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A \(q\)-extension of the Erkus-Srivastava polynomials in several variables. (English) Zbl 1152.33010

The author generalizes the unified presentation of some multivariable polynomials introduced by E. Erkuş and H. M. Srivastava [Integral Transform Spec. Funct. 17, 267–273 (2006; Zbl 1098.33016)] to a \(q\)-analogue. Using the generating function \[ \prod_{i=1}^r\,{1\over (x_i t^{m_i};q)_{\alpha_i}} = \sum_{n=0}^{\infty} u_{n,q}^{(\alpha_1,\ldots,\alpha_r)}(x_1,\ldots,x_r) t^n, \] the author derives the polynomials \[ u_{n,q}^{(\alpha_1,\ldots,\alpha_r)}(x_1,\ldots,x_r)=\sum_{m_1 k_1+\cdots+m_r k_r=n}\, (q^{\alpha_{k_1}};q)_{k_1}\cdots (q^{\alpha_{k_r}};q)_{k_r}\, {x^{k_1}_1\over (q,q)_{k_1}}\cdots {x^{k_r}_r\over (q,q)_{k_r}}.\eqno{(*)} \] For \(m_j=1\;(1\leq j\leq r)\) an analogue of the Chan-Chyan-Srivastava multivariable polynomials is obtained and for \(m_j=j\;(1\leq j\leq r)\) a \(q\)-analogue of the Lagrange-Hermite polynomials introduced by the author [Internat. J. Comput. Numer. Anal. Appl. 6, 143–135 (2004; Zbl 1101.33013)] is found. Moreover, two theorems on families of bilinear and bilateral generating functions for the \(q\)-analogue \((*)\) are given.

MSC:

33D70 Other basic hypergeometric functions and integrals in several variables
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D50 Orthogonal polynomials and functions in several variables expressible in terms of basic hypergeometric functions in one variable
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