Related papers: Generating functions for the Bannai-Ito polynomial…
We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind $ \JS(n+k,n;z)$ by generalizing the analogous results for the Stirling and Legendre-Stirling numbers. More precisely, letting…
Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials -- the complementary Bannai-Ito…
Connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves. For classical orthogonal polynomials whose weights are invariant under the action of the symmetric group, connection…
In this paper, we present a new type of generating function of generalized Apostol-Bernoulli poly Daehee polynomial (GABPDP). By using the generating function of GABPDP, we discuss some special cases and useful identities of generalized…
A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little…
This work addresses a full characterization of three new q-polynomials derived from the $q-$oscillator algebra. Related matrix elements and generating functions are deduced. Further, a connection between Hahn factorial and q-Gaussian…
We generalize generating functions for hypergeometric orthogonal polynomials, namely Jacobi, Gegenbauer, Laguerre, and Wilson polynomials. These generalizations of generating functions are accomplished through series rearrangement using…
We propose a new method of calculation of generating functions of Chebyshev polynomials in several variables associated with root systems of simple Lie algebras. We obtain the generating functions of the polynomials in two variables…
The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators…
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed…
The $q$-Racah polynomials are expressed in terms of certain ratios of scalar products of Bethe states associated with Bethe equations of either homogeneous or inhomogeneous type. This result is obtained by combining the theory of Leonard…
We construct a $k$-fold $q$-series as a generating function of $k$-regular partitions for each positive integer $k$. The $k=1$ case is one of Euler's $q$-series identities pertaining to the partitions into distinct parts. The construction…
The $q$-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra $\mathfrak{osp}_q(1\vert 2)$. It turned out to be isomorphic to the Askey-Wilson algebra. In the present paper these…
We consider Problem 6.94 posed in the book Concrete Mathematics by Graham, Knuth, and Patashnik, and solve it by using bivariate exponential generating functions. The family of recurrence relations considered in the problem contains many…
In this note, we give an alternative proof of the generating function of $p$-Bernoulli numbers. Our argument is based on the Euler's integral representation.
It has been known since 2007 that the Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere,…
Let $H_{m}(z)$ be a sequence of polynomials whose generating function $\sum_{m=0}^{\infty}H_{m}(z)t^{m}$ is the reciprocal of a bivariate polynomial $D(t,z)$. We show that in the three cases $D(t,z)=1+B(z)t+A(z)t^{2}$,…
We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one free parameter to them. In particular, we generalize generating functions for the Askey-Wilson, continuous…
In this work we define a unified generating functions for 9 different kinds of set partitions including cyclically ordered set partitions. Such generating function depends on 4 parameters. We consider property of this function and provide…
Using realisations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner-Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of…