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

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.

21 pages. v4: typos corrected in eq. (3.24), Example 5.2, Appendix B compared with published version

References in corpus (11)

• Quantum symmetric Kac-Moody pairs
• Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory
• The half-infinite XXZ chain in Onsager's approach
• An integrable structure related with tridiagonal algebras
• Superintegrability in two dimensions and the Racah-Wilson algebra
• Bispectrality of multivariable Racah-Wilson polynomials
• A deformed analogue of Onsager's symmetry in the XXZ open spin chain
• A new current algebra and the reflection equation
• Bispectral commuting difference operators for multivariable Askey-Wilson polynomials
• Algebraization of difference eigenvalue equations related to $U_q(s{l}_2)$
• A family of tridiagonal pairs and related symmetric functions

Cited by in corpus (5)

• Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
• Degenerate Sklyanin algebras, Askey-Wilson polynomials and Heun operators
• The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials
• Gaudin model for the multinomial distribution
• Universal TT- and TQ-relations via centrally extended q-Onsager algebra