scholarly journals Some Types of Identities Involving the Legendre Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 114 ◽  
Author(s):  
Shimeng Shen ◽  
Li Chen
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Computational Problem ◽  
Non Linear ◽  
Recursive Sequence ◽  
Interesting Identity ◽  
Combinatorial Methods

In this paper, a new non-linear recursive sequence is firstly introduced. Then, using this sequence, a computational problem involving the convolution of the Legendre polynomial is studied using the basic and combinatorial methods. Finally, we give an interesting identity.

scholarly journals A New Identity Involving the Chebyshev Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 244 ◽  
Author(s):  
Yixue Zhang ◽  
Zhuoyu Chen
Keyword(s):  
Chebyshev Polynomials ◽  
Second Order ◽  
Simple Problem ◽  
Computational Problem ◽  
Non Linear ◽  
Recursive Sequence ◽  
Interesting Identity ◽  
Combinatorial Methods

In this paper, firstly, we introduced a second order non-linear recursive sequence, then we use this sequence and the combinatorial methods to perform a deep study on the computational problem concerning one kind sums, which includes the Chebyshev polynomials. This makes it possible to simplify a class of complex computations involving the second type Chebyshev polynomials into a very simple problem. Finally, we give a new and interesting identity for it.

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.

scholarly journals Some Identities Involving the Fubini Polynomials and Euler Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 300 ◽  
Author(s):  
Guohui Chen ◽  
Li Chen
Keyword(s):  
Power Series ◽  
Second Order ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Non Linear ◽  
Combinatorial Methods

In this paper, we first introduce a new second-order non-linear recursive polynomials U h , i ( x ) , and then use these recursive polynomials, the properties of the power series and the combinatorial methods to prove some identities involving the Fubini polynomials, Euler polynomials and Euler numbers.

scholarly journals Numerical study of heat transfer of a micropolar fluid through a porous medium with radiation

2018 ◽  
Vol 22 (1 Part B) ◽  
pp. 557-565 ◽  
Author(s):  
Fakhrodin Mohammadi ◽  
Mohammad Rashidi
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Differential Equations ◽  
Collocation Method ◽  
Micropolar Fluid ◽  
Legendre Polynomials ◽  
Spectral Collocation Method ◽  
Spectral Collocation ◽  
Shifted Legendre Polynomials ◽  
Non Linear

An efficient Spectral Collocation method based on the shifted Legendre polynomials was applied to get solution of heat transfer of a micropolar fluid through a porous medium with radiation. A similarity transformation is applied to convert the governing equations to a system of non-linear ordinary differential equations. Then, the shifted Legendre polynomials and their operational matrix of derivative are used for producing an approximate solution for this system of non-linear differential equations. The main advantage of the proposed method is that the need for guessing and correcting the initial values during the solution procedure is eliminated and a stable solution with good accuracy can be obtained by using the given boundary conditions in the problem. A very good agreement is observed between the obtained results by the proposed Spectral Collocation method and those of previously published ones.

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 Some Identities Involving Fibonacci Polynomials and Fibonacci Numbers

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 334 ◽  
Author(s):  
Yuankui Ma ◽  
Wenpeng Zhang
Keyword(s):  
Power Series ◽  
Structural Properties ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Fibonacci Polynomials ◽  
Recursive Sequence ◽  
Combinatorial Methods

The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using this new sequence, the properties of the power series, and the combinatorial methods.

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.

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 A New Identity Involving Balancing Polynomials and Balancing Numbers

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1141 ◽  
Author(s):  
Yuanyuan Meng
Keyword(s):  
Power Series ◽  
Structural Properties ◽  
Second Order ◽  
Balancing Numbers ◽  
Recursive Sequence ◽  
Combinatorial Methods

In this paper, a second-order nonlinear recursive sequence M ( h , i ) is studied. By using this sequence, the properties of the power series, and the combinatorial methods, some interesting symmetry identities of the structural properties of balancing numbers and balancing polynomials are deduced.

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