scholarly journals Filter integrals for orthogonal polynomials

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
T Amdeberhan ◽  
Adriana Duncan ◽  
Victor H Moll ◽  
Vaishavi Sharma
Keyword(s):  
Orthogonal Polynomials ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Gegenbauer Polynomials

Motivated by an expression by Persson and Strang on an integral involving Legendre polynomials, stating that the square of $P_{2n+1}(x)/x$ integrated over $[-1,1]$ is always $2$, we present analog results for Hermite, Chebyshev, Laguerre and Gegenbauer polynomials as well as the original Legendre polynomial with even index.

scholarly journals Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
David W. Pravica ◽  
Njinasoa Randriampiry ◽  
Michael J. Spurr
Keyword(s):  
Fourier Transform ◽  
Theta Function ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Inverse Fourier Transform ◽  
Recursion Relations ◽  
Jacobi Theta Function ◽  
Convergence Results ◽  
The Family ◽  
Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

LEGENDRE POLYNOMIALS AS THE NORMALIZATION OF PHOTON-SUBTRACTED SQUEEZED STATES

2009 ◽  
Vol 24 (20) ◽  
pp. 1597-1603 ◽  
Author(s):  
HONG-YI FAN ◽  
LI-YUN HU ◽  
XUE-XIANG XU
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Hermite Polynomial ◽  
Vacuum State ◽  
Squeezed States ◽  
Squeezed State ◽  
Normalization Factor ◽  
Squeezed Vacuum State ◽  
Squeezed Vacuum ◽  
Excitation State

By converting the photon-subtracted squeezed state (PSSS) to a squeezed Hermite-polynomial excitation state we find that the normalization factor of PSSS is an m-order Legendre polynomial of the squeezing parameter, where m is the number of subtracted photons. Some new relations about the Legendre polynomials are obtained by this analysis. We also show that the PSSS can also be treated as a Hermite-polynomial excitation on squeezed vacuum state.

scholarly journals Random regression models to estimate genetic parameters for milk production of Guzerat cows using orthogonal Legendre polynomials

2014 ◽  
Vol 49 (5) ◽  
pp. 372-383 ◽  
Author(s):  
Maria Gabriela Campolina Diniz Peixoto ◽  
Daniel Jordan de Abreu Santos ◽  
Rusbel Raul Aspilcueta Borquis ◽  
Frank Ângelo Tomita Bruneli ◽  
João Cláudio do Carmo Panetto ◽  
...  
Keyword(s):  
Milk Production ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Regression Models ◽  
Environmental Effect ◽  
Genetic Parameters ◽  
Genetic Effect ◽  
Random Regression ◽  
Additive Genetic Effect ◽  
Fifth Order

The objective of this work was to compare random regression models for the estimation of genetic parameters for Guzerat milk production, using orthogonal Legendre polynomials. Records (20,524) of test-day milk yield (TDMY) from 2,816 first-lactation Guzerat cows were used. TDMY grouped into 10-monthly classes were analyzed for additive genetic effect and for environmental and residual permanent effects (random effects), whereas the contemporary group, calving age (linear and quadratic effects) and mean lactation curve were analized as fixed effects. Trajectories for the additive genetic and permanent environmental effects were modeled by means of a covariance function employing orthogonal Legendre polynomials ranging from the second to the fifth order. Residual variances were considered in one, four, six, or ten variance classes. The best model had six residual variance classes. The heritability estimates for the TDMY records varied from 0.19 to 0.32. The random regression model that used a second-order Legendre polynomial for the additive genetic effect, and a fifth-order polynomial for the permanent environmental effect is adequate for comparison by the main employed criteria. The model with a second-order Legendre polynomial for the additive genetic effect, and that with a fourth-order for the permanent environmental effect could also be employed in these analyses.

scholarly journals Bayesian B-spline mapping for dynamic quantitative traits

2012 ◽  
Vol 94 (2) ◽  
pp. 85-95 ◽  
Author(s):  
JUN XING ◽  
JIAHAN LI ◽  
RUNQING YANG ◽  
XIAOJING ZHOU ◽  
SHIZHONG XU
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Simulated Dataset ◽  
Model Parameters ◽  
B Spline ◽  
B Splines ◽  
Bayesian Mapping ◽  
Bayesian Shrinkage ◽  
Dynamic Traits ◽  
Over Time

