Related papers: A physicist's guide to the solution of Kummer's eq…
We consider an equation $$ L_{\alpha ,\beta ,\gamma} (u) \equiv u_{xx} + u_{yy} + u_{zz} + \displaystyle \frac{{2\alpha}}{x}u_x + \displaystyle \frac{{2\beta}}{y}u_y + \displaystyle \frac{{2\gamma}}{z}u_z = 0 $$ in a domain ${\bf R}_3^ +…
We present a detailed derivation of the continuity, Euler, and energy balance equations from many particle Schrodinger equation. Interparticle interaction is explicitly considered as the Coulomb interaction. We show the QHD equations in a…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
In a recent paper \cite{Temme:2021:AKH} new asymptotic expansions are given for the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed and special attention for the case $a\sim b$. In this…
The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formul\ae\ for Gauss' hypergeometric functions. We define the tau-function of such a system as the…
In this article we describe the novel method to construct fundamental solutions for operators with variable coefficients. That method was introduced in "A note on the fundamental solution for the Tricomi-type equation in the hyperbolic…
We first show that hypergeometric functions appear naturally as spectral functions when applying pseudo-differential calculus to decipher heat kernel asymptotic in the situation where the symbol algebra is noncommutative. Such observation…
In this article, we undertake the study of the function $\mathscr{F}(x;u,v)$, which we refer to as the Herglotz-Zagier-Novikov function. This function appears in Novikov's work on the Kronecker limit formula, which was motivated by Zagier's…
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,…
The mathematical model representing the equation of motion of a pendulum is nonlinear. Solutions that satisfy the equation cannot be represented by elementary functions, such as trigonometric functions. To solve such problems, it is common…
A `trinomial hypersurface' is a hypersurface that is defined by a single polynomial having 3 non-constant terms in it and no constant term. A `disjoint-term trinomial hypersurface' is a trinomial hypersurface whose defining polynomial has…
In Slater's 1960 standard work on confluent hypergeometric functions, also called Kummer functions, a number of asymptotic expansions of these functions can be found. We summarize expansions derived from a differential equation for large…
The main aim of the present work is to give some interesting the $q$-analogues of various $q$-recurrence relations, $q$-recursion formulas, $q$-partial derivative relations, $q$-integral representations, transformation and summation…
The Tricomi equation is a second-order partial differential equation of mixed elliptic-hyperbolic type. It was first analyzed in the work by Francesco Giacomo Tricomi (1923) on the well-posedness of a boundary value problem. The Tricomi…
In this paper hyperbolic partial differential equations with random coefficients are discussed. Such random partial differential equations appear for instance in traffic flow problems as well as in many physical processes in random media.…
The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the Lie algebra $sl_2$ is a system of linear difference equations with values in a tensor product of $sl_2$ Verma modules. We solve the equation in terms…
By making use of some techniques based upon certain inverse new pairs of symbolic operators, the author investigate several decomposition formulas associated with Humbert hypergeometric functions $\Phi_1 $, $\Phi_2 $, $\Phi_3 $, $\Psi_1 $,…
The great success of the theory of hypergeometric series in one variable has stimulated the development of a corresponding theory in two and more variables. Horn has investigated the convergence of 34 (14 complete and 20 confluent)…
Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the…
The paper is an investigation of the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These…