Remarks on the Zeros of the Associated Legendre Functions with Integral Degree
J. F. van Diejen
本文给出了积分度相关勒让德函数零点的计算公式。
We present some formulas for the computation of the zeros of the integral-degree associated Legendre functions with respect to the order.
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Legendre's relation for the complete elliptic integrals of the first and second kinds is generalized. The proof depends on an application of the generalized trigonometric functions and is alternative to the proof for Elliott's identity.
Integral representations of the Legendre chi function
利用初等变元推导出Legendre chi函数$χ_{s}(x)$对$|z|<1$和$\Re(s)>1$有效的整表示.我们早先建立的关于黎曼ζ函数$ζ(2 n+1)$和Dirichlet β函数$β(2 n)$，$ n\in\mathbb{N}$的整表示的结果是这些表的直接结果。
A simple derivation of the integrals of products of Legendre polynomials with logarithmic weight
该研究针对带对数权的勒让德多项式乘积积分问题，给出了阶数为m、n的勒让德多项式对应区间[0,1]上含对数权乘积积分的简单推导方法。
Generalizations and specializations of generating functions for Jacobi, Gegenbauer, Chebyshev and Legendre polynomials with definite integrals
本文推广和特殊化了经典正交多项式的生成函数，即Jacobi、Gegenbauer、Chebyshev和勒让德多项式。通过推广雅可比多项式的二元生成函数序列，得到了Gegenbauer多项式生成函数的一个推广。生成函数的特殊化是通过高斯超几何函数用不太一般的函数重新表示来完成的。给出了与所提出的正交多项式级数展开式相对应的定积分。
The complete Lipschitz-Hankel integrals (LHIs) include the Laplace transforms of the Bessel functions, multiplied by powers. Such Laplace transforms can be evaluated using associated Legendre functions. It is noted that there are errors in published versions of these evaluations, and a merged and emended list of seven transforms is given. Errata for standard reference works, such as the table of Gradshteyn and Ryzhik, are also given. Most of the errors are attributable to inconsistent normalization of the Legendre functions. These transforms can be viewed as limits of incomplete LHIs, which fi…
On Some Integrals Over the Product of Three Legendre Functions
定积分$ \int_{-1}^1(1-x^2)^{(ν-1)/2}[P_ν(x)]^3\D x$、$ \int_{-1}^1(1-x^2)^{(ν-1)/2}[P_ν(x)]^2P_ν(-x)\D x$、$\int_{-1}^1x(1-x^2)^{(ν-1)/2}[P_{ν+1}(x)]^3\D x $和$\int_{-1}^1x(1-x^2)^{(ν-1)/2}[P_{ν+1}(x)]^2P_{ν+1}(-x)\D x $以封闭形式计算，其中$P_ν$是次数$ν$的Legendre函数，而$ \Rν>-1$。这些公式的特殊情况与具有算术意义的椭圆积分上的某些积分有关。
Two point Gauss-Legendre Quadrature Rule for Riemann-Stieltjes integrals
为了用$2$ -点高斯求积规则来逼近黎曼-斯蒂尔切斯积分$\int_a^b {f\left( t \right)dg\left( t \right)}$，我们引入了求积规则\begin{align*} \int_{ - 1}^1 {f\left( t \right)dg\left( t \right)} \approx A f\left( { - \frac{\sqrt 3 }{3}} \right) + B f\left( {\frac{\sqrt 3 }{3}} \right), \end{align*}，以便适当地选择$A$和$B$ .在各种假设下，这种近似的误差估计所涉及的功能提供。
Definite Integrals using Orthogonality and Integral Transforms
利用正交性和积分变换，我们得到了相关勒让德函数与贝塞尔函数、相关勒让德函数和第一类切比雪夫多项式乘积的定积分.
Generalizations of an integral for Legendre polynomials by Persson and Strang
Ewesson和Strang（2003）计算了平方奇数次勒让德多项式除以x^2的[-1，1]上的积分等于2。我们考虑了正交多项式关于一般偶正交测度的类似积分，Gegenbauer和Hermite多项式作为显式特殊情况。接下来，经过二次变换，我们被引导到一般的非对称的情况下，雅可比和Laguerre多项式作为显式的特殊情况。在这种情况下，也会出现不确定求和的例子。本文的结论与哈恩多项式的早期结果的推广。为了达到相对较好的明确评价，必须进行一些调整。
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.