SummaryOwing to their ability and flexibility to describe individual gene expression at different time points, random regression (RR) analyses have become a popular procedure for the genetic analysis of dynamic traits whose phenotypes are collected over time. Specifically, when modelling the dynamic patterns of gene expressions in the RR framework, B-splines have been proved successful as an alternative to orthogonal polynomials. In the so-called Bayesian B-spline quantitative trait locus (QTL) mapping, B-splines are used to characterize the patterns of QTL effects and individual-specific time-dependent environmental errors over time, and the Bayesian shrinkage estimation method is employed to estimate model parameters. Extensive simulations demonstrate that (1) in terms of statistical power, Bayesian B-spline mapping outperforms the interval mapping based on the maximum likelihood; (2) for the simulated dataset with complicated growth curve simulated by B-splines, Legendre polynomial-based Bayesian mapping is not capable of identifying the designed QTLs accurately, even when higher-order Legendre polynomials are considered and (3) for the simulated dataset using Legendre polynomials, the Bayesian B-spline mapping can find the same QTLs as those identified by Legendre polynomial analysis. All simulation results support the necessity and flexibility of B-spline in Bayesian mapping of dynamic traits. The proposed method is also applied to a real dataset, where QTLs controlling the growth trajectory of stem diameters in Populus are located.

scholarly journals On an Inequality for Legendre Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2044
Author(s):  
Florin Sofonea ◽  
Ioan Ţincu
Keyword(s):  
Orthogonal Polynomials ◽  
Lower Bounds ◽  
Legendre Polynomials ◽  
Probability Distributions ◽  
Upper And Lower Bounds ◽  
Discrete Probability ◽  
Discrete Probability Distributions

This paper is concerned with the orthogonal polynomials. Upper and lower bounds of Legendre polynomials are obtained. Furthermore, entropies associated with discrete probability distributions is a topic considered in this paper. Bounds of the entropies which improve some previously known results are obtained in terms of inequalities. In order to illustrate the results obtained in this paper and to compare them with other results from the literature some graphs are provided.

scholarly journals Old and New Identities for Bernoulli Polynomials via Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Luis M. Navas ◽  
Francisco J. Ruiz ◽  
Juan L. Varona
Keyword(s):  
Fourier Series ◽  
Linear Combination ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Gegenbauer Polynomials ◽  
Linear Combinations ◽  
The Given

The Bernoulli polynomialsBkrestricted to[0,1)and extended by periodicity haventh sine and cosine Fourier coefficients of the formCk/nk. In general, the Fourier coefficients of any polynomial restricted to[0,1)are linear combinations of terms of the form1/nk. If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.

Numerical Solution of Eighth-order Boundary Value Problems by Using Legendre Polynomials

2017 ◽  
Vol 15 (02) ◽  
pp. 1750083 ◽  
Author(s):  
Anna Napoli ◽  
Waleed M. Abd-Elhameed
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Boundary Value ◽  
Operational Matrix ◽  
Harmonic Numbers ◽  
Eighth Order ◽  
Numerical Examples ◽  
Derivatives Of

The main aim of this paper is to present and analyze a numerical algorithm for the solution of eighth-order boundary value problems. The proposed solutions are spectral and they depend on a new operational matrix of derivatives of certain shifted Legendre polynomial basis, along with the application of the collocation method. The nonzero elements of the operational matrix are expressed in terms of the well-known harmonic numbers. Numerical examples provide favorable comparisons with other existing methods and ascertain the efficiency and applicability of the proposed algorithm.

scholarly journals Derivation of New Class of Orthogonal Polynomials with Recurrence Relation

2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Emmanuel Adeyefa ◽  
Raphael Adeniyi ◽  
Olatunde Olanayo ◽  
Yahaya Haruna ◽  
John Oladunjoye
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Legendre Polynomials ◽  
Basic Properties ◽  
New Class

This paper presents the derivation of a new class of orthogonal polynomials named ADEM-B orthogonal polynomials, valid in the interval [-1, 1] with respect to weight function. The analysis of some basic properties of the polynomials shows that the polynomials are symmetrical depending on whether index n in  is even or odd. The recurrence relation of the class of the polynomials is presented and a brief review of the formulation of existing scheme is considered to test the applicability of the polynomials. Findings reveal that these polynomials produce the same results as in zeros of Chebyshev and Legendre polynomials.

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