Related papers: On $q$-Middle Convolution and $q$-Hypergeometric E…
Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a…
The metohod of ortogonal rotations introduced in the previous papers of the author is used for construction of the explicit form the generators of the simple roots for quantum (and ussual) semisimple algebras. All calculations are presented…
We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric…
In this paper, we would like to study the linear Cauchy problems for semi-linear $\sigma$-evolution models with mixing a parabolic like damping term corresponding to $\sigma_1 \in [0,\sigma/2)$ and a $\sigma$-evolution like damping…
Earlier work introduced a method for obtaining indefinite $q$-integrals of $q$-special functions from the second-order linear $q$-difference equations that define them. In this paper, we reformulate the method in terms of $q$-Riccati…
In this current article, we introduce the quadruple Shehu transform and its inverse. We also introduce some properties of quadruple Shehu transform. The Convolution theorem and its proof are also discussed. Further, to solve homogeneous and…
We introduce a kind of finite truncation of the hypergeometric series and provide its discretized integral representation. This is motivated by recent results of Maesaka-Seki-Watanabe and Hirose-Matsusaka-Seki on the identity between…
In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is…
The duality relation of one-variable multiple polylogarithms was proved by Hirose, Iwaki, Sato and Tasaka by means of iterated integrals. In this paper, we give a new proof using the method of connected sums, which was recently invented by…
In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric…
We give a Jackson integral representation for Kajihara's $q$-hypergeometric series $W^{M,2}$. We construct a $q$-difference system that corresponds to this integral. This system is an extension of the variant of $q$-hypergeometric equation…
This paper uses the convolution theorem of the Laplace transform to derive new inverse Laplace transforms for the product of two parabolic cylinder functions in which the arguments may have opposite sign. These transforms are subsequently…
Notions of the orthogonality and convolution orthogonality are explored with the use of the Kontorovich-Lebedev transform and its convolution. New classes of the corresponding orthogonal polynomials and functions are investigated. Integral…
In this work, generalized hypergeometric functions for bicomplex argument is introduced and its convergence criteria is derived. Furthermore, integral representation of this function has been established. Moreover, quadratic transformation,…
Let $(Q,\sigma)$ be a symmetric quiver, where $Q=(Q_0,Q_1)$ is a finite quiver without oriented cycles and $\sigma$ is a contravariant involution on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $<,>$ on…
We introduce a family of algebras which are multiplicative analogues of preprojective algebras, and their deformations, as introduced by M. P. Holland and the first author. We show that these algebras provide a natural setting for the…
In this paper, we establish a $q$-integral formula by using the orthogonality relation, and also provide a new proof of the $q$-orthogonality relation for the continuous $q$-ultraspherical polynomials. A new $q$-beta integral with five…
For infinite-dimensional groups $G\supset K$ the double cosets $K\setminus G/K$ quite often admit a structure of a semigroup; these semigroups act in $K$-fixed vectors of unitary representations of $G$. We show that such semigroups can be…
As part of our study of the $q$-tetrahedron algebra $\boxtimes_q$ we introduce the notion of a $q$-inverting pair. Roughly speaking, this is a pair of invertible semisimple linear transformations on a finite-dimensional vector space, each…
In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\binom{n}{2}}$ with $q\in(0,1)$ fixed. To make sense of these formal…