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Gasper, George; Rahman, Mizan

Some systems of multivariable orthogonal \(q\)-Racah polynomials. (English) Zbl 1121.33019

Ramanujan J. 13, No. 1-3, 389-405 (2007).
The authors employ certain \(q\)-identities and \(q\)-summation formulae, for example, summation formula for terminating \({_6\phi_5}\) to derive basic analogs of extended Racah polynomials and second system of Racah’s multivariate orthogonal polynomials as considered by M. V. Tratnik [J. Math. Phys. 32, No. 9, 2337–2342 (1991; Zbl 0742.33007)]. Special and limit cases of their results provide interesting known results.
Reviewer: Ajendra Nath Srivastava (Puna)

Cited in 42 Documents

MSC:

33D50 Orthogonal polynomials and functions in several variables expressible in terms of basic hypergeometric functions in one variable
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Keywords:

Multivariable discrete orthogonal polynomials; Multivariable basic hypergeometric orthogonal polynomials; Several variables

Citations:

Zbl 0742.33007
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\textit{G. Gasper} and \textit{M. Rahman}, Ramanujan J. 13, No. 1--3, 389--405 (2007; Zbl 1121.33019)
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References:

[1] 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
[2] Askey, R., Wilson, J.A.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoirs Amer. Math. Soc. 319, (1985) · Zbl 0572.33012
[3] van Diejen, J.F., Stokman, J.V.: Multivariable q-Racah polynomials. Duke Math. J. 91, 89–136 (1998) · Zbl 0951.33010
[4] Dunkl, C.F.: Orthogonal polynomials in two variables of q-Hahn and q-Jacobi type. SIAM J. Alg. Disc. Meth. 1, 137–151 (1980) · Zbl 0499.33007
[5] Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge Univ. Press (2001) · Zbl 0964.33001
[6] Gasper, G.: Positivity and special functions. In: Askey, R. (ed.) Theory and Applications of Special Functions, pp. 375–433. Academic Press, New York (1975) · Zbl 0326.33009
[7] Gasper, G., Rahman, M.: Nonnegative kernels in product formulas for q-Racah polynomials. J. Math. Anal. Appl. 95, 304–318 (1983) · Zbl 0518.33006
[8] Gasper, G., Rahman, M.: Product formulas of Watson, Bailey and Bateman types and positivity of the Poisson kernel for q-Racah polynomials. SIAM J. Math. Anal. 15, 768–789 (1984) · Zbl 0545.33013
[9] Gasper, G., Rahman, M.: Basic Hypergeometric Series. 2nd ed. Cambridge Univ. Press (2004) · Zbl 1129.33005
[10] Gasper, G., Rahman, M.: q-Analogues of some multivariable biorthogonal polynomials. In: Ismail, M.E.H., Koelink, E. (eds.) Theory and Applications of Special Functions. A volume dedicated to Mizan Rahman. Dev. Math. vol. 13, pp. 185–208 (2005) · Zbl 1219.33023
[11] Gasper, G., Rahman, M.: Some systems of multivariable orthogonal Askey–Wilson polynomials. In: Ismail, M.E.H., Koelink, E. (eds.) Theory and Applications of Special Functions. A volume dedicated to Mizan Rahman. Dev. Math. vol. 13, pp. 209–219 (2005) · Zbl 1219.33027
[12] Granovskii, Ya.I., Zhedanov, A.S.: ’Twisted’ Clebsch-Gordan coefficients for SU q (2). J. Phys. A 25, L1029–L1032 (1992)
[13] Gustafson, R.A.: A Whipple’s transformation for hypergeometric series in U(n) and multivariable hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 18, 495–530 (1987) · Zbl 0607.33015
[14] Karlin, S., McGregor, J.: Linear growth models with many types and multidimensional Hahn polynomials. In: Askey, R. (ed.) Theory and Applications of Special Functions, pp. 261–288. Academic Press, New York (1975) · Zbl 0361.60071
[15] Koekoek, R., Swarttouw, R.F.: The Askey–scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 98–17, Delft Univ. of Technology, o http://aw.twi.tudelft.nl/\(\sim\)koekoek/research.html (1998)
[16] Koelink, H.T., Van der Jeugt, J.: Convolutions for orthogonal polynomials from Lie and quantum algebra representations. SIAM J. Math. Anal. 29, 794–822 (1998) · Zbl 0977.33013
[17] Koornwinder, T.H.: Askey–Wilson polynomials for root systems of type BC. Contemp. Math. 138, 189–204 (1998) · Zbl 0797.33014
[18] Rosengren, H.: Multivariable q-Hahn polynomials as coupling coefficients for quantum algebra representations. Int. J. Math. Math. Sci. 28, 331–358 (2001) · Zbl 1049.33014
[19] Spiridonov, V.P.: Theta hypergeometric integrals. Algebra i Analiz (St. Petersburg Math. J.) 15, 161–215 (2003)
[20] Stokman, J.V.: On BC type basic hypergeometric orthogonal polynomials. Trans. Amer. Math. Soc. 352, 1527–1579 (2000) · Zbl 0936.33008
[21] Tratnik, M.V.: Multivariable Wilson polynomials. J. Math. Phys. 30, 2001–2011 (1989) · Zbl 0687.33014
[22] Tratnik, M.V.: Multivariable biorthogonal continuous–discrete Wilson and Racah polynomials. J. Math. Phys. 31, 1559–1575 (1990) · Zbl 0707.33007
[23] Tratnik, M.V.: Some multivariable orthogonal polynomials of the Askey tableau–continuous families. J. Math. Phys. 32, 2065–2073 (1991) · Zbl 0746.33007
[24] Tratnik, M.V.: Some multivariable orthogonal polynomials of the Askey tableau–discrete families. J. Math. Phys. 32, 2337–2342 (1991) · Zbl 0742.33007
[25] Wilson, J.A.: Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11, 690–701 (1980) · Zbl 0454.33007
[26] Xu, Y.: On discrete orthogonal polynomials of several variables. Adv. in Appl. Math. 33, 615–632 (2004) · Zbl 1069.33011
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