q-Analogues of Some Multivariable Biorthogonal Polynomials

Abstract

In 1989, M. V. Tratnik found a pair of multivariable biorthogonal polynomials P_n(x) and ${\bar P}$_m(x), which is not necessarily the complex conjugate of P_m(x), such that

$$\int\limits_\infty ^\infty { \cdot \cdot \cdot } \int\limits_\infty ^\infty {w(x)P_n (x)\bar P_m (x)\mathop \Pi \limits_{j - 1}^p } dx_j = \mu _{n,m} \delta _{N,M} ,$$

where $x = (x_1 , \ldots ,x_p ),\;n = (n_1 , \ldots ,n_p ),\;m = (m_1 , \ldots ,m_p ),\;N = \mathop \sum \limits_{j = 1}^p \;n_j ,\;M = \mathop \sum \limits_{j = 1}^p \;m_j ,\;\mu _{n,m}x = (x_1 , \ldots ,x_p ),\;n = (n_1 , \ldots ,n_p ),\;m = (m_1 , \ldots ,m_p ),\;N = \mathop \sum \limits_{j = 1}^p \;n_j ,\;M = \mathop \sum \limits_{j = 1}^p \;m_j ,\;\mu _{n,m}$ is the constant of biorthogonality (which Tratnik did not evaluate),

$$w(x) = \Gamma (A - iX)\Gamma (B + iX)\left| {\frac{{\Gamma (c + iX)\Gamma (d + iX)}} {{\Gamma (2iX)}}} \right|^2 \mathop \Pi \limits_{k = 1}^p \Gamma (a_k + ix_k )\Gamma (b_k - ix_k ),$$

$$X = \mathop \sum \limits_{k = 1}^p x_k ,\quad A = \mathop \sum \limits_{k = 1}^p a_k ,\quad B = \mathop \sum \limits_{k = 1}^p b_k ,$$

and the a's, b's, x's, c and d are real. In the q-case we find that the appropriate weight function is a product of a multivariable version of the integrand in the Askey-Roy integral and of the Askey-Wilson weight function in a single variable that depends on x₁,…, x_p.

Supported in part by the NSERC grant #A6197.