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Proceedings of the American Mathematical Society

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Reproducing kernels for $ q$-Jacobi polynomials


Authors: Waleed A. Al-Salam and Mourad E. H. Ismail
Journal: Proc. Amer. Math. Soc. 67 (1977), 105-110
MSC: Primary 33A65
MathSciNet review: 0454104
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Abstract: We derive a family of reproducing kernels for the q-Jacobi polynomials $ \Phi _n^{(\alpha ,\beta )}(x){ = _2}{\Phi _1}({q^{ - n}},{q^{n - 1 + \beta }};{q^\alpha };q,qx)$. This is achieved by proving that the polynomials $ \Phi _n^{(\alpha ,\beta )}(x)$ satisfy a discrete Fredholm integral equation of the second kind with a positive symmetric kernel, then applying Mercer's theorem.


References [Enhancements On Off] (What's this?)

  • [1] Waleed A. Al-Salam and Mourad E. H. Ismail, Polynomials orthogonal with respect to discrete convolution, J. Math. Anal. Appl. 55 (1976), no. 1, 125–139. MR 0410236
  • [2] G. Andrews and R. Askey, The classical and discrete orthogonal polynomials and their q-analogues (in preparation).
  • [3] Wolfgang Hahn, Über Orthogonalpolynome, die 𝑞-Differenzengleichungen genügen, Math. Nachr. 2 (1949), 4–34 (German). MR 0030647
  • [4] Einar Hille, Introduction to general theory of reproducing kernels, Rocky Mountain J. Math. 2 (1972), no. 3, 321–368. MR 0315109
  • [5] Mourad E. H. Ismail, Connection relations and bilinear formulas for the classical orthogonal polynomials, J. Math. Anal. Appl. 57 (1977), no. 3, 487–496. MR 0430361
  • [6] M. Rahman, A five-parameter family of positive kernels from Jacobi polynomials, SIAM J. Math. Anal. 7 (1976), no. 3, 386–413. MR 0407342
  • [7] M. Rahman, Some positive kernels and bilinear sums for Hahn polynomials, SIAM J. Math. Anal. 7 (1976), no. 3, 414–435. MR 0407343
  • [8] Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
  • [9] F. G. Tricomi, Integral equations, Pure and Applied Mathematics. Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0094665

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0454104-5
Keywords: q-Jacobi polynomials, q-Laguerre polynomials, connection relations, bilinear forms, reproducing kernels, discrete integral equations, Mercer's theorem, symmetric kernels
Article copyright: © Copyright 1977 American Mathematical Society