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.

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].

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 Power of a determinant with two physical applications

1999 ◽  
Vol 22 (4) ◽  
pp. 745-759 ◽  
Author(s):  
James D. Louck
Keyword(s):  
Hall Effect ◽  
Generating Function ◽  
Quantum Hall Effect ◽  
Unitary Group ◽  
Hypergeometric Series ◽  
Quantum Hall ◽  
Wave Functions ◽  
Vandermonde Determinant ◽  
Magic Square

An expression for thekth power of ann×ndeterminant inn2indeterminates(zij)is given as a sum of monomials. Two applications of this expression are given: the first is the Regge generating function for the Clebsch-Gordan coefficients of the unitary groupSU(2), noting also the relation to the 3 F2hypergeometric series; the second is to the even powers of the Vandermonde determinant, or, equivalently, all powers of the discriminant. The second result leads to an interesting map between magic square arrays and partitions and has applications to the wave functions describing the quantum Hall effect. The generalization of this map to arbitrary square arrays of nonnegative integers, having given row and column sums, is also given.

scholarly journals A Lie-theoretic interpretation of multivariate hypergeometric polynomials

2012 ◽  
Vol 148 (3) ◽  
pp. 991-1002 ◽  
Author(s):  
Plamen Iliev
Keyword(s):  
Lie Algebra ◽  
Generating Function ◽  
Hypergeometric Series ◽  
Recurrence Relations ◽  
Multinomial Distribution ◽  
Discrete Variables ◽  
Hypergeometric Polynomials

AbstractIn 1971, Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004, Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach, they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra $\mathfrak {sl}_{d+1}(\mathbb {C})$. Our approach yields yet another proof of the orthogonality. It also shows that the polynomials satisfy d independent recurrence relations each involving d2+d+1 terms. This, combined with the duality, establishes their bispectrality. We illustrate our results with several explicit examples.

scholarly journals The Generating Function of Ternary Trees and Continued Fractions

10.37236/1079 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Ira M. Gessel ◽  
Guoce Xin
Keyword(s):  
Generating Function ◽  
Continued Fraction ◽  
Hypergeometric Series ◽  
Combinatorial Proof ◽  
Hankel Determinant ◽  
Simple Method ◽  
Alternating Sign Matrices ◽  
Hankel Determinants ◽  
Fraction Method ◽  
Special Case

Michael Somos conjectured a relation between Hankel determinants whose entries ${1\over 2n+1}{3n\choose n}$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of $3$, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.

HYPERGEOMETRIC BERNOULLI POLYNOMIALS AND APPELL SEQUENCES

2008 ◽  
Vol 04 (05) ◽  
pp. 767-774 ◽  
Author(s):  
ABDUL HASSEN ◽  
HIEU D. NGUYEN
Keyword(s):  
Generating Function ◽  
Hypergeometric Series ◽  
Bernoulli Polynomials ◽  
Moment Condition ◽  
Classical Counterpart ◽  
Equivalent Definition ◽  
Appell Sequences ◽  
Zero Moment ◽  
Complete Analogy

There are two analytic approaches to Bernoulli polynomials Bn(x): either by way of the generating function zexz/(ez - 1) = ∑ Bn(x)zn/n! or as an Appell sequence with zero mean. In this article, we discuss a generalization of Bernoulli polynomials defined by the generating function zN exz/(ez - TN-1(z)), where TN(z) denotes the Nth Maclaurin polynomial of ez, and establish an equivalent definition in terms of Appell sequences with zero moments in complete analogy to their classical counterpart. The zero-moment condition is further shown to generalize to Bernoulli polynomials generated by the confluent hypergeometric series.

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 The colored Jones polynomial and Kontsevich–Zagier series for double twist knots

2021 ◽  
pp. 2150031
Author(s):  
Jeremy Lovejoy ◽  
Robert Osburn
Keyword(s):  
Generating Function ◽  
Hypergeometric Series ◽  
Jones Polynomial ◽  
Roots Of Unity ◽  
Colored Jones Polynomial ◽  
Unimodal Sequences ◽  
Jones Polynomials ◽  
Positive Integers ◽  
Colored Jones Polynomials

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots [Formula: see text] and [Formula: see text] where [Formula: see text] and [Formula: see text] are positive integers. In the [Formula: see text] case, this leads to new families of [Formula: see text]-hypergeometric series generalizing the Kontsevich–Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of [Formula: see text] gives a generalization of a duality at roots of unity between the Kontsevich–Zagier function and the generating function for strongly unimodal sequences.

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