Reproducing kernels for Jacobi polynomials
Authors:
Waleed A. AlSalam and Mourad E. H. Ismail
Journal:
Proc. Amer. Math. Soc. 67 (1977), 105110
MSC:
Primary 33A65
MathSciNet review:
0454104
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Abstract: We derive a family of reproducing kernels for the qJacobi polynomials . This is achieved by proving that the polynomials satisfy a discrete Fredholm integral equation of the second kind with a positive symmetric kernel, then applying Mercer's theorem.
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 [1]
 W. AlSalam and M. E. H. Ismail, Polynomials orthogonal with respect to a discrete convolution, J. Math. Anal. Appl. 55 (1976), 125139. MR 0410236 (53:13986)
 [2]
 G. Andrews and R. Askey, The classical and discrete orthogonal polynomials and their qanalogues (in preparation).
 [3]
 W. Hahn, Über Orthogonalpolynome, die qdifferenzengleichungen, Math. Nachr. 2 (1949), 434. MR 0030647 (11:29b)
 [4]
 E. Hille, Introduction to general theory of reproducing kernels, Rocky Mountain J. Math. 2 (1972), 321368. MR 0315109 (47:3658)
 [5]
 M. E. H. Ismail, Connection relations and bilinear formulas for the classical orthogonal polynomials, J. Math. Anal. Appl. 57 (1977), 487496. MR 0430361 (55:3366)
 [6]
 M. Rahman, A five parameter family of positive kernels from Jacobi polynomials, SIAM J. Math. Anal. 7 (1976), 386413. MR 0407342 (53:11118)
 [7]
 , Some positive kernels and bilinear sums for Hahn polynomials, SIAM J. Math. Anal. 7 (1976), 414435. MR 0407343 (53:11119)
 [8]
 L. J. Slater, Generalized hypergeometric function, Cambridge Univ. Press, Cambridge, 1966. MR 0201688 (34:1570)
 [9]
 F. G. Tricomi, Integral equations, Interscience, New York, 1957. MR 0094665 (20:1177)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197704541045
PII:
S 00029939(1977)04541045
Keywords:
qJacobi polynomials,
qLaguerre polynomials,
connection relations,
bilinear forms,
reproducing kernels,
discrete integral equations,
Mercer's theorem,
symmetric kernels
Article copyright:
© Copyright 1977
American Mathematical Society
