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We give an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, we list a number of summations, transformations and explicit…

We introduce sequences of functions orthogonal on a finite interval: proper orthogonal rational functions, orthogonal exponential functions, orthogonal logarithmic functions, and transmuted orthogonal polynomials

This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on…

We introduce new classes of right quaternionic Hilbert spaces of Bargmann-Fock type $\mathcal{GB}_{m}^{2}(\mathbb{H})$, labeled by nonnegative integer $m$, generalizing the so-called slice hyperholomorphic Bargmann-Fock space introduced…

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their…

In this paper, bivariate Szasz-Mirakjan type operators are introduced along with the estimation of its approximation properties and its rate of convergence. Furthermore, to check the asymptotic behavior of the said bivariate operators, the…

In 1916, MacMahon showed that permutations in $S_n$ with a fixed descent set $I$ are enumerated by a polynomial $d_I(n)$. Diaz-Lopez, Harris, Insko, Omar, and Sagan recently revived interest in this descent polynomial, and suggested the…

We study the algebraic properties of plethystic vertex operators, introduced in J. Phys. A: Math. Theor. 43 405202 (2010), underlying the structure of symmetric functions associated with certain generalized universal character rings of…

We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation of GL(m) over a number field with unitary central character. We investigate the decay rate of…

Our main aim is to investigate the approximation properties for the summation integral type operators in a statistical sense. In this regard, we prove the statistical convergence theorem using well known Korovkin theorem and the degree of…

We study non-Hermitian integrable fermion and boson systems from the perspectives of Grothendieck polynomials. The models considered in this article are the five-vertex model as a fermion system and the non-Hermitian phase model as a boson…

In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic…

In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\to1$. These $q$-series generalize Ramanujan function $A_{q}(z)$; Jackson's $q$-Bessel function $J_{\nu}^{(2)}$(z;q), Ismail-Masson orthogonal…

We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$.…

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K…

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical…

Let $\mathfrak g$ be an infinite-dimensional Lie algebra, and $G$ be the algebraic completion of a $\mathfrak g$-module. Using the geometric model of Schottky uniformization of Riemann sphere to obtain a higher genus Riemann surface, we…

A class of pseudo-hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum…

We derive the asymptotic distribution of ordinal-pattern frequencies under weak dependence conditions and investigate the long-run covariance matrix not only analytically for moving-average, Gaussian, and the novel generalized coin-tossing…

Distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive Hermitian matrices. The inverse Fourier transform of the distribution is shown to be a Fredholm determinant of a…