula for a fundamental solution matrix of the associated system of $q$-difference equations.

This is the second part of the project "unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of Hermitian matrices." In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional Hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big q-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended '2 Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schrodinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of the Jackson integral constitutes the eigenvector of each of the two-unbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the q-Meixner (q-Charlier) polynomials do not form a complete basis of the '2 Hilbert space, based on the fact that the dual q-Meixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the q-Meixner polynomials is obtained by constructing the duals of the dual q-Meixner polynomials which require the two-component Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big q-Jacobi family and their duals and q-Meixner (q-Charlier) polynomials. Published by AIP Publishing., Article, JOURNAL OF MATHEMATICAL PHYSICS. 59(1):013504 (2018)

We present three types of summation formulas for the root system \(G_{2}^{\vee}\), which are generalized from Bailey’s summation formula for a very-well-poised balanced 6ψ6 basic hypergeometric series.

We derive a $$q$$ -generalization of Euler’s decomposition formula for the $$q$$ MZVs recently introduced by Y. Ohno, J. Okuda, and W. Zudilin. This answers a question posed by these authors in J Number Theory 132, 144–155 (2012).

In this paper we study the q-analogue of the jα Bessel function (see (1)) which results after minor changes from the so-called Exton function studied by Koornwinder and Swarttow. Our objective is first to establish, using only the q -Jackson integral and the q-derivative, some properties of this function with proofs similar to the classical case; second to construct the associated q-Fourier analysis which will be used in a coming work to construct the q-analogue of the Bessel-hypergroup.

A multiple summation formula for the root system F4 originated from Bailey's very-well-poised 6ψ6 summation theorem is studied. The multiple sum in this paper is included in a classification list (M. Ito, Compositio Math., 129 (2001), 325–340) of Jackson integral associated with irreducible reduced root system which contains sums investigated by Gustafson, Macdonald, Aomoto, and the author.

The Ramanujan $_{1} \psi_{1}$ summation theorem is studied from the perspective of Jackson integrals, $q$-difference equations and connection formulae. This is an approach which has previously been shown to yield Bailey’s very-well-poised $_{6} \psi_{6}$ summation. Bilateral Jackson integral generalizations of the Dixon–Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_{1} \psi_{1}$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulae for Jackson integrals relating to other roots systems obtained from the same perspective.

We state certain product formulae for Jackson integrals associated with irreducible reduced root systems. The Jackson integral is defined here as a sum over any full-rank sublattice of the coweight lattice for the root system. In particular, a Weyl group symmetry classification of the Jackson integrals is done when they have an expression of a product of the Jacobi elliptic theta functions. Most of the product formulae investigated by Aomoto, Macdonald and Gustafson appear in the list of classifications. A new product formula for an F4 root system is included in it.

A study is made of functions on the lattice generated by the integer powers of , . A -analogue of the Fourier transformation is constructed based on the Jackson integral in the space of generalized functions on the lattice.

We review the construction of a q-analogue of the Gaussian measure. We apply that construction to obtain a q-analogue of Feynman integrals and to compute explicitly an example of such integrals., Comment: 5 figures, 10 pages

A. Selberg evaluated an important multivariable extension of the beta function integral. Andrews found a related integral and evaluated it using a result of Dyson, Gunson and Wilson. Basic hypergeometric, or q-series, extensions of these integrals are considered and evaluated in the two-dimensional case. Conjectures are given for the values of these integrals in the n-dimensional case.