Related papers: Rational solutions of polynomail-exponential equat…
The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger like one. Earlier results are…
For a rational function of several variables with nonnegative imaginary part on the upper poly-half-plane, the matrix representations are obtained.
This note gives a simple approach to q-analogues of some results associated with Abel polynomials.
We introduce sequences of functions orthogonal on a finite interval: proper orthogonal rational functions, orthogonal exponential functions, orthogonal logarithmic functions, and transmuted orthogonal polynomials
We study nonconstant rational solutions of \[ x'=A_3(t)x^{n_3}+A_2(t)x^{n_2}+A_1(t)x^{n_1}, \qquad 1<n_1<n_2<n_3, \] with $A_i\in\Bbbk[t]$, $\Bbbk\in\{\mathbb R,\mathbb C\}$. We prove that every such solution is of the form $x=1/p(t)$, and…
The paper concerns singular solutions of nonlinear elliptic equations.
We establish the boundedness character of solutions of a system of rational difference equations with a variable coefficient
This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear…
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…
The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…
A determinant expression for the rational solutions of the Painlev\'e III (P$_{\rm III}$) equation whose entries are the Laguerre polynomials is given. Degeneration of this determinant expression to that for the rational solutions of…
We consider algebraic ordinary differential equations (AODEs) and study their polynomial and rational solutions. A sufficient condition for an AODE to have a degree bound for its polynomial solutions is presented. An AODE satisfying this…
We provide an analytical closed-form solution of the exponential equation $a^x+a^{-x}=x$ for a specific value $a$, discuss the number of roots in general case, and provide bounds on the roots.
We claim to resolve the P=?NP problem via a formal argument for P=NP.
We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.
A formulation of the Maxwell equations in terms of the split octonions is presented.
Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
Combinatorial formulas expressing cyclic rook polynomials and cyclic permanents of rectangular matrices in terms of expansions along rows are presented
We give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.