Related papers: Differential equations associated with Lambda-Chan…
We discuss the construction of oscillator-like systems associated with orthogonal polynomials on the example of the Fibonacci oscillator. In addition, we consider the dimension of the corresponding lie algebras.
We construct 2x2-matrix linear problems with a spectral parameter for the Painleve equations I-V by means of the degeneration processes from the elliptic linear problem for the Painleve VI equation. These processes supplement the known…
I present a $q$-analog of the discrete Painlev\'e I equation, and a special realization of it in terms of $q$-orthogonal polynomials.
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
In this paper we presents an algorithm for finding a solution of the linear nonhomogeneous quaternionic-valued differential equations. Moveover, several examples shows the feasibility of our algorithm.
In this paper we first construct an analytic realization of the $C_\lambda$-extended oscillator algebra with the help of difference-differential operators. Secondly, we study families of $d$-orthogonal polynomials which are extensions of…
In a previous paper we deformed Hermite polynomials to three associated polynomials .Here we apply the same deformation to Laguerre polynomials .
Many authors have recently studied the degenerate harmonic numbers. This paper makes two main contributions. First, we derive several explicit expressions for these numbers, which are a degenerate version of the ordinary harmonic numbers.…
We use a recently found method to characterise all the invertible fourth-order difference equations linear in the extremal values based on the existence of a discrete Lagrangian. We also give some result on the integrability properties of…
Rational solutions of the fourth order analogue to the Painlev'e equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulas for their coefficients…
The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study…
A new kind of deformed calculus was introduced recently in studying of parabosonic coordinate representation. Based on this deformed calculus, a new deformation of Legendre polynomials is proposed in this paper, some properties and…
In this research paper, structured bi-matrix variate, matrix quadratic equations are considered. Some lemmas related to determining the eigenvalues of unknown matrices are proved. Also, a method of determining the diagonalizabe unknown…
We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…
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…
We derive some identities and relations and extremal problems and minimization and Fourier development involving of integral Legendre polynomials.
We study multiple orthogonal polynomials exploiting their explicit determinantal representation in terms of moments. Our reasoning follows that applied to solve the Hermite-Pad\'{e} approximation and interpolation problems. We study also…
In this note we extend the classical relation between the equilibrium configurations of unit movable point charges in a plane electrostatic field created by these charges together with some fixed point charges and the polynomial solutions…
The construction of stochastic solutions for nonlinear partial differential equations is a powerful method to obtain new exact results and to develop efficient numerical algorithms, in particular when domain decomposition techniques are…