S. Sadov
Double integrals that represent matrix elements of the power and logarithmic potentials in the Legendre polynoiomial basis on [-1,1] are found in a closed form. Several proofs are given, which involve different special functions and identities. A connection of the new formulae and Whipple's hypergeometric summation formula is shown.
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Expressions for the derivatives of the Legendre polynomials of the first kind with respect to the order of these polynomials are given. An explicit form for the fourth derivative is presented.
We show that the multiwavelets, introduced by Alpert in 1993, are related to type I Legendre-Angelesco multiple orthogonal polynomials. We give explicit formulas for these Legendre-Angelesco polynomials and for the Alpert multiwavelets. The multiresolution analysis can be done entirely using Legendre polynomials, and we give an algorithm, using Cholesky factorization, to compute the multiwavelets and a method, using the Jacobi matrix for Legendre polynomials, to compute the matrices in the scaling relation for any size of the multiplicity of the multiwavelets.
We give a remarkable additional orthogonality property of the classical Legendre polynomials on the real interval $[-1,1]$: polynomials up to degree $n$ from this family are mutually orthogonal under the arcsine measure weighted by the degree-$n$ normalized Christoffel function.
In this paper, we study the scaling properties of Legendre polynomials Pn(x). We show that Pn(ax), where a is a constant, can be expanded as a sum of either Legendre polynomials Pn(x) or their multiple derivatives dkPn(x)/dxk, and we derive a general expression for the expansion coefficients. In addition, we demonstrate that the multiple derivative dkPn(x)/dxk can also be expressed as a sum of Legendre polynomials and we obtain a recurrence relation for the coefficients.
In this paper, we show that the Dickson polynomials of the third kind satisfy a nonhomogeneous second order linear ordinary differential equation whose general solution contains Legendre functions.
We present a geometric approach to the asymptotics of the Legendre polynomials $P_{k,n+1}$, based on the Szegö kernel of the Fermat quadric hypersurface, and leading to complete asymptotic expansions holding on expanding subintervals of $[-1,1]$.
In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given.
A simple derivation of the integrals of products of Legendre polynomials with logarithmic weight
该研究针对带对数权的勒让德多项式乘积积分问题，给出了阶数为m、n的勒让德多项式对应区间[0,1]上含对数权乘积积分的简单推导方法。
A Mellin transform representation is derived for the associated Legendre functions of the first kind on the cut
Clifford-Legendre and Clifford-Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on $m$-dimensional euclidean space ${\mathbb R}^m$ and taking values in the associated Clifford algebra ${\mathbb R}_m$. New recurrence and Bonnet type formulae for these polynomials are proved, as their Fourier transforms are computed. Explicit representations in terms of spherical monogenics and Jacobi polynomials are given, with consequences including the interlacing of the zeros. In the case $m=2$ we describe a degeneracy between the even- and odd-indexed p…
On a pointwise inequality for even Legendre polynomials in high dimensional spheres
本文针对高维球面建立含谱间隙效应的相邻偶 Legendre 多项式逐点不等式，从而改进并支撑 Boltzmann 方程 Fisher 信息研究中的关键结果。
Persson and Strang (2003) evaluated the integral over [-1,1] of a squared odd degree Legendre polynomial divided by x^2 as being equal to 2. We consider a similar integral for orthogonal polynomials with respect to a general even orthogonality measure, with Gegenbauer and Hermite polynomials as explicit special cases. Next, after a quadratic transformation, we are led to the general nonsymmetric case, with Jacobi and Laguerre polynomials as explicit special cases. Examples of indefinite summation also occur in this context. The paper concludes with a generalization of the earlier results for H…
Addition formula for big q-Legendre polynomials
从Koornwinder关于大$q$ -勒让德多项式为量子$SU(2)$群上的球元的解释出发，导出了大$q$ -勒让德多项式的加法公式.该公式涉及Al-Salam-Carlitz多项式、小$q$ -雅可比多项式和对偶$q$ -Krawtchouk多项式。对于小的$q$ -超球多项式，一个关于大的$q$ -勒让德多项式的乘积公式后面是$q$ -积分.当$q$趋于1时，得到了勒让德多项式的加法和乘积公式。
In the present paper, we deal mainly with arithmetic properties of Legendre polynomials by using their orthogonality property. We show that Legendre polynomials are proportional with Bernoulli, Euler, Hermite and Bernstein polynomials.
