q-Analogues of Some Multivariable Biorthogonal Polynomials

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


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.


Weight Function Orthogonal Polynomial Hypergeometric Series Basic Hypergeometric Series Wilson Polynomial 
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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • George Gasper
    • 1
  • Mizan Rahman
    • 2
  1. 1.Department of MathematicsNorthwestern UniversityEvanston
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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