Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
$d$-orthogonal polynomials; matrix element; coherent state; hypergeometric function; Meixner polynomials; $d$-dimensional linear functional vector
Summary:
This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of $d$-orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when $d=1$. The underlying algebraic framework allowed a systematic derivation of the recurrence relations, difference equation, lowering and rising operators and generating functions which these polynomials satisfy.
References:
[1] Aptekarev, A. I.: Multiple orthogonal polynomials. J. Comput. Appl. Math. 99 (1998), 423-447. DOI 10.1016/S0377-0427(98)00175-7 | MR 1662713 | Zbl 0958.42015
[2] Arvesú, J., Coussement, J., Assche, W. Van: Some discrete multiple orthogonal polynomials. J. Comput. Appl. Math. 153 (2003), 19-45. DOI 10.1016/S0377-0427(02)00597-6 | MR 1985676 | Zbl 1021.33006
[3] Cheikh, Y. Ben, Lamiri, I.: On obtaining dual sequences via inversion coefficients. Proc. of the 4th workshop on advanced special functions and solutions of PDE's Sabaudia, Italy, 2009, Lecture Notes of Seminario Interdisciplinare di Mathematica {\it 9} A. Cialdea et al. (2010), 41-56. Zbl 1216.44003
[4] Cheikh, Y. Ben, Zaghouani, A.: $d$-orthogonality via generating functions. J. Comput. Appl. Math. 199 (2007), 2-22. DOI 10.1016/j.cam.2005.01.051 | MR 2267527 | Zbl 1119.42009
[5] Bouzeffour, F., Zagouhani, A.: $q$-oscillator algebra and $d$-orthogonal polynomials. J. Nonlinear Math. Phys. 20 (2013), 480-494. DOI 10.1080/14029251.2013.868262 | MR 3196458
[6] Genest, V. X., Miki, H., Vinet, L., Zhedanov, A.: The multivariate Charlier polynomials as matrix elements of the Euclidean group representation on oscillator states. J. Phys. A, Math. Theor. 47 (2014), Article ID 215204, 16 pages. DOI 10.1088/1751-8113/47/21/215204 | MR 3207168 | Zbl 1296.33025
[7] Genest, V. X., Vinet, L., Zhedanov, A.: $d$-orthogonal polynomials and $\frak {su}$ (2). J. Math. Anal. Appl. 390 (2012), 472-487. DOI 10.1016/j.jmaa.2012.02.004 | MR 2890531 | Zbl 1238.33004
[8] Koekoek, R., Lesky, P. A., Swarttouw, R. F.: Hypergeometric Orthogonal Polynomials and Their $q$-analogues. Springer Monographs in Mathematics, Springer, Berlin (2010). DOI 10.1007/978-3-642-05014-5 | MR 2656096 | Zbl 1200.33012
[9] Lamiri, I., Ouni, A.: $d$-orthogonality of some basic hypergeometric polynomials. Georgian Math. J. 20 (2013), 729-751. DOI 10.1515/gmj-2013-0039 | MR 3139281 | Zbl 1282.33027
[10] Maroni, P.: L'orthogonalité et les récurrences de polynômes d'ordre supérieur à deux. Ann. Fac. Sci. Toulouse, Math. (5) 11 French (1989), 105-139. DOI 10.5802/afst.672 | MR 1425747 | Zbl 0707.42019
[11] Plyushchay, M. S.: Deformed Heisenberg algebra with reflection. Nuclear Physics B 491 (1997), 619-634. DOI 10.1016/S0550-3213(97)00065-5 | MR 1449322 | Zbl 0937.81034
[12] Rosenblum, M.: Generalized Hermite polynomials and the Bose-like oscillators calculus. Nonselfadjoint Operators and Related Topics. Workshop on Operator Theory and Its Applications Beersheva, Israel, 1992, Oper. Theory Adv. App. 73, Birkhäuser, Basel (1994), 369-396 A. Feintuch et al. DOI 10.1007/978-3-0348-8522-5_15 | MR 1320555 | Zbl 0826.33005
[13] Assche, W. Van, Coussement, E.: Some classical multiple orthogonal polynomials. J. Comput. Appl. Math. 127 (2001), 317-347. DOI 10.1016/S0377-0427(00)00503-3 | MR 1808581 | Zbl 0969.33005
[14] Iseghem, J. Van: Laplace transform inversion and Padé-type approximants. Appl. Numer. Math. 3 (1987), 529-538. DOI 10.1016/S0377-0427(00)00503-3 | MR 0918793 | Zbl 0634.65129
[15] Vinet, L., Zhedanov, A.: Automorphisms of the Heisenberg-Weyl algebra and $d$-orthogonal polynomials. J. Math. Phys. 50 (2009), Article No. 033511, 19 pages. DOI 10.1063/1.3087425 | MR 2510916 | Zbl 1202.33018
Partner of
EuDML logo