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On a multivariable extension for the extended Jacobi polynomials. (English) Zbl 1172.33002

In this paper a multivariable extension \(F^{( \alpha, \beta)}_{\mathbf n}(\mathbf x)\) of the extended Jacobi polynomials (EJP) is investigated and some relations satisfied by these polynomials are given. The extended Jacobi polynomials (EJP) are defined in [I. Fujiwara, Math. Jap. 11, 133–148 (1966; Zbl 0154.06402)] by the Rodrigues formula \[ F^{(\alpha,\beta)}_{n}(x;a,b,c)=\frac{(-c)^n}{n!}(x-a)^{-\alpha} (b-x)^{-\beta}D_x^n\left((x-a)^{n+\alpha}(b-x)^{n+\beta}\right). \] In the paper with help of the product of these EJPs the multivariable EJPs are defined \(F^{(\alpha, \beta)}_{\mathbf n} (\mathbf x)\equiv F^{(\alpha_1, \beta_1)}_{n_1} (x_1; a_1,b_1,c_1)\dots F^{(\alpha_s, \beta_s)}_{n_s} (x_s; a_s,b_s,c_s)\).
In Theorem 2.1 a relation between the polynomials \(F^{(\alpha, \beta)}_{\mathbf n}(\mathbf x)\) and the classical Jacobi polynomials \(P^{(\alpha_i, \beta_i)}_{n_i} (x_i)\) is obtained. In Theorem 2.4 the orthogonality of multivariable EJPs \(F^{( \alpha, \beta)}_{\mathbf n}(\mathbf x)\) with respect to the weight function \(\prod_{i=1}^s (x_i-a_i)^{\alpha_i} (b_i-x_i)^{\beta_i}\) is proved. In section 3 relations between EJPs and Chan-Chyan-Srivastava multivariable polynomials [see W.-Ch. C. Chan, Ch.-J. Chyan and H. M. Srivastava, Integral Transforms Spec. Funct. 12, No. 2, 139–148 (2001; Zbl 1057.33003)] are given. Generating functions and recurrence relations for multivariable EJPs are considered in section 4. Partial differential equations for the product of EJPs are found in section 5 and integral representations for the multivariable EJPs are obtained in section 6. In section 7 several families of multilinear and multilateral generating functions for multivariable EJPs are derived.

MSC:

33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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[1] Chan, W.-C. C.; Chyan, C.-J.; Srivastava, H. M., The Lagrange polynomials in several variables, Integral Transforms Spec. Funct., 12, 139-148 (2001) · Zbl 1057.33003
[2] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions, vol. III (1955), McGraw-Hill Book Company: McGraw-Hill Book Company New York/Toronto/London · Zbl 0064.06302
[3] Feldheim, E., Relations entre les polynômes de Jacobi, Laguerre et Hermite, Acta Math., 74, 117-138 (1941) · JFM 68.0152.04
[4] Fujiwara, I., A unified presentation of classical orthogonal polynomials, Math. Japon., 11, 133-148 (1966) · Zbl 0154.06402
[5] Lee, D. W., Partial differential equations for products of two classical orthogonal polynomials, Bull. Korean Math. Soc., 42, 1, 179-188 (2005) · Zbl 1077.33017
[6] Pittaluga, G.; Sacripante, L.; Srivastava, H. M., Some families of generating functions for the Jacobi and related orthogonal polynomials, J. Math. Anal. Appl., 238, 385-417 (1999) · Zbl 0944.33008
[7] Rainville, E. D., Special Functions (1960), The Macmillan Company: The Macmillan Company New York · Zbl 0050.07401
[8] Srivastava, H. M.; Manocha, H. L., A Treatise on Generating Functions (1984), Halsted Press (Ellis Horwood Limited, Chichester)/John Wiley and Sons: Halsted Press (Ellis Horwood Limited, Chichester)/John Wiley and Sons New York · Zbl 0535.33001
[9] Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23 (1975) · JFM 65.0278.03
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