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Area, I.; Godoy, E.; Ronveaux, A.; Zarzo, A.

Solving connection and linearization problems within the Askey scheme and its \(q\)-analogue via inversion formulas. (English) Zbl 0988.33008

J. Comput. Appl. Math. 133, No. 1-2, 151-162 (2001).
Given a sequence of terminating basic hypergeometric series of the form \[ F_k(x) = {}_r\phi_s\left(\left. {q^{-k},a_2,\ldots, a_r \atop b_1,\ldots, b_s} \right|q; qx \right) \;\text{or} _r\phi_s\left(\left. {q^{-k},a_2q^k,a_3,\ldots, a_r \atop b_1,\ldots, b_s} \right|q; qx \right), \] the authors use an identity of Verma to expand \(x^n\) in terms of these polynomials. They then show how this result can be used to obtain connection coefficients and linearization coefficients for any of the orthogonal polynomials or their \(q\)-analogs within the Askey scheme.
Reviewer: David M.Bressoud (Saint Paul)

Cited in 31 Documents

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

linearization coefficients; connection coefficients
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References:

[1] Abramowitz, M.; Stegun, I., Handbook of Mathematical Functions (1965), Dover: Dover New York
[2] Álvarez-Nodarse, R.; Marcellán, F., The modification of classical Hahn polynomials of a discrete variable, Integral Transforms Special Funct., 4, 3, 243-262 (1995) · Zbl 0849.33007
[3] Álvarez-Nodarse, R.; Ronveaux, A., Recurrence relation for connection coefficients between \(q\)-orthogonal polynomials of discrete variables in the non-uniform lattice \(x(s)=q^{2s}\), J. Phys. A: Math. Gen., 29, 22, 7165-7175 (1996) · Zbl 0932.33026
[4] Álvarez-Nodarse, R.; Yáñez, R. J.; Dehesa, J. S., Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials, J. Comput. Appl. Math., 89, 171-197 (1998) · Zbl 0909.33006
[5] Andrews, G. E.; Askey, R., Classical orthogonal polynomials, (Brezinski, C.; etal., Polynômes Orthogonaux et Applications. Polynômes Orthogonaux et Applications, Lecture Notes in Mathematics, vol. 1171 (1985), Springer: Springer Berlin), 36-62 · Zbl 0596.33016
[6] 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
[7] Area, I.; Godoy, E.; Ronveaux, A.; Zarzo, A., Inversion problems in the \(q\)-Hahn tableau, J. Symbolic Comput., 28, 6, 767-776 (1999) · Zbl 0953.33005
[8] Artés, P. L.; Dehesa, J. S.; Martı́nez-Finkelshtein, A.; Sánchez-Ruiz, J., Linearization and connection coefficients for hypergeometric-type polynomials, J. Comput. Appl. Math., 99, 15-26 (1998) · Zbl 0927.33005
[9] Askey, R., Orthogonal Polynomials and Special Functions. Orthogonal Polynomials and Special Functions, CBMS-NSF Reg. Conferences Series in Applied Mathematics, vol. 21 (1975), SIAM: SIAM Philadelphia, Pennsylvania · Zbl 0298.26010
[10] Askey, R.; Wilson, J. A., A set of orthogonal polynomials that generalize the Racah coefficients or \(6-j\) symbols, SIAM J. Math. Anal., 10, 1008-1016 (1979) · Zbl 0437.33014
[11] Askey, R.; Wilson, J. A., Some Basic Hypergeometric Polynomials that Generalize Jacobi Polynomials. Some Basic Hypergeometric Polynomials that Generalize Jacobi Polynomials, Memoirs of the American Mathematical Society, vol. 319 (1985), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 0572.33012
[12] Dimitrov, P., Connection coefficients and zeros of orthogonal polynomials, this issue, J. Comput. Appl. Math., 133, 331-340 (2001) · Zbl 1010.42016
[13] Fields, J. L.; Wimp, J., Expansions of hypergeometric functions in hypergeometric functions, Math. Comp., 15, 390-395 (1961) · Zbl 0107.05902
[14] Gasper, G., Projection formulas for orthogonal polynomials of a discrete variable, J. Math. Anal. Appl., 45, 176-198 (1974) · Zbl 0276.33026
