Second order $$q$$-difference equations solvable by factorization method.(English)Zbl 1119.39017

The factorization method to solve ordinary differential equations due to Darboux has many applications to orthogonal polynomials and quantum mechanics. The authors propose an extension to $$q$$-difference equations. It is a $$q$$-analogue, since, in the continuous limit $$q \to 1$$, it boils down to the classical factorization method (at least in the usual intuitive description of “continuous limit”). The paper contains the proof in a particular important case (indeed, a generalisation of a famous study by L. Infeld and T. E. Hull, Rev. Mod. Phys. 23, 21–68 (1951; Zbl 0043.38602) that the $$q$$-Hahn orthogonal polynomials are among the solutions. They also consider other interesting examples.

MSC:

 39A13 Difference equations, scaling ($$q$$-differences) 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 39A12 Discrete version of topics in analysis

Zbl 0043.38602
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References:

 [1] Álvarez-Nodarse, R.; Atakishiyev, N.M.; Costas-Santos, R.S., Factorization of the hypergeometric-type difference equation on non-uniform lattices: dynamical algebra, J. phys. A: math. gen., 38, 153-174, (2005) · Zbl 1079.33017 [2] Álvarez-Nodarse, R.; Costas-Santos, R.S., Factorization method for difference equations of hypergeometric type on nonuniform lattices, J. phys. A: math. gen., 34, 5551-5569, (2001) · Zbl 0994.39001 [3] Álvarez-Nodarse, R.; Smirnov, Y.F., The dual Hahn q-polynomials in the lattice x(s)=[s]q[s+1]q and the q-algebras suq(2) and suq(1,1), J. phys. A: math. gen., 29, 1435-1451, (1996) · Zbl 0912.33011 [4] Atakishiyev, N.M.; Frank, A.; Wolf, K.B., A simple difference realization of the Heisenberg q-algebra, J. math. phys., 35, 3253-3260, (1994) · Zbl 0809.17014 [5] Atakishiyev, N.M.; Suslov, S.K., Difference analogs of the harmonic oscillator, Theoret. and math. phys., 85, 64-73, (1990) · Zbl 1189.81099 [6] Atakishiyev, N.M.; Suslov, S.K., Explicit realization of the $$q$$-harmonic oscillator, Theoret. and math. phys., 87, 154-156, (1991) [7] Bangerezako, G., The factorization method for the askey – wilson polynomials, J. comput. appl. math., 107, 219-232, (1999) · Zbl 0933.39042 [8] Bangerezako, G.; Hounkonnou, M.N., The factorization method for the general second order q-difference equation and the laguerre – hahn polynomials on the general q-lattice, J. phys. A: math. gen., 36, 765-773, (2003) · Zbl 1051.39018 [9] Chihara, T.S., An introduction to orthogonal polynomials, (1978), Gordon and Breach New York · Zbl 0389.33008 [10] Darboux, G., Sur une proposition relative aux equations lineaires, C.R. acad. sci. Paris, 94, 1456, (1882) · JFM 14.0264.01 [11] Dirac, P.A.M., Principles of quantum mechanics, (1947), Clarendon Press Oxford · Zbl 0030.04801 [12] Gasper, G.; Rahman, M., Basic hypergeometric series, (1990), Cambridge University Press Cambridge · Zbl 0695.33001 [13] Goliński, T.; Odzijewicz, A., General difference calculus and its application to functional equations of the second order, Czechoslovak J. phys., 52, 1219-1224, (2002) · Zbl 1051.39023 [14] Goliński, T.; Odzijewicz, A., Factorization method for second order functional equations, J. comput. appl. math., 176, 331-355, (2005) · Zbl 1067.39034 [15] Hahn, W., Uber orthogonalpolynome die q-differenzengleichungeg genüngen¨, Math. nachr., 2, 4-34, (1949) [16] Infeld, L.; Hull, T.E., The factorization method, Rev. mod. phys., 23, 21-68, (1951) · Zbl 0043.38602 [17] R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report no. 98-17, TUDelft, webpage http://aw.twi.tudelft.nl/koekoek/askey.html, 1998. [18] de Lange, O.L.; Raab, R.E., Operator methods in quantum mechanics, (1991), Clarendon Press Oxford · Zbl 1300.78004 [19] Mielnik, B.; Nieto, L.M.; Rosas-Ortiz, O., The finite difference algorithm for higher order supersymmetry, Phys. lett. A, 269, 70-78, (2000) · Zbl 1115.81350 [20] Miller, W., Lie theory and special functions, (1968), Academic Press New York, London · Zbl 0174.10502 [21] Miller, W., Lie theory and q-difference equations, SIAM J. math. anal., 1, 2, 171-188, (1970) · Zbl 0206.39001 [22] Odzijewicz, A.; Horowski, H.; Tereszkiewicz, A., Integrable multi-boson systems and orthogonal polynomials, J. phys. A: math. gen., 34, 4353-4376, (2001) · Zbl 0978.81084 [23] Schrödinger, E., Proc. roy. irish acad. A, 46, (1940) [24] Spiridonov, V., Universal superpositions of coherent states and self-similar potentials, Phys. rev. A, 52, 1909-1935, (1995) [25] Veselov, A.P.; Shabat, A.B., Funct. anal. appl., 27, 1-21, (1993)
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