相关论文: Hypergeometric solutions for variants of the $q$-h…
In recent years, Karr's difference field theory has been extended to the so-called $R\Pi\Sigma$-extensions in which one can represent not only indefinite nested sums and products that can be expressed by transcendental ring extensions, but…
We consider the q-derivatives of the Srivastava and Daoust basic multivariable hypergeometric function with respect to the parameters. This function embodies a entire number of various q-hypergeometric series of one and several variables.…
General structure of the multivariate plain and q-hypergeometric terms and univariate elliptic hypergeometric terms is described. Some explicit examples of the totally elliptic hypergeometric terms leading to multidimensional integrals on…
In this paper, we mainly establish two supercongruences involving truncated hypergeometric series by using some hypergeometric transformation formulas. The first supercongruence confirms a recent conjecture of the second author. The second…
A $q$-difference analog of the sixth Painlev\'e equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear…
The paper discusses the summability of formal solutions of some linear q-difference-differential equations, and improves the previous result in [Tahara-Yamazawa, Opsucula Math. 35 (2015), 713-738].
In this paper, using the theory of the so-called fractional calculus we show that it is possible to easily obtain the solutions for the confluent hypergeometric equation. Our approach is to be compared with the standard one (Frobenius)…
In this work we establish new forms of Heun-to-Heun transformations and Heun-to-Hypergeometric transformations. The transformations are realised by changing the independent variable in a non-linear way. Using these we also point out some…
We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for…
We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in…
It is suggested that the (p,q)-hypergeometric series studied by Burban and Klimyk (in Integral Transforms and Special Functions, 2 (1994) 15 - 36) can be considered as a special case of a more general (P,Q)-hypergeometric series.
We introduce derivations on the algebra of multiple harmonic q-series and show that they generate linear relations among the q-series which contain the derivation relations for a q-analogue of multiple zeta values due to Bradley. As a…
We study the divergent basic hypergeometric series which is a $q$-analog of divergent hypergeometric series. This series formally satisfies the linear $q$-difference equation. In this paper, for that equation, we give an actual solution…
We show that the Schr\"odinger and Klein-Gordon equations can both be derived from an Hypergeometric differential equation. The same applies to non linear generalizations of these equations.
In this note we solve a problem about the rational representablility of hupergeometric terms which represent hypergeometric sums. This problem was proposed by Koornwinder in [4].
We describe a new approach to the notion of general hypergeometric functions
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
A $q$-deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear $q$-difference equation associated with the $q$-Painlev\'e VI equation. Then we obtain integral transformations. We investigate the…
We aim to introduce a new extension of beta function and to study its important properties. Using this definition, we introduce and investigate new extended hypergeometric and confluent hypergeometric functions. Further, some hybrid…
We obtain some properties of a class $\mathcal{A}$ of $q$-hypergeometric orthogonal polynomials with $q=-1$, described by a uniform parametrization of the recurrence coefficients. We construct a class $\mathcal{C}$ of complementary $-1$…