Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Mourad E. H. Ismail ; Erik Koelink ; Pablo Román

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

Burchnall's method to invert the Feldheim–Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey–Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner–Pollaczek and big $q$-Jacobi polynomials and big $q$-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner–Pollaczek and Krawtchouk polynomials.

Détail

Keywords: orthogonal polynomials; Askey scheme and its $q$-analogue; expansion formulas; Toda lattice.

Bibliographie

[1] Aharmim B., Amal E. H., Fouzia E. W., Ghanmi A., “Generalized Zernike polynomials: operational formulae and generating functions”, Integral Transforms Spec. Funct., 26 (2015), 395–410, arXiv: 1312.3628 | DOI | MR | Zbl

[2] Al-Salam W. A., “Operational representations for the Laguerre and other polynomials”, Duke Math. J., 31 (1964), 127–142 | DOI | MR | Zbl

[3] Andrews G. E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999 | DOI | Zbl

[4] Askey R., Orthogonal polynomials and special functions, Reg. Conf. Ser. Appl. Math., 21, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975

[5] Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003 | DOI | Zbl

[6] Basor E., Chen Y., Ehrhardt T., “Painlevé V and time-dependent Jacobi polynomials”, J. Phys. A: Math. Theor., 43 (2010), 015204, 25 pp., arXiv: 0905.2620 | DOI | MR | Zbl

[7] Burchnall J. L., “A note on the polynomials of Hermite”, Quart. J. Math., Oxford Ser., 12 (1941), 9–11 | DOI | MR

[8] Carlitz L., “A note on the Laguerre polynomials”, Michigan Math. J., 7 (1960), 219–223 | DOI | MR | Zbl

[9] Casas F., Murua A., Nadinic M., “Efficient computation of the Zassenhaus formula”, Comput. Phys. Commun., 183 (2012), 2386–2391, arXiv: 1204.0389 | DOI | MR | Zbl

[10] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | Zbl

[11] Gekhtman M., “Hamiltonian structure of non-abelian Toda lattice”, Lett. Math. Phys., 46 (1998), 189–205 | DOI | MR | Zbl

[12] Gould H. W., Hopper A. T., “Operational formulas connected with two generalizations of Hermite polynomials”, Duke Math. J., 29 (1962), 51–63 | DOI | MR | Zbl

[13] Ismail M. E. H., “The Askey–Wilson operator and summation theorems”, Mathematical Analysis, Wavelets, and Signal Processing (Cairo, 1994), Contemp. Math., 190, eds. M. E. H. Ismail, M. Z. Nashed, A. I. Zayed, A. F. Ghaleb, Amer. Math. Soc., Providence, RI, 1995, 171–178 | DOI | Zbl

[14] Ismail M. E. H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, 98, Cambridge University Press, Cambridge, 2009 | Zbl

[15] Ismail M. E. H., Koelink E., Román P., in preparation

[16] Kametaka Y., “On the Euler–Poisson–Darboux equation and the Toda equation. I”, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 145–148 | DOI | MR | Zbl

[17] Kametaka Y., “On the Euler–Poisson–Darboux equation and the Toda equation. II”, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 181–184 | DOI | MR | Zbl

[18] Koekoek R., Lesky P. A., Swarttouw R. F., “Hypergeometric orthogonal polynomials and their $q$-analogues”, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010 | DOI | Zbl

[19] Koekoek R., Swarttouw R. F., The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998 http://aw.twi.tudelft.nl/k̃oekoek/askey/

[20] Koelink H. T., Van Der Jeugt J., “Convolutions for orthogonal polynomials from Lie and quantum algebra representations”, SIAM J. Math. Anal., 29 (1998), 794–822, arXiv: q-alg/9607010 | DOI | MR | Zbl

[21] Koornwinder T. H., “Compact quantum groups and $q$-special functions”, Representations of Lie Groups and Quantum Groups (Trento, 1993), Pitman Res. Notes Math. Ser., 311, Longman Sci. Tech., Harlow, 1994, 46–128 | Zbl

[22] Magnus W., “On the exponential solution of differential equations for a linear operator”, Comm. Pure Appl. Math., 7 (1954), 649–673 | DOI | MR | Zbl

[23] Nielsen N., “Recherches sur les polynomes d'Hermite”, Det Kgl. Danske Videnskabernes Selskab. Math.-Fys. Meddelelser I, 6 (1918), 1–78

[24] Rainville E. D., Special functions, The Macmillan Co., New York, 1960 | Zbl

[25] Singh R. P., “Operational formulae for Jacobi and other polynomials”, Rend. Sem. Mat. Univ. Padova, 35 (1965), 237–244 | MR | Zbl

[26] Zhedanov A., “The Toda chain: solutions with dynamical symmetry and classical orthogonal polynomials”, Theoret. and Math. Phys., 82 (1990), 6–11 | DOI | MR

[27] Zhedanov A., “Elliptic solutions of the Toda chain and a generalization of the Stieltjes–Carlitz polynomials”, Ramanujan J., 33 (2014), 157–195, arXiv: 0712.0058 | DOI | MR | Zbl