A characterization of the classical orthogonal discrete and \(q\)-polynomials. (English) Zbl 1108.33007

Alfaro, M.; Álvarez-Nodarse, R.

J. Comput. Appl. Math. 201, No. 1, 48-54 (2007).

Let \(P\) be the linear space of polynomial functions in \(\mathbb C\) with complex coefficients and let \(P^*\) be its algebraic dual space. The forward difference operator \(\Delta\) and the \(q\)-derivative operator \({\mathcal D}_q\) on \(P\) are given by \[ \Delta y(x)=y(x+1)-y(x) \quad \text{and}\quad {\mathcal D}_q\pi(x)={\pi(qx)-\pi(x)\over (q-1)x },\quad | q| \not=0,1. \] Let \(u\in P^*\) be a quasi-define functional and let \(\{P_n\}_{n\geq 0}\) be the monic orthogonal polynomial sequence (MOPS, for shorter) with respect to the functional \(u\). We say \(\{P_n\}_{n\geq 0}\) is a \(\Delta\)-classical (resp. \(q\)-classical) MOPS iff the sequence \(\{\Delta P_n\}_{n\geq 1}\) (resp. \(\{{\mathcal D}_q P_n\}_{n\geq 1}\)) is also orthogonal. In the literature there exist many characterization theorems for the \(\Delta\)-classical and \(q\)-classical MOPSs. In this paper, the authors present another one which asserts that \(\{P_n\}_{n\geq 0}\) is a \(\Delta\)-classical (resp. \(q\)-classical) MOPS iff for every \(n\geq 1\)
\[ P_nu=\Delta(\alpha_{n-1}\phi u)\quad \text{(resp.}\;\;P_nu={\mathcal D}_q(\alpha_{n-1}\phi u)) \] where \(\alpha_{n-1}\) is a polynomial of degree \(n-1\) and \(\phi \) is a polynomial of degree less or equal to 2.

Reviewer: Youssef Ben Cheikh (Monastir)

MSC:

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

Keywords:

classical orthogonal polynomials; discrete orthogonal polynomials; \(q\)-polynomials; characterization theorems

References:

[1]	R. Álvarez-Nodarse, On characterizations of classical polynomials, J. Comput. Appl. Math. 2005, in press.; R. Álvarez-Nodarse, On characterizations of classical polynomials, J. Comput. Appl. Math. 2005, in press. · Zbl 1108.33008
[2]	Álvarez-Nodarse, R.; Medem, J. C., \(q\)-Classical polynomials and the \(q\)-Askey and Nikiforov-Uvarov Tableaus, J. Comput. Appl. Math., 135, 197-223 (2001) · Zbl 1024.33013
[3]	García, A. G.; Marcellán, F.; Salto, L., A distributional study of discrete classical orthogonal polynomials, J. Comput. Appl. Math., 57, 147-162 (1995) · Zbl 0853.33009
[4]	Hahn, W., Über orthogonalpolynomen die \(q\)-differentialgleichungen genügen, Math. Nachr., 2, 4-34 (1949) · Zbl 0031.39001
[5]	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 \(q\)
[6]	Lesky, P., Über Polynomsysteme, die Sturm-Liouvilleschen differenzengleichungen genügen, Math. Zeit., 78, 439-445 (1962) · Zbl 0107.05403
[7]	Marcellán, F.; Álvarez-Nodarse, R., On the Favard Theorem and their extensions, J. Comput. Appl. Math., 127, 231-254 (2001) · Zbl 0970.33008
[8]	Marcellán, F.; Branquinho, A.; Petronilho, J., Classical orthogonal polynomials: a functional approach, Acta Appl. Math., 34, 283-303 (1994) · Zbl 0793.33009
[9]	Marcellán, F.; Petronilho, J., On the solution of some distributional differential equations: existence and characterizations of the classical moment functionals, Integral Transform. Special Funct., 2, 185-218 (1994) · Zbl 0832.33006
[10]	Medem, J. C.; Álvarez-Nodarse, R.; Marcellán, F., On the \(q\)-polynomials: a distributional study, J. Comput. Appl. Math., 135, 157-196 (2001) · Zbl 0991.33007