| Title:
|
Deformed Heisenberg algebra with reflection and $d$-orthogonal polynomials (English) |
| Author:
|
Bouzeffour, Fethi |
| Author:
|
Ben Mansour, Hanen |
| Author:
|
Zaghouani, Ali |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
1 |
| Year:
|
2017 |
| Pages:
|
57-71 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
$d$-orthogonal polynomials |
| Keyword:
|
matrix element |
| Keyword:
|
coherent state |
| Keyword:
|
hypergeometric function |
| Keyword:
|
Meixner polynomials |
| Keyword:
|
$d$-dimensional linear functional vector |
| MSC:
|
22E47 |
| MSC:
|
33C45 |
| MSC:
|
33D15 |
| idZBL:
|
Zbl 06738504 |
| idMR:
|
MR3632998 |
| DOI:
|
10.21136/CMJ.2017.0358-15 |
| . |
| Date available:
|
2017-03-13T12:05:13Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146040 |
| . |
| Reference:
|
[1] Aptekarev, A. I.: Multiple orthogonal polynomials.J. Comput. Appl. Math. 99 (1998), 423-447. Zbl 0958.42015, MR 1662713, 10.1016/S0377-0427(98)00175-7 |
| Reference:
|
[2] Arvesú, J., Coussement, J., Assche, W. Van: Some discrete multiple orthogonal polynomials.J. Comput. Appl. Math. 153 (2003), 19-45. Zbl 1021.33006, MR 1985676, 10.1016/S0377-0427(02)00597-6 |
| Reference:
|
[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 |
| Reference:
|
[4] Cheikh, Y. Ben, Zaghouani, A.: $d$-orthogonality via generating functions.J. Comput. Appl. Math. 199 (2007), 2-22. Zbl 1119.42009, MR 2267527, 10.1016/j.cam.2005.01.051 |
| Reference:
|
[5] Bouzeffour, F., Zagouhani, A.: $q$-oscillator algebra and $d$-orthogonal polynomials.J. Nonlinear Math. Phys. 20 (2013), 480-494. MR 3196458, 10.1080/14029251.2013.868262 |
| Reference:
|
[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. Zbl 1296.33025, MR 3207168, 10.1088/1751-8113/47/21/215204 |
| Reference:
|
[7] Genest, V. X., Vinet, L., Zhedanov, A.: $d$-orthogonal polynomials and $\frak {su}$ (2).J. Math. Anal. Appl. 390 (2012), 472-487. Zbl 1238.33004, MR 2890531, 10.1016/j.jmaa.2012.02.004 |
| Reference:
|
[8] Koekoek, R., Lesky, P. A., Swarttouw, R. F.: Hypergeometric Orthogonal Polynomials and Their $q$-analogues.Springer Monographs in Mathematics, Springer, Berlin (2010). Zbl 1200.33012, MR 2656096, 10.1007/978-3-642-05014-5 |
| Reference:
|
[9] Lamiri, I., Ouni, A.: $d$-orthogonality of some basic hypergeometric polynomials.Georgian Math. J. 20 (2013), 729-751. Zbl 1282.33027, MR 3139281, 10.1515/gmj-2013-0039 |
| Reference:
|
[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. Zbl 0707.42019, MR 1425747, 10.5802/afst.672 |
| Reference:
|
[11] Plyushchay, M. S.: Deformed Heisenberg algebra with reflection.Nuclear Physics B 491 (1997), 619-634. Zbl 0937.81034, MR 1449322, 10.1016/S0550-3213(97)00065-5 |
| Reference:
|
[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. Zbl 0826.33005, MR 1320555, 10.1007/978-3-0348-8522-5_15 |
| Reference:
|
[13] Assche, W. Van, Coussement, E.: Some classical multiple orthogonal polynomials.J. Comput. Appl. Math. 127 (2001), 317-347. Zbl 0969.33005, MR 1808581, 10.1016/S0377-0427(00)00503-3 |
| Reference:
|
[14] Iseghem, J. Van: Laplace transform inversion and Padé-type approximants.Appl. Numer. Math. 3 (1987), 529-538. Zbl 0634.65129, MR 0918793, 10.1016/S0377-0427(00)00503-3 |
| Reference:
|
[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. Zbl 1202.33018, MR 2510916, 10.1063/1.3087425 |
| . |