Mutually inverse series relating Ferrers and associated Legendre functions and generating functions pertaining to them
本文利用超几何及广义超几何多项式的生成函数，建立联系任意复指标 Ferrers 函数与连带 Legendre 函数的三类互反级数，并给出相关多项式的高阶渐近式及新连接公式。
We prove {\em sign equidistribution} of Legendre polynomials: the ratio between the lengths of the regions in the interval $[-1, 1]$ where the Legendre polynomial assumes positive versus negative values, converges to one as the degree grows. The proof method also has application to the symmetry conjecture for a basis of eigenfunctions in the sphere.
Addition formula for 2-parameter family of Askey-Wilson polynomials
对于一类两参数的Askey-Wilson多项式族，即勒让德多项式的基本类似物，导出了一个加法公式。加法公式是Koornwinder的小$q$ -勒让德多项式的加法公式的双参数推广。导出了相应的乘积公式。当基趋于1时，加法与乘积公式转化为勒让德多项式的加法与乘积公式。加法公式是从量子$SU(2)$群上的广义矩阵元的Askey-Wilson多项式的解释中推导出来的。
The use of operational methods of different nature is shown to be a fairly powerful tool to study different problems regarding the theory of Legendre and Legendre-like polynomials. We show how the use of the well known integral representations linking Hermite and Legendre like polynomials and of operational technique allow the derivation of new properties regarding the generating functions, repeated derivatives of irrational functions and so on. We finally extend the method of the integral transform to functions with more than one variable and derive new expansion criteria of a given function …
We introduce a new class of polynomials $\{P_{n}\}$, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with $n+1$ unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuous Sobolev type inner product.
The main objective of this paper is to give a wide study on the conformable fractional Legendre polynomials (CFLPs). This study is assumed to be a generalization and refinement, in an easy way, of the scalar case into the context of the conformable fractional differentiation. We introduce the CFLPs via different generating functions and provide some of their main properties and convergence results. Subsequently, some pure recurrence and differential recurrence relations, Laplace's first integral formula and orthogonal properties are then developed for CFLPs. We append our study with presenting…
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.
Curious identities involving Legendre polynomials and Apéry-like numbers
研究涉及勒让德多项式与类阿佩里数的恒等式，通过建立恒等式推导出新的超级同余式。
We show how one can obtain rational approximants for $q$-extensions of the harmonic series and the logarithm (and many other similar quantities) by Padé approximation using little $q$-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for the harmonic series and a better upper bound than the one given by Matala-aho and Väänänen for the logarithm.
Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of "exceptional" degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explic…
In a series of recent works, we have provided a number of explicit expressions for the derivative of the associated Legendre function of the first kind with respect to its degree, $[\partial P_ν^{m}(z)/\partialν]_{ν=n}$, with $m,n\in\mathbb{N}$. In this communication, we use some of those expressions to obtain several, we believe new, explicit formulas for the derivatives $\mathrm{d}^{m}[P_{n}(z)\ln(z\pm1)]/\mathrm{d}z^{m}$, where $P_{n}(z)$ is the Legendre polynomial.
We 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 Shaun Cooper. By using this modular parametrisation we resolve a subfamily of identities involving $1/π$ which was experimentally observed by Zhi-Wei Sun.
