The $q-$Onsager algebra and multivariable $q-$special functions

TL;DR

This paper introduces mutually commuting $q$-difference operators linked to the $q$-Onsager algebra, explores their polynomial eigenfunctions, and expresses their overlaps using entangled products of $q$-special functions, revealing new algebraic and orthogonality properties.

Contribution

It constructs explicit polynomial eigenfunctions and overlap coefficients for operators related to the $q$-Onsager algebra, connecting them with multivariable $q$-special functions and their orthogonality relations.

Findings

Eigenfunctions are entangled products of elementary Pochhammer functions.

Overlap coefficients are expressed as entangled products of $q$-Racah polynomials.

Dual bases form orthogonal systems with block tridiagonal operator actions.

Abstract

Two sets of mutually commuting $q-$difference operators $x_i$ and $y_j$, $i,j=1, ...,N$ such that $x_i$ and $y_i$ generate a homomorphic image of the $q-$Onsager algebra for each $i$ are introduced. The common polynomial eigenfunctions of each set are found to be entangled product of elementary Pochhammer functions in $N$ variables and $N+3$ parameters. Under certain conditions on the parameters, they form two `dual' bases of polynomials in $N$ variables. The action of each operator with respect to its dual basis is block tridiagonal. The overlap coefficients between the two dual bases are expressed as entangled products of $q-$Racah polynomials and satisfy an orthogonality relation. The overlap coefficients between either one of these bases and the multivariable monomial basis are also considered. One obtains in this case entangled products of dual $q-$Krawtchouk polynomials. Finally, the `split' basis in which the two families of operators act as block bidiagonal matrices is also provided.