scholarly journals Legendre-Gould Hopper-Based Sheffer Polynomials and Operational Methods

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2051
Author(s):  
Nabiullah Khan ◽  
Mohd Aman ◽  
Talha Usman ◽  
Junesang Choi
Keyword(s):  
Generating Function ◽  
Legendre Polynomials ◽  
Sheffer Polynomials ◽  
Operational Methods

A remarkably large of number of polynomials have been presented and studied. Among several important polynomials, Legendre polynomials, Gould-Hopper polynomials, and Sheffer polynomials have been intensively investigated. In this paper, we aim to incorporate the above-referred three polynomials to introduce the Legendre-Gould Hopper-based Sheffer polynomials by modifying the classical generating function of the Sheffer polynomials. In addition, we investigate diverse properties and formulas for these newly introduced polynomials.

scholarly journals Moments of the weighted Cantor measures

2019 ◽  
Vol 52 (1) ◽  
pp. 256-273
Author(s):  
Steven N. Harding ◽  
Alexander W. N. Riasanovsky
Keyword(s):  
Probability Measure ◽  
Generating Function ◽  
Exponential Decay ◽  
Legendre Polynomials ◽  
Borel Probability Measure ◽  
Moment Generating Function ◽  
Unit Interval ◽  
Uniform Error ◽  
Natural Case ◽  
Borel Probability

AbstractBased on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying {\mu ^\alpha }(E) = \sum\nolimits_{n = 0}^{N - 1} {{\alpha _n}{\mu ^\alpha }(\varphi _n^{ - 1}(E))} where ϕn : x ↦ (x + n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L_{{\mu ^\alpha }}^2 [0, 1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xm dμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first m moments within uniform error ε in O((log log(1/ε)) · m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2, 1/2].

Two new operational methods for solving a kind of fractional volterra integral equations

2016 ◽  
Vol 09 (02) ◽  
pp. 1650032 ◽  
Author(s):  
Nasibeh Mollahasani ◽  
Mahmoud Mohseni Moghadam
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Legendre Polynomials ◽  
Singular Integral Equations ◽  
Volterra Integral Equations ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations ◽  
Cas Wavelets ◽  
Operational Methods

In this paper, two methods based on CAS wavelets and Legendre polynomials are applied to approximate the solutions of a kind of fractional Volterra integral equations called weakly singular integral equations. The methods are compared presenting some examples.

scholarly journals Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments

2017 ◽  
Vol 2017 ◽  
pp. 1-4 ◽  
Author(s):  
Wei Li ◽  
Chang-Yuan Chen ◽  
Shi-Hai Dong
Keyword(s):  
Differential Equation ◽  
Generating Function ◽  
Legendre Polynomials ◽  
Present Method ◽  
Special Functions ◽  
Analytical Result

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is,∫-11Pl′m′xt-1/1+t2-2xtPk′m′(x)/(1+t2-2tx)(l′+1)/2dx, wheret∈(0,1). The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument∫0∞Jn(αx2+z2)/(x2+z2)nx2m+1dx.

scholarly journals Certain results associated with hybrid relatives of the q-Sheffer sequences

2022 ◽  
Vol 40 ◽  
pp. 1-15
Author(s):  
Subuhi Khan ◽  
Tabinda Nahid
Keyword(s):  
Numerical Simulations ◽  
Generating Function ◽  
Orthogonality Condition ◽  
Appell Polynomials ◽  
New Class ◽  
Sheffer Sequences ◽  
Sheffer Polynomials

The intended objective of this paper is to introduce a new class of the hybrid q-Sheffer polynomials by means of the generating function and series definition. The determinant definition and other striking properties of these polynomials are established. Certain results for the continuous q-Hermite-Appell polynomials are obtained. The graphical depictions are performed for certain members of the hybrid q-Sheffer family. The zeros of these members are also explored using numerical simulations. Finally, the orthogonality condition for the hybrid q-Sheffer polynomials is established.

Operational rules and generalized special polynomials

2020 ◽  
Vol 32 (2) ◽  
pp. 269-277
Author(s):  
Mahvish Ali
Keyword(s):  
Generating Function ◽  
Special Polynomials ◽  
Sheffer Polynomials

AbstractIn this work, the generalized family of Hermite–Sheffer polynomials is introduced by using Euler’s integral and operational rules. Furthermore, some properties are established. In particular, generating function and determinant definition for the generalized Hermite–Sheffer polynomials are obtained. Some examples are also considered and their corresponding results are established.

scholarly journals A GENERATING FUNCTION OF THE SQUARES OF LEGENDRE POLYNOMIALS

2013 ◽  
Vol 89 (1) ◽  
pp. 125-131 ◽  
Author(s):  
WADIM ZUDILIN
Keyword(s):  
Generating Function ◽  
Legendre Polynomials ◽  
Hypergeometric Series ◽  
Modular Function

AbstractWe relate a one-parametric generating function for the squares of Legendre polynomials to an arithmetic hypergeometric series whose parametrisation by a level 7 modular function was recently given by Cooper. By using this modular parametrisation we resolve a subfamily of identities involving$1/ \pi $which was experimentally observed by Sun.

scholarly journals Discrete Hypergeometric Legendre Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2546
Author(s):  
Tom Cuchta ◽  
Rebecca Luketic
Keyword(s):  
Generating Function ◽  
Difference Equations ◽  
Legendre Polynomials ◽  
Hypergeometric Series ◽  
Recurrence Relations ◽  
Discrete Analog

A discrete analog of the Legendre polynomials defined by discrete hypergeometric series is investigated. The resulting polynomials have qualitatively similar properties to classical Legendre polynomials. We derive their difference equations, recurrence relations, and generating function.

Sign in / Sign up

Export Citation Format

Share Document