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

Pascal Baseilhac, Luc Vinet, Alexei Zhedanov

Published 2016-11-28Version 1

Two mutually commuting families of $N-$variable $q-$difference operators that generate the $q-$Onsager algebra are given. The common polynomial eigenfunctions of each family 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.