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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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References

  1. Gasper, G. and Rahman, M. (1990). Basic Hypergeometric Series. Cambridge University Press, Cambridge.Google Scholar
  2. Gasper, G. and Rahman, M. (2003). Some systems of multivariable orthogonal Askey-Wilson polynomials. This Proceedings.Google Scholar
  3. Granovskĭ Y. I. and Zhedanov, A. S. (1992). 'Twisted’ Clebsch-Gordan coefficients for suq(2). J. Phys. A, 25:L1029–L1032.CrossRefGoogle Scholar
  4. Gustafson, R. A. (1987). A Whipple's transformation for hypergeometric series in u(n) and multivariable hypergeometric orthogonal polynomials. SIAM J. Math. Anal., 18:495–530.CrossRefzbMATHMathSciNetGoogle Scholar
  5. Koelink, H. T. and Van der Jeugt, J. (1998). Convolutions for orthogonal polynomials from Lie and quantum algebra representations. SIAM J. Math. Anal., 29:794–822.CrossRefMathSciNetGoogle Scholar
  6. Rahman, M. (1981). Discrete orthogonal systems corresponding to Dirichlet distribution. Utilitas Mathematica, 20:261–272.zbMATHMathSciNetGoogle Scholar
  7. Rosengren, H. (2001). Multivariable q-Hahn polynomials as coupling coefficients for quantum algebra representations. Int. J. Math. Sci., 28:331–358.CrossRefzbMATHMathSciNetGoogle Scholar
  8. Tratnik, M. V. (1991a). Some multivariable orthogonal polynomials of the Askey tableau—continuous families. J. Math. Phys., 32:2065–2073.CrossRefzbMATHMathSciNetGoogle Scholar
  9. Tratnik, M. V. (1991b). Some multivariable orthogonal polynomials of the Askey tableau—discrete families. J. Math. Phys., 32:2337–2342.CrossRefzbMATHMathSciNetGoogle Scholar
  10. van Diejen, J. F. and Stokman, J. V. (1998). Multivariable q-Racah polynomials. Duke Math. J., 91:89–136.CrossRefMathSciNetGoogle Scholar
  11. Wilson, J. A. (1980). Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal., 11:690–701.CrossRefzbMATHMathSciNetGoogle Scholar

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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