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We study the asymptotics of fundamental solutions of p-adic pseudo-differential equations connected with homogeneous polynomials by using techniques of local zeta functions theory.
In this paper we present a decision procedure for computing pFq hypergeometric solutions for third order linear ODEs, that is, solutions for the classes of hypergeometric equations constructed from the 3F2, 2F2, 1F2 and 0F2 standard…
In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit…
We show that in four particular cases the derivative of the solution of Heun general equation can be expressed in terms of a solution to another Heun equation. Starting from this property, we use the Gauss hypergeometric functions to…
A method for converting the geometrical problem of rectangle packing to an algebraic problem of solving a system of polynomial equations is described.
We give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
We consider the Hirota equation (the discrete analog of the generalized Toda system) over a finite field. We present the general algebro-geometric method of construction of solutions of the equation. As an example we construct analogs of…
A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure…
In this paper, we give the analytic expression for the homogeneous part of solutions of arbitrary tree-level cosmological correlators, including massive propagators and time-derivative interaction cases. The solutions are given in the form…
We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.
We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems A_n, C_n and D_n. In the special cases of classical and q-series, our approach leads to new elementary proofs of the…
Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution…
In this short note, we describe a problem in algebraic geometry where the solution involves Catalan numbers. More specifically, we consider the derived category of coherent sheaves on an elliptic surface, and the action of its…
We provide an explicit formulation for the solution to the Catalan's triangle system using Catalan's trapezoids and a specified boundary condition. Additionally, we study this system with various boundary conditions obtained by utilizing…
In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=\Delta u+v^p,\qquad & x\in\Omega,\,\,\,t>0,\\ \partial_t…
In this paper we present applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. According to variational approach in the general case we have the solution as a…
This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation and…
Using a mixture of classical and probabilistic techniques we investigate the convexity of solutions to the elliptic pde associated with a certain generalized Ornstein-Uhlenbeck process.
Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey-Wilson, Wilson and continuous Hahn families.