×

zbMATH — the first resource for mathematics

\(q\)-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.

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.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alfaro, M., Álvarez-Nodarse, R.: A characterization of the classical orthogonal discrete and q-polynomials. J. Comput. Appl. Math. 201, 48–54 (2007) · Zbl 1108.33007 · doi:10.1016/j.cam.2006.01.031
[2] Al-Salam, W.A., Chihara, T.S.: Another characterization of the classical orthogonal polynomials. SIAM J. Math. Anal. 3, 65–70 (1992) · Zbl 0238.33010 · doi:10.1137/0503007
[3] Álvarez-Nodarse, R.: Polinomios Hipergeométricos y q-Polinomios. Monografías del Seminario Matemático ”Garcia de Galdeano”, vol. 26. Prensas Universitarias de Zaragoza, Zaragoza (2003) (in Spanish)
[4] Álvarez-Nodarse, R.: On characterizations of classical polynomials. J. Comput. Appl. Math. 196, 320–337 (2006) · Zbl 1108.33008 · doi:10.1016/j.cam.2005.06.046
[5] Álvarez-Nodarse, R., Arvesú, J.: On the q-polynomials on the exponential lattice x(s)=c 1 q s +c 3. Integral Trans. Special Funct. 8, 299–324 (1999) · Zbl 0956.33009 · doi:10.1080/10652469908819236
[6] Álvarez-Nodarse, R., Smirnov, Yu.F., Costas-Santos, R.S.: A q-analog of the Racah polynomials and q-algebra su q (2) in quantum optics. J. Russ. Laser Res. 27(1), 1–32 (2006) · doi:10.1007/s10946-006-0001-4
[7] Andrews, G.E.: q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, CBMS Regional Conference Series in Mathematics, vol. 66. Amer. Math. Soc., Providence (1996)
[8] Askey, R., Suslov, S.K.: The q-harmonic oscillator and the Al-Salam and Carlitz polynomials. Lett. Math. Phys. 29(2), 123–132 (1993) · Zbl 0919.33010 · doi:10.1007/BF00749728
[9] Atakishiev, N.M., Suslov, S.K.: Difference analogs of the harmonic oscillator. Theor. Math. Phys. 85(1), 1055–1062 (1991) · Zbl 1189.81099 · doi:10.1007/BF01017247
[10] Atakishiev, N.M., Suslov, S.K.: A realization of the q-harmonic oscillator. Theor. Math. Phys. 87(1), 442–444 (1991) · Zbl 1189.81100 · doi:10.1007/BF01016585
[11] Atakishiyev, N.M., Rahman, M., Suslov, S.K.: On classical orthogonal polynomials. Constr. Approx. 11, 181–226 (1995) · Zbl 0837.33010 · doi:10.1007/BF01203415
[12] Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978) · Zbl 0389.33008
[13] Fine, N.J.: Hypergeometric Series and Applications. Mathematical Surveys and Monographs. Amer. Math. Soc., Providence (1988)
[14] 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 · doi:10.1016/0377-0427(93)E0241-D
[15] Hahn, W.: Über die Jacobischen polynome und zwei verwandte Polynomklassen. Math. Zeit. 39, 634–638 (1935) · Zbl 0011.06202 · doi:10.1007/BF01201380
[16] Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, vol. 98-17. Reports of the Faculty of Technical Mathematics and Informatics, Delft, The Netherlands (1998)
[17] Koornwinder, T.H.: Orthogonal polynomials in connection with quantum groups. In: Nevai, P. (ed.) NATO Advanced Study Institute Series C, vol. 294, pp. 257–292. Kluwer Academic, Dordrecht (1990) · Zbl 0697.42019
[18] Koornwinder, T.H.: Compact quantum groups and q-special functions. In: Baldoni, V., Picardello, M.A. (eds.) Pitman Research Notes in Mathematics Series, pp. 257–292. Longman Scientific & Technical, New York (1994) · Zbl 0821.17015
[19] Koornwinder, T.H.: The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 27, 214–226 (2007) · Zbl 1120.33018 · doi:10.1016/j.cam.2006.10.015
[20] Malashin, A.A.: q-analog of Racah polynomials on the lattice x(s)=[s]q[s+1] q and its connections with 6j-symbols for the su q (2) and su q (1,1) quantum algebras. Master’s thesis, Moscow State University (1992) (in Russian)
[21] Marcellán, F., Branquinho, A., Petronilho, J.C.: Classical orthogonal polynomials: a functional approach. Acta Appl. Math. 34, 283–303 (1994) · Zbl 0793.33009 · doi:10.1007/BF00998681
[22] 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 · doi:10.1016/S0377-0427(00)00584-7
[23] Nagiyev, Sh.M.: Difference Schrödinger equation and q-oscillator model. Theor. Math. Phys. 102, 180–187 (1995) · Zbl 0853.39009 · doi:10.1007/BF01040399
[24] Nikiforov, A.F., Uvarov, V.B.: The Special Functions of Mathematical Physics. Birkhäuser, Basel (1988) · Zbl 0624.33001
[25] Nikiforov, A.F., Suslov, S.K., Uvarov, V.B.: Classical Orthogonal Polynomials of a Discrete Variable. Springer Series in Computational Physics. Springer, Berlin (1991) · Zbl 0743.33001
[26] Routh, E.: On some properties of certain solutions of a differential equation of the second order. Proc. Lond. Math. Soc. 16, 245–261 (1884) · JFM 17.0315.02 · doi:10.1112/plms/s1-16.1.245
[27] Vilenkin, N.Ja., Klimyk, A.U.: Representations of Lie Groups and Special Functions, vols. I, II, III. Kluwer Academic, Dordrecht (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.