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\(q\)-classical orthogonal polynomials: A general difference calculus approach. (English) Zbl 1204.33011
One of the many ways to characterize a family \((p_n)\) of classical polynomials (Hermite, Laguerre, Jacobi, and Bessel) is the structure relation \[ \varphi(x)p'_n(x)=a_np_{n+1}(x)+b_np_{n}(x)+c_np_{n-1}(x),\quad n\geq0,\eqno(1) \] where \(\varphi\) is a fixed polynomial of degree at most 2 and \(c_n\neq0\), \(n\geq1\). The relation (1) also characterizes the discrete classical orthogonal polynomials (Hahn, Krawtchouk, Meixner) when the derivative is replaced by the forward difference operator \(\Delta f\equiv f(x+1)-f(x)\). In [J. C. Medem, R. Alvarez-Nodarse and F. Marcellan, J. Comput. Appl. Math. 135, No. 2, 157–196 (2001; Zbl 0991.33007)] the orthogonal polynomials (the \(q\)-Hahn class) are characterized by a structure relation obtained from (1) replacing the derivative by the \(q\)-difference operator \(D_qf=\frac{f(qx)-f(x)}{(q-1)x}\), \(q\in \mathbb C\), \(|q|\neq0,1\). In the present paper two new structure relations for the \(q\)-polynomials which belong to the \(q\)-Askey tableau and the lattice is \(q\)-quadratic \(x(s)=c_1q^s+c_2q^{-s}+ c_3\), with \(c_1c_2\neq0\), being \(q\in\mathbb C\setminus(\{0\}\cup (\cup_{n\geq1}\{z\in\mathbb C:z^n=1\}))\). It is proved that the following relation characterizes the \(q\)-polynomials \[ Mp_{n}(x(s))=e_n\frac{\Delta p_{n+1}(s)}{\Delta x(s)}+ f_n\frac{\Delta p_{n}(s)}{\Delta x(s)}+ g_n\frac{\Delta p_{n-1}(s)}{\Delta x(s)},\quad n\geq0, \] where \(Mf=\frac12(f(x+1)-f(x))\) is the forward arithmetic mean operator, \((e_n)\), \((f_n)\), and \((g_n)\) are sequences of complex numbers such that \(e_n\neq0\) and \(g_n\neq c_n\). A characterization theorem for classical orthogonal polynomials in a more general framework is given. The Rodrigues operator is defined and a unified expression for the linear differential (difference resp.) hypergeometric operators related to the classical families, and for their polynomial eigenfunctions are deduced. Hahn’s theorem for the \(q\)-polynomials of the \(q\)-Askey tableau is proved.

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