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.
On Parameter Differentiation for Integral Representations of Associated Legendre Functions
对于相关勒让德函数用修正贝塞尔函数表示的整表示，我们证明了在积分符号下关于参数微分的合理性.有了这个理由，第一和第二类相关的勒让德函数的导数相对于程度评估在奇数半整数度，一般复杂的订单，和导数相对于订单评估在整数阶，一般复杂的程度。我们还讨论了由$f(z)=z/(\sqrt{z+1}\sqrt{z-1})$给出的复变函数$f:{\mathbb C}\setminus\{-1,1\}\to{\mathbb C}$的性质。
Associated Legendre functions of fractional degree appear in the solution of boundary value problems in wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer, they can be expressed using complete elliptic integrals. In this study, many transformations are derived, which reduce the case when the degree differs from an integer by one-third, one-fourth or one-sixth to the classical case. These transformations, or identities, facilitate the symbolic manipulation and evaluation of Legendre and Ferrers functions. They gen…
Legendre Hyperelliptic integrals, π new formulae and Lauricella functions through the elliptic singular moduli
本文在文[10]、[11]的基础上，通过第一类椭圆积分的比值和劳里切拉型三元或三元超几何函数的某些值，得到了π的六个新公式，见方程。这将是通过减少一些超椭圆积分椭圆的方法教勒让德在他的论文。最后，利用超几何Lauricella函数对某些超椭圆积分进行了计算，得到了它们在某些特殊点上的进一步的计算，以及它们的解析延拓.
The definite integrals $ \int_{-1}^1x[P_ν(x)]^4\,\mathrm{d} x$ and $ \int_{0}^1x[P_ν(x)]^2\{[P_ν(x)]^2-[P_ν(-x)]^2\}\,\mathrm{d} x$ are evaluated in closed form, where $ P_ν$ stands for the Legendre function of degree $ ν\in\mathbb C$. Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green's functions and Epstein zeta functions.
Generalized Elliptic Integrals and the Legendre M-function
我们研究了在椭圆积分理论中出现的函数的单调性和凸性，特别是从半平面到梯形的Schwarz-Christoffel共形映射的情形.我们得到尖锐的单调性和凸性的结果，这些功能的组合，以及功能的不等式和线性化性质。
For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis…
Coupling of the Legendre polynomials with kernels $|x-y|^α$ and $\ln|x-y|$
在[-1，1]的勒让德多项式基上，用二重积分表示幂和对数势的矩阵元，得到了一个封闭的形式.给出了几个证明，涉及到不同的特殊函数和恒等式。新公式和惠普尔的超几何求和公式的连接。
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 …
Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions
表达式的衍生物相对于修改贝塞尔函数的顺序评估在整数阶和某些整表示的相关勒让德函数的模参数大于单位被用来计算导数的相关勒让德函数相对于它们的参数。对于第一类和第二类的相关联的勒让德函数，对于一般的复数阶数，关于阶数的导数在奇数半整数阶数处被评估，并且对于一般的复数阶数，关于阶数的导数在整数阶数处被评估。
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…
Discrete analogs of the classical Mehler-Fock transforms are introduced and investigated. It involves series with the associated Legendre function $P^μ_{in-1/2}(x), x > 1,\ {\rm Re} μ< 1/2, \ n \in \mathbb{N}, i $ is the imaginary unit, and the so-called incomplete Legendre integrals. Several expansions of arbitrary functions and sequences in terms of these series and integrals are established.
Legendre Functions, Spherical Rotations, and Multiple Elliptic Integrals
本文导出了广义Clebsch-Gordan积分$ \int_{-1}^1 {[}P_ν(x){]}^2P_ν(-x)\D x$的一个封闭式公式，其中$ P_ν$是任意复次数的Legendre函数$ ν\in\mathbb C$ .计算了$ P_ν(x)P_ν(-x),-1<x<1$的有限希尔伯特变换。给出了一个新近证明的包含第一类完全椭圆积分$ \mathbf K(k)$和Euler伽玛函数$ Γ(z)$的恒等式$\int_0^1[\mathbf K(\sqrt{1-k^2})]^{3}\D k=6\int_0^1[\mathbf K(k)]^2\mathbf K(\sqrt{1-k^2})k\D k=[Γ(1/4)]^{8}/(128π^2) $的解析证明.
Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among generalized complete elliptic integrals, Fourier-Legendre (FL) series expansions, and ${}_p F_q$ series. We produce new hypergeometric transformations and closed-form evaluations for new series involving harmonic numbers, through the use of the integration method outlined as follows: Letting $K$ denote the complete elliptic integral of the first kind, for a suitable function $g$ we evaluate integrals such as $$ \int_{0}^{1} K\left( \sqrt{x} \right) g(x)…
Numerical computation of real or complex elliptic integrals
本文对这三类对称椭圆积分的数值计算算法作了若干改进，并推广到复变量（对第三类积分有一些限制）。数值检查值，一致性检查，并与勒让德积分和Bulirsch积分的关系。
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。
Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut (-1,1)
我们使用Cohl [arXiv：1105.2735]的最新发现并评估涉及Gegenbauer多项式的两个积分：$\int_{-1}^{x}\mathrm{d}t\:(1-t^{2})^{λ-1/2}(x-t)^{-κ-1/2}C_{n}^λ(t)$和$\int_{x}^{1}\mathrm{d}t\:(1-t^{2})^{λ-1/2}(t-x)^{-κ-1/2}C_{n}^λ(t)$，两者都具有$\Realλ>-1/2$、$\Realκ<1/2$、$-1<x<1$。结果表示在切割相关的勒让德函数$P_{n+λ-1/2}^{κ-λ}(\pm x)$和$Q_{n+λ-1/2}^{κ-λ}(x)$。此外，我们发现封闭形式表示的系列$\sum_{n=0}^{\infty}(\pm)^{n}[(n+λ)/λ]P_{n+λ-1/2}^{κ-λ}(\pm x)C_{n}^λ(t)$和$\sum_{n=0}^{\infty}(\pm)^{n}[(n+λ)/λ]Q_{n+λ-1/2}^{κ-λ}(\pm x)C_{n}^λ(t)$，都与$\Realλ>-1/2$，$\Realκ<1/2$，$-1<t<1$，$-1<x<1$。
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 …
The hyperbola rectification from Maclaurin to Landen and the Lagrange-Legendre transformation for the elliptic integrals
本文介绍了主要的数学研究进行，在英国和大陆之间的1742年至1827年，对主题的双曲线整流，从而增加了我们的一些贡献。我们开始与马克劳林发明微积分及其显着的作用，在早期中期1700年;接下来，我们重点有点他的评价，1742年，双曲过剩，解释背后的真正动机，他的研究。对于他的几何分析的处理，我们附上我们的，一个纯粹的分析的选择。我们的双曲线调查然后切换到约翰·兰登，一个业余数学家，他可能写更多的是为了确定他的优先事项，而不是解释他的非凡发现。我们跟随他在模糊的证明，他的定理对双曲线整流，解释联系马克劳林等。与链的几何结构，我们重视我们的解释兰登的治疗。我们的现代分析证明他的双曲极限过剩，通过椭圆积分的第一和第二类也提供了，我们证明了为什么所谓的Landen变换的椭圆积分不能归因于他。接下来，这个问题离开英国的大陆：字符拉格朗日介绍，即使我们的兴趣只涉及他1785年的回忆录无理积分，其中算术几何平均，AGM，是由他建立的。然而，我们的目标不是AGM，而是检测所谓的椭圆积分的Landen变换的真正来源。事实上，拉格朗日的文件包含一个微分身份源于AGM：整合它，我们展示了如何可能达到著名的勒让德递归计算的第一类椭圆积分，这出现在他的Traité，1827年，后拉格朗日的文件。
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…
We derive sufficient conditions for the existence of the Weber formal solution of the corresponding integral equation, related to the familiar Weber-Orr integral transforms. This gives a solution to the old Weber-Titchmarsh problem (posed in Proc. Lond. Math. Soc. 22(2) (1924), pp.15, 16.) Our method involves properties of the inverse Mellin transform of integrable functions. The Mellin-Parseval equality and some integrals with the associated Legendre functions are used.
Asymptotic approximations for symmetric elliptic integrals
对称椭圆积分是三个或四个变量的齐次函数，在最近的积分表和计算机程序中被用作勒让德积分的替代。当某些变量远大于其他变量时，给出了具有误差界的渐近近似。在大多数情况下，它们是从被积函数的一致逼近导出的。作为应用，证明了第一、第二、第三类对称椭圆积分关于系数为有理函数的函数是线性无关的。
We recast Byerly's formula for integrals of products of Legendre polynomials. Then we adopt the idea to the case of Jacobi polynomials. After that, we use the formula to derive an asymptotic formula for integrals of products of Jacobi polynomials. The asymptotic formula is similar to an analogous one recently obtained by the first author and Jeff Geronimo for a different case. Thus, it suggests that such an asymptotic behavior is rather generic for integrals of products of orthogonal polynomials.
A new form of the generalized complete elliptic integrals
将广义三角函数应用于完全椭圆积分的Legendre-Jacobi标准式，给出了Borweins广义完全椭圆积分的一种新形式.根据形式，可以很容易地证明这些积分具有与经典积分相似的性质。特别地，可以按照算术-几何平均值建立广义$π$的计算公式，如Salamin和Brent的$π$的Gauss-Legendre算法。此外，还给出了Ramanujan三次变换的一个初等新证明。
A recurrence relation for generalised connection coefficients
本文针对正交多项式及类似函数的积分，提出并证明了一个通用递推关系，可统一处理标准正交多项式连接系数与 Legendre 函数乘积积分等情形。
On the nonoscillatory phase function for Legendre's differential equation
本文通过将Legendre微分方程的复解表示为振荡指数函数与非振荡初等函数的积分乘积，证明了该方程存在非振荡相函数，并推导了用于大阶Legendre函数和其非振荡相函数导数的渐近展开。
Multiple elliptic integrals related to the generalized Clebsch-Gordan (CG) integral are of importance in many areas in physics and special functions theory. Zhou has introduced and applied Legendre function-based techniques to prove symbolic evaluations for integrals of CG form involving twofold and threefold products of complete elliptic integral expressions, and this includes Zhou's remarkable proof of an open problem due to Wan. The foregoing considerations motivate the results introduced in this article, in which we prove closed-form evaluations for new CG-type integrals that involve three…
We study the Hp-Lq boundedness of certain integral operators of fractional type.
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral is derived which involves an arbitrary function, and therefore yields an infinite number of indefinite integrals for any special function which obeys such a differential equation. Techniques are presented to obtain the more interesting integrals generated by such an approach, and many integrals, both previously known and completely new are derived using the…
More remarkable sinc integrals and sums
我们用泊松求和公式计算sinc函数乘积的积分（参见[4])[5]和[3]中的相关积分。我们还推广了文[5]中的结果，并引入了其它一些值得注意的积分.最后给出了Siegel型下界的一个求和形式。（参见：[2]，定理3）
A Remarkable Definite Integral
一个积分公式的开发，适用于一个基本上是任意的功能。应用程序是由黎曼ζ函数。
Some remarks about Cauchy integrals and fractal sets
简要讨论了维数大于1的集合上柯西积分的一些问题。
Some remarks about Cauchy integrals
本文介绍了柯西积分的某些方面以及与之相关的欧几里得空间中的集合几何和测度几何等。
We prove that the $L^2$ bound of an oscillatory integral associated with a polynomial depends only on the number of monomials that this polynomial consists of.
Properties of generalized Jacobi elliptic functions with three parameters
本文用三个参数推广雅可比椭圆函数与完全椭圆积分，构造Wallis型积分公式，推导等价于Elliott恒等式的Legendre型关系，并给出二项展开式的不等式。
In this paper a double integral containing two Gaussian hypergeometric functions is discussed. The integral is not found in the literature and a direct computation is not (yet) possible. Therefore, a complete different integral is computed by two methods. The first method gives a combination of hypergeometric functions and the requested original integral, while with the second method this different integral can be computed directly. Equalization of the results from both methods gives the requested integral.
Different types of sinc integrals are investigated when the standard sine function is replaced by the generalised $\sin_{p,q}$ in two parameters. A striking generalisation of the improper Dirichlet integral is achieved. A second surprising but interesting generalisation of the identity between the Dirichlet integral and that of the integrand $\text{sinc}^2$ is discovered. Moreover, an asymptotically sharpened form of Ball's integral inequality is obtained in terms of two parameters.
Multi-integral representations for Jacobi functions of the first and second kind
该研究将雅可比多项式与第二类雅可比函数推广为指标可取复数的雅可比函数，按解析延拓来源分为两类，借助高斯超几何函数性质推导了两类雅可比函数的多重导数与多重积分表示。
A remark concerning sinc integrals
本文给出了Hanspeter Schmid关于$K_n:=2\int_0^\infty\cos t\,\prod_{k=0}^n\mathrm{sinc}\left(\frac{t}{2k+1}\right)\mathrm{d}t=π/2$当$n\in\{0,1,\ldots,55\}$，$K_n<π/2$当$n\geq 56$的结果的一个简单证明.此外，我们提出了两个sinc积分，其中值$π/2$被削弱尽快$n\geq 418$和$n\geq 3091$，分别。
Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity
我们找到两个收敛级数展开勒让德的第一不完全椭圆积分$F(λ,k)$的递归计算初等函数。这两种展开式在单位正方形$0<λ,k<1$的每一点上都有效。截断展开式产生$F(λ,k)$的渐近近似，因为$λ$和/或$k$趋于1，包括从任何方向接近对数奇点$λ=k=1$的情况。在每一个近似阶给出了显式的误差界。为了读者的方便，我们提出了明确的表达低阶近似和数值例子来说明其准确性。我们的推导是基于重排的一些已知的二重级数展开，超几何求和算法和不等式的超几何函数。
Some remarks about Cauchy integrals and totally real surfaces in C^m
这里似乎有相当多的空间，来讨论与C^m中的，实维数为m的，完全实的曲面M，以及C^m上，围绕M的，几个复变数的方面有关的有趣的事情。当m = 1时，有一个基本的情形，即曲线上的Cauchy积分算子，曲线的补曲线上的全纯函数，等等，本文简要地指出有关这些问题的几点。
Some remarks concerning integrals of curvature on curves and surfaces
在这些注记中，我们讨论了复分析（柯西积分的导数）与曲线曲面曲率之间的某些关系。在更高的维度上，柯西积分是基于使用四元数或Clifford代数的复分析的推广。这些主题出现在“分析和一致可求积集”（与盖伊大卫，美国数学学会，1993年），在这里我们描述一些基本点。在AMS特别会议“表面几何和形状感知”（Hoboken，2001年）的基础上提出了这篇论文的介绍。
Li et al. give an integral formula for the Catalan-Qi number of the second kind. They show that this integral can be written as a summation with double factorials. In this paper the integral is reduced to a product of the Catalan number and a hypergeometric function. This hypergeometric function can be written as an associated Legendre polynomial of the first kind. There are also connections with the incomplete Beta function and the Gegenbauer polynomials. In the last section a related integral is calculated.
On Burkholder function for orthogonal martingales and zeros of Legendre polynomials
Burkholder通过$\E|Z|^p$得到了$\E|W|^p$的精确估计，其中$W$是$Z$的鞅变换，或者换句话说，对于鞅$W$微分从属于鞅$Z$。他的结果是$\E|W|^p\le (p^*-1)^p\E|Z|^p$，其中$p^* =\max (p, \frac{p}{p-1})$。如果鞅有一个额外的性质是正交鞅，会发生什么？这一性质类似于函数的柯西-黎曼方程（对于鞅），它自然地出现在奇异积分的问题中（见第1节末尾的参考文献）。我们在这里确定，在这种情况下，常数是完全不同的。实际上，是$\E|W|^p\le (\frac{1+z_p}{1-z_p})^p\E|Z|^p$，$p\ge 2$，其中$z_p$是Legendre常微分方程的某个解的特定零点。我们还证明了这一估计的锐度。渐近地，$(1+z_p)/(1-z_p)=(4j^{-2}_0+o(1))p$，$p\to\infty$，其中$j_0$是零阶贝塞尔函数的第一个正零点。与特殊函数的零点（以及$p=n(n+1)$的正交多项式）的这种联系是相当出乎意料的。
On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree
在我们最近的作品[R。Szmytkowski，J.Phys.A39（2006）15147; corrigendum：40（2007）7819; addendum：40（2007）14887]，我们研究了第一类Legendre函数$P_ν(z)$关于其次数$ν$的导数。在目前的工作中，我们扩展了这些研究，并构建了几个表示的衍生物的第一类相关的勒让德函数，$P_ν^{\pm m}(z)$，关于程度$ν$，为$m\in\mathbb{N}$。首先，我们建立了$\partial P_ν^{\pm m}(z)/\partialν$的围道积分表示.然后，他们被用来推导Rodrigues型公式的$[\partial P_ν^{\pm m}(z)/\partialν]_{ν=n}$与$n\in\mathbb{N}$。其次，得到了$[\partial P_ν^{\pm m}(z)/\partialν]_{ν=n}$的一些封闭形式的表达式。这些结果被施加到找到几个表示，明确和罗德里格斯型，为相关的勒让德函数的第二种整数度和顺序，$Q_{n}^{\pm m}(z)$ ;明确表示是适合用于数值计算的目的，在各个地区的复杂$z$平面。最后，衍生物$[\partial^{2}P_ν^{m}(z)/\partialν^{2}]_{ν=n}$，$[\partial Q_ν^{m}(z)/\partialν]_{ν…
A summation formula over the zeros of a combination of the associated Legendre functions with a physical application
利用推广的Abel-Plana公式，导出了相应Legendre函数组合的零点级数关于次数的求和公式.作为一个极限情况，获得了先前在文献中讨论过的贝塞尔函数的组合的零点上的级数的求和公式。作为应用，在常负曲率空间背景下，我们计算了同心球壳间具有一般曲率耦合参数的标量场的Wightman函数.对于Dirichlet边界条件，相应的模式和包含相关Legendre函数的组合的零上的系列。求和公式的应用允许我们以两个积分之和的形式来表示Wightman函数。第一个对应于Wightman函数的几何形状的一个单一的球壳和第二个是由第二壳的存在引起的。研究了真空场平方期望值中的边界诱导部分。对于远离边界的点，相应的重整化过程被简化为无边界部分。
We provide examples of vector fields on $(\mathbb{C}^3, 0)$ admitting a formal first integral but no holomorphic first integral. These examples are related to a question raised by D. Cerveau and motivated by the celebrated theorems of Malgrange and Mattei-Moussu.
Some remarks on the $L^p-L^q$ boundedness of trigonometric sums and oscillatory integrals
本文讨论了三角多项式L ^p-L ^q估计中最佳常数的渐近性态以及与非齐次Schrodinger方程解有关的积分算子的渐近性态。这使我们有机会回顾一些基本事实的振荡积分和稳定相位法，并作出一些评论，在连接与哈茨估计。
Some remarks on sinc integrals and their connection with combinatorics, geometry and probability
我们给出了一个替代的，组合/几何评价一类不当sinc积分研究的Borweins。一个概率的解释也注意到，并用于阐明相关的组合身份。
Remark on the Boundedness of the Cauchy Singular Integral Operator on Variable Lebesgue Spaces with Radial Oscillating Weights
最近，V. Kokilashvili，N. Samko和S. Samko证明了Carleson曲线上Cauchy奇异积分算子在带径向振动权的变Lebesgue空间上有界的一个充分条件。这个条件是制定的Matuszewska-Orlicz指数的权重。我们证明了他们的结果的部分逆。
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