Related papers: Solving conformable Gegenbauer differential equati…
We introduce the definition of conformable derivative on time scales and develop its calculus. Fundamental properties of the conformable derivative and integral on time scales are proved. Linear conformable differential equations with…
A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
It is known that difference equations generated as the Newton-Raphson iteration for quadratic equations are solvable in closed form, and the solution can be constructed from linear three-term recurrence relations with constant coefficients.…
We investigate a class of power series occurring in some problems in quantum optics. Their coefficients are either Gegenbauer or Laguerre polynomials multiplied by binomial coefficients. Although their sums have been known for a long time,…
We develop the theory of conformal blocks in CFT_d expressing them as power series with Gegenbauer polynomial coefficients. Such series have a clear physical meaning when the conformal block is analyzed in radial quantization: individual…
Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function…
Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the…
This paper presents a somewhat exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric equation (CFGHE) about the fractional regular singular…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions.…
Convergence of the Gauss resolution process for a complex singular foliation of dimension r is shown to be equivalent to finite type of a graded sheaf which is built using base (r+2) expansions of integers. As applications it is calculated…
We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras.…
We solve a problem of Marshal Hall by proving the convergence of the catalan number generating function directly on the basis of its recursion formula.
Using the direct relation between the Gegenbauer polynomials and the Ferrers function of the first kind, we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and the Ferrers function of the first kind. We then…
In this article, we study the set of all solutions of linear differential equations using Hurwitz series. We first obtain explicit recursive expressions for solutions of such equations and study the group of differential automorphisms of…
In this work, we illustrate and explore the use of Taylor series as solutions of differential equations. For a large a number of classes of differential equations in the literature, there are plenty of sources where the well known Taylor…
The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions and equivariant forms.
A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little…
In this paper, we study the existence of positive solutions for a class of conformable fractional differential equations with integral boundary conditions. By using the properties of the Green's function and the fixed point theorem in a…