We consider a generalization of the classical Laplace operator, which includes the Laplace-Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, which are generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher rank Bannai-Ito algebra.

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers.  Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce some properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof.

Abstract. We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester-Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models.

We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel-Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant k and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to ∞, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.

The Howe dual pair (sl (2), O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a wellchosen element of sl(2) such that the Helmholtz relations are satisfied.In this paper we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized Fourier transforms, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We obtain operator exponential expressions for all these transforms by making extensive use of the theory of integer-valued polynomials. We also find a plane wave decomposition of their integral kernel and establish uncertainty principles. In important special cases we even obtain closed formulas for the integral kernels.In the second part of the paper, the same problem is considered for the dual pair (osp(1|2), Spin(m)), in the context of the Dirac operator on R m . This connects our results with the Clifford-Fourier transform studied in previous work.with ∆ x the Laplace operator and E x the Euler operator, one haswhich relates the transform with the representation theory of the Lie algebra sl(2) as the oper-[23, 28] for a detailed mathematical treatment. By now well-established fields of research based on this observation include the fractional FT [39] and the family of linear canonical transforms (LCTs) [45]. Both have extensive practical applications in the design of optical systems and signal processing.The observation in formula (1.1) has also lead to many other theoretically oriented generalizations of the FT, by considering alternative realizations of sl(2) in terms of differential or difference operators (or combinations thereof). Once such a realization is obtained, it is possible to define a generalized FT based on formula (1.1). It is then a challenging question to find a concrete integral transform expression for this operator. Research on this topic has a long history: it has been investigated in both the continuous and the discrete case, and also for more complicated algebras than sl (2), such as the superalgebra osp(1|2). Important examples in the continuous case include the Dunkl transform [19,2] and the radially deformed Fourier transform [35,36,3,14] as well as its Clifford deformation [16,17]. In more complicated geometries, the super Fourier transform [13] and a q-deformed version [10] have been investigated. In the discrete case we mention the fractional Fourier-Kravchuk transform [1] and various further deformations [32,33]. For a more detailed review of the followed strategy and the results in this line of investigation we refer the reader to [37,12].The present paper has a related yet different aim. We do not intend to change the realization of sl(2) in formula (1.2), but instead wish to study precisely how unique the operator realization of the FT in (1.1) is and to what extent the interplay of sl(2)...