[15] Gasper, G.; Rahman, M., Basic Hypergeometric Series. Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, vol. 35 (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0695.33001
[16] Godoy, E.; Ronveaux, A.; Zarzo, A.; Area, I., Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case, J. Comput. Appl. Math., 84, 257-275 (1997) · Zbl 0909.65008
[17] Godoy, E.; Ronveaux, A.; Zarzo, A.; Area, I., On the limit relations between classical continuous and discrete orthogonal polynomials, J. Comput. Appl. Math., 91, 1, 97-105 (1998) · Zbl 0934.33013
[18] Godsil, C. D., Algebraic Combinatorics (1993), Chapman & Hall: Chapman & Hall New York · Zbl 0814.05075
[19] Hylleraas, E., Linearization of products of Jacobi polynomials, Math. Scand., 10, 189-200 (1962) · Zbl 0109.29603
[20] R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\); R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)
[21] Koepf, W.; Schmersau, D., Representations of orthogonal polynomials, J. Comput. Appl. Math., 90, 57-94 (1998) · Zbl 0907.65017
[22] Koornwinder, T., Compact quantum groups and \(q\)-special functions, (Baldoni, V.; Picardello, M. A., Representations of Lie Groups and Quantum Groups. Representations of Lie Groups and Quantum Groups, Pitman Research Notes in Mathematics Series, vol. 311 (1994), Longman Scientific & Technical: Longman Scientific & Technical Harlow), 46-128 · Zbl 0821.17015
[23] Lasser, R., Linearization of the product of associated Legendre polynomials, SIAM J. Math. Anal., 14, 403-408 (1983) · Zbl 0509.33007
[24] Lewanowicz, S., Recurrence relations for the connection coefficients of the orthogonal polynomials of a discrete variable, J. Comput. Appl. Math., 76, 213-229 (1996) · Zbl 0877.33003
[25] Y.L. Luke, The Special Functions and their Approximations, 2 Vols. Academic Press, New York, 1969.; Y.L. Luke, The Special Functions and their Approximations, 2 Vols. Academic Press, New York, 1969. · Zbl 0193.01701
[26] Miller, W., Special functions and the complex Euclidean group in 3-space, II, J. Math. Phys., 9, 1175-1187 (1968) · Zbl 0159.09503
[27] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, 3 Vols. Gordon and Breach, New York, 1986.; A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, 3 Vols. Gordon and Breach, New York, 1986. · Zbl 0606.33001
[28] Rainville, E. D., Special Functions (1960), Macmillan: Macmillan New York · Zbl 0050.07401
[29] Ronveaux, A.; Zarzo, A.; Godoy, E., Recurrence relation for connection coefficients between two families of orthogonal polynomials, J. Comput. Appl. Math., 62, 67-73 (1995) · Zbl 0876.65005
[30] Ronveaux, A.; Zarzo, A.; Area, I.; Godoy, E., Transverse limits in the Askey tableau, J. Comput. Appl. Math., 99, 1-2, 327-335 (1998) · Zbl 0936.33004
[31] Szwarc, R., Linearization and connection coefficients of orthogonal polynomials, Mh. Math., 113, 319-329 (1992) · Zbl 0766.33008
[32] N.M. Temme, J.L. López, The role of Hermite polynomials in asymptotics analysis, CWI Report MAS-R9926, 1999, Amsterdam. Methods Appl. Anal., to appear.; N.M. Temme, J.L. López, The role of Hermite polynomials in asymptotics analysis, CWI Report MAS-R9926, 1999, Amsterdam. Methods Appl. Anal., to appear.
[33] Verma, A., Certain expansions of the basic hypergeometric functions, Math. Comp., 20, 151-157 (1966) · Zbl 0192.15104
[34] Wilson, J. A., Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal., 11, 690-701 (1980) · Zbl 0454.33007
[35] Zarzo, A.; Area, I.; Godoy, E.; Ronveaux, A., Results for some inversion problems for classical continuous and discrete orthogonal polynomials, J. Phys. A: Math. Gen., 30, L35-L40 (1997) · Zbl 0992.33003
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