We provide closed-form expressions for the degree-derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$ and $[\partial Q_ν(z)/\partialν]_{ν=n}$, with $z\in\mathbb{C}$ and $n\in\mathbb{N}_{0}$, where $P_ν(z)$ and $Q_ν(z)$ are the Legendre functions of the first and the second kind, respectively. For $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$, we find that $$\displaystyle\frac{\partial^{2}P_ν(z)}{\partialν^{2}}\bigg|_{ν=n}=-2P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}+B_{n}(z)\ln\frac{z+1}{2}+C_{n}(z),$$ where $\textrm{Li}_{2}[(1-z)/2]$ is the dilogarithm function, $P_{n}(z)$ is the Legendre polynomia…
Sparse Legendre expansions via $\ell_1$ minimization
研究从少量随机样本中恢复关于勒让德多项式基稀疏的多项式，通过验证预处理随机勒让德矩阵的受限等距性质，证明了ℓ1最小化方法可以高效恢复，并推广到雅可比多项式等正交多项式系统及无限维函数空间的近似恢复。
We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.
Earlier work introduced a method for obtaining indefinite $q$-integrals of $q$-special functions from the second-order linear $q$-difference equations that define them. In this paper, we reformulate the method in terms of $q$-Riccati equations, which are nonlinear and first order. We derive $q$-integrals using fragments of these Riccati equations, and here only two specific fragment types are examined in detail. The results presented here are for $q$-Airy function, Ramanujan function, Jackson $q$-Bessel functions, discrete $q$-Hermite polynomials, $q$-Laguerre polynomials, Stieltjes-Wigert pol…
Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for Ferrers functions of the argument tanh(a+b) are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for …
We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this technique is to provide a solution to a problem recently raised by M. E. H. Ismail.
Burkholder obtained a sharp estimate of $\E|W|^p$ via $\E|Z|^p$, where $W$ is a martingale transform of $Z$, or, in other words, for martingales $W$ differentially subordinated to martingales $Z$. His result is that $\E|W|^p\le (p^*-1)^p\E|Z|^p$, where $p^* =\max (p, \frac{p}{p-1})$. What happens if the martingales have an extra property of being orthogonal martingales? This property is an analog (for martingales) of the Cauchy-Riemann equation for functions, and it naturally appears from a problem on singular integrals (see the references at the end of Section~1). We establish here that in th…
Harmonic analysis of little $q$-Legendre polynomials
本文研究 little $q$-Legendre 多项式的调和分析，利用 Fourier 变换与连分数等工具证明其 $L^1$-代数具有弱顺从性与右特征顺从性但不顺从，提供了首个此类多项式超群实例。
Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it a…
We use the recent findings of Cohl [arXiv:1105.2735] and evaluate two integrals involving the Gegenbauer polynomials: $\int_{-1}^{x}\mathrm{d}t\:(1-t^{2})^{λ-1/2}(x-t)^{-κ-1/2}C_{n}^λ(t)$ and $\int_{x}^{1}\mathrm{d}t\:(1-t^{2})^{λ-1/2}(t-x)^{-κ-1/2}C_{n}^λ(t)$, both with $\Realλ>-1/2$, $\Realκ<1/2$, $-1<x<1$. The results are expressed in terms of the on-the-cut associated Legendre functions $P_{n+λ-1/2}^{κ-λ}(\pm x)$ and $Q_{n+λ-1/2}^{κ-λ}(x)$. In addition, we find closed-form representations of the series $\sum_{n=0}^{\infty}(\pm)^{n}[(n+λ)/λ]P_{n+λ-1/2}^{κ-λ}(\pm x)C_{n}^λ(t)$ and $\sum_{n=0…
A hyperelliptic saga on a generating function of the squares of Legendre polynomials
本文将 Legendre 多项式平方的生成函数分解为超椭圆曲线周期乘积，发现其满足定义于 $\mathbb{Q}$ 上、单值群在 $\operatorname{SL}_2(\mathbb{R})$ 中稠密的二阶算术微分方程，并给出类似构造方法。
For the oscillator-like systems, connected with the Laguerre, Legendre and Chebyshev polynomials coherent states of Glauber-Barut-Girardello type are defined. The suggested construction can be applied to each system of orthogonal polynomials including classical ones as well as deformed ones.
Complementary polynomials of Legendre polynomials are briefly presented, as well as those for the confluent and hypergeometric functions, relativistic Hermite polynomials and corresponding new pre-Laguerre polynomials. The generating functions are all given in closed form and are much simpler than the standard ones. Some are simply polynomials in two variables. New recursions and addition formulas are derived.
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