## $$q$$-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.)

Zbl 0991.33007
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### 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)
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