V. Heikkala, H. Lindén, M. K. Vamanamurthy, M. Vuorinen
We study monotonicity and convexity properties of functions arising in the theory of elliptic integrals, and in particular in the case of a Schwarz-Christoffel conformal mapping from a half-plane to a trapezoid. We obtain sharp monotonicity and convexity results for combinations of these functions, as well as functional inequalities and a linearization property.
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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.
For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for associated Legendre functions of the first and second kind with respect to the degree are evaluated at odd-half-integer degrees, for general complex-orders, and derivatives with respect to the order are evaluated at integer-orders, for general complex-degrees. We also discuss the properties of the complex function $f:{\mathbb C}\setminus\{-1,1\}\to{\mathbb …
We present some formulas for the computation of the zeros of the integral-degree associated Legendre functions with respect to the order.
This paper, pursuing the work started in [10] and [11], holds six new formulae for π, see equations, through ratios of first kind elliptic integrals and some values of hypergeometric functions of three or four variables of Lauricella type. This will be accomplished by reducing some hyperelliptic integrals to elliptic by the methods taught by Legendre in his treatise. Eventually, evaluating some hyperelliptic integrals by means of hypergeometric Lauricella functions, we obtain some further evaluations of themselves in some particular points and also in their analytic continuation.
We, by making use of elementary arguments, deduce integral representations of the Legendre chi function $χ_{s}(x)$ valid for $|z|<1$ and $\Re(s)>1$. Our earlier established results on the integral representations for the Riemann zeta function $ζ(2 n+1)$ and the Dirichlet beta function $β(2 n)$ ,$ n\in\mathbb{N}$, are direct consequence of these representations.
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。
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.
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…
The definite integrals $ \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 $ and $\int_{-1}^1x(1-x^2)^{(ν-1)/2}[P_{ν+1}(x)]^2P_{ν+1}(-x)\D x $ are evaluated in closed form, where $P_ν$ is the Legendre function of degree $ν$, and $ \Rν>-1$. Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.
A closed-form formula is derived for the generalized Clebsch-Gordan integral $ \int_{-1}^1 {[}P_ν(x){]}^2P_ν(-x)\D x$, with $ P_ν$ being the Legendre function of arbitrary complex degree $ ν\in\mathbb C$. The finite Hilbert transform of $ P_ν(x)P_ν(-x),-1<x<1$ is evaluated. An analytic proof is provided for a recently conjectured identity $\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) $ involving complete elliptic integrals of the first kind $ \mathbf K(k)$ and Euler's gamma function $ Γ(z)$.
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…
In order to approximate the Riemann--Stieltjes integral $\int_a^b {f\left( t \right)dg\left( t \right)}$ by $2$--point Gaussian quadrature rule, we introduce the quadrature rule \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*} for suitable choice of $A$ and $B$. Error estimates for this approximation under various assumptions for the functions involved are provided as well.
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.
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.
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…
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…
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 …
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)…
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.
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…
This article describes the main mathematical researches performed, in England and in the Continent between 1742-1827, on the subject of hyperbola rectification, thereby adding some of our contributions. We start with the Maclaurin inventions on Calculus and their remarkable role in the early mid 1700s; next we focus a bit on his evaluation, 1742, of the hyperbolic excess, explaining the true motivation behind his research. To his geometrical-analytical treatment we attach ours, a purely analytical alternative. Our hyperbola inquiry is then switched to John Landen, an amateur mathematician, who…
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.
Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can be easily shown that these integrals have similar properties to the classical ones. In particular, it is possible to establish a computation formula of the generalized $π$ in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for $π$ by Salamin and Brent. Moreover, an elementary new proof of Ramanujan's cubic transfor…
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。
Expressions for the derivatives with respect to order of modified Bessel functions evaluated at integer orders and certain integral representations of associated Legendre functions with modulus argument greater than unity are used to compute derivatives of the associated Legendre functions with respect to their parameters. For the associated Legendre functions of the first and second kind, derivatives with respect to the degree are evaluated at odd-half-integer degrees, for general complex orders, and derivatives with respect to the order are evaluated at integer orders, for general complex de…
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.
As a contribution to the Ramanujan theory of elliptic functions to alternative bases, Li-Chien Shen has shown how analogues of the Jacobian elliptic functions may be derived from incomplete hypergeometric integrals in signatures three and four. We determine precisely the signatures in which the Jacobian analogues or their squares are indeed elliptic.
Jacobi's elliptic integrals and elliptic functions arise naturally from the Schwarz-Christoffel conformal transformation of the upper half plane onto a rectangle. In this paper we study generalized elliptic integrals which arise from the analogous mapping of the upper half plane onto a quadrilateral and obtain sharp monotonicity and convexity properties for certain combinations of these integrals, thus generalizing analogous well-known results for classical conformal capacity and quasiconformal distortion functions.
We describe, in terms of generalized elliptic integrals, the hyperbolic metric of the twice-punctured sphere with one conical singularity of prescribed order. We also give several monotonicity properties of the metric and a couple of applications.
Associated Legendre functions of the first kind give a family of BCOV rings on elliptic curves. We prove that the family is parametrized by $q$-exponents of the eta function $η(q^{24})$. Our method involves a classification of rational solutions of a Riccati equation under some constraints.
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。
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。
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。
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。
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 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.
The table of Gradshteyn and Ryzhik contains many entries that are related to elliptic integrals. We present a systematic derivation of some of them.
We prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems. Using the (A,C) inversion formula to invert one of the known C-type elliptic beta integrals, we obtain a new elliptic beta integral for the root system of type A. Validity of this integral is established by a different method as well.
Various integrals over elliptic integrals are evaluated as couplings on spheres, resulting in some integral and series representations for the mathematical constants $π$, $G$ and $ζ(3)$.
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…
This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.
In the paper, by using Lupaş integral inequality, the authors find some new inequalities for the complete elliptic integrals of the first and second kinds. These results improve some known inequalities.
In this paper the authors investigate a power mean inequality for a special function which is defined by the complete elliptic integrals.
We give elementary proofs of the univariate elliptic beta integral with bases $|q|, |p|<1$