Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 57-71, 2017

# Deformed Heisenberg algebra with reflection and \$d\$-orthogonal polynomials

## Fethi Bouzeffour, Hanen Ben Mansour, Ali Zaghouani

#### Received July 3, 2015.   First published February 24, 2017.

Fethi Bouzeffour, Department of Mathematics, College of Sciences, King Saud University, P. O. Box \$2455\$, Riyadh \$11451\$, Saud Arabia, e-mail: fbouzaffour@ksu.edu.sa; Hanen Ben Mansour, Ali Zaghouani, Department of mathematics, Faculty of Sciences of Bizerte, Carthage University, P. O. Box 64, Jarzouna 7021, Bizerte, Tunisia, and University of Tunis El Manar, Faculty of Sciences of Tunis, B. P. 248 2092, Tunis, Tunisia, e-mail: benmansourhanen52@yahoo.com, ali.zaghouani@gmail.com

Abstract: 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.

Keywords: \$d\$-orthogonal polynomials; matrix element; coherent state; hypergeometric function; Meixner polynomials; \$d\$-dimensional linear functional vector

Classification (MSC 2010): 33C45, 33D15, 22E47

DOI: 10.21136/CMJ.2017.0358-15

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