Generalizations of generating functions for hypergeometric orthogonal polynomials with definite integrals. (English) Zbl 1306.33016

Cohl, Howard S.; MacKenzie, Connor; Volkmer, Hans

J. Math. Anal. Appl. 407, No. 2, 211-225 (2013).

Summary: We generalize generating functions for hypergeometric orthogonal polynomials, namely Jacobi, Gegenbauer, Laguerre, and Wilson polynomials. These generalizations of generating functions are accomplished through series rearrangement using connection relations with one free parameter for these orthogonal polynomials. We also use orthogonality relations to determine corresponding definite integrals.

MSC:

33C45

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

orthogonal polynomials; generating functions; connection coefficients; generalized hypergeometric functions; eigenfunction expansions; definite integrals

Software:

DLMF

References:

[1]	Abramowitz, M.; Stegun, I. A., (Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55 (1972), US Government Printing Office: US Government Printing Office Washington, DC) · Zbl 0543.33001
[2]	Andrews, G. E.; Askey, R.; Roy, R., (Special Functions. Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71 (1999), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0920.33001
[3]	Askey, R., Orthogonal Polynomials and Special Functions (1975), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, Pa · Zbl 0298.26010
[5]	Cohl, H. S., On a generalization of the generating function for Gegenbauer polynomials, Integral Transforms Spec. Funct. (2013) · Zbl 1280.35003
[7]	Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions. Vol. I (1981), Robert E. Krieger Publishing Co. Inc.: Robert E. Krieger Publishing Co. Inc. Melbourne, Fla · Zbl 0051.30303
[8]	Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (2007), Elsevier, Academic Press: Elsevier, Academic Press Amsterdam · Zbl 1208.65001
[9]	Ismail, M. E.H., (Classical and Quantum Orthogonal Polynomials in One Variable. Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications, vol. 98 (2005), Cambridge University Press: Cambridge University Press Cambridge), With Two Chapters by Walter Van Assche, With a Foreword by Richard A. Askey · Zbl 1082.42016
[10]	Koekoek, R.; Lesky, P. A.; Swarttouw, R. F., (Hypergeometric Orthogonal Polynomials and their \(q\)-Analogues. Hypergeometric Orthogonal Polynomials and their \(q\)-Analogues, Springer Monographs in Mathematics (2010), Springer-Verlag: Springer-Verlag Berlin), With a Foreword by Tom H. Koornwinder · Zbl 1200.33012
[11]	Magnus, W.; Oberhettinger, F.; Soni, R. P., (Formulas and Theorems for the Special Functions of Mathematical Physics. Formulas and Theorems for the Special Functions of Mathematical Physics, Die Grundlehren der mathematischen Wissenschaften, vol. Band 52 (1966), Springer-Verlag New York, Inc.: Springer-Verlag New York, Inc. New York) · Zbl 0143.08502
[12]	(Olver, F. W.J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1198.00002
[14]	Srivastava, H. M.; Manocha, H. L., (A Treatise on Generating Functions. A Treatise on Generating Functions, Ellis Horwood Series: Mathematics and its Applications (1984), Ellis Horwood Ltd.: Ellis Horwood Ltd. Chichester) · Zbl 0535.33001
[15]	Szegő, G., (Orthogonal Polynomials. Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23 (1959), American Mathematical Society: American Mathematical Society Providence, RI) · JFM 61.0386.03
[16]	Wilson, J. A., Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal., 11, 4, 690-701 (1980) · Zbl 0454.33007
[17]	Wilson, J. A., Asymptotics for the \({}_4 F_3\) polynomials, J. Approx. Theory, 66, 1, 58-71 (1991) · Zbl 0729.33006