# q-Analogues of Some Multivariable Biorthogonal Polynomials

• George Gasper
• Mizan Rahman
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 13)

## Abstract

In 1989, M. V. Tratnik found a pair of multivariable biorthogonal polynomials Pn(x) and $${\bar P}$$m(x), which is not necessarily the complex conjugate of Pm(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,…, xp.

## Keywords

Weight Function Orthogonal Polynomial Hypergeometric Series Basic Hypergeometric Series Wilson Polynomial
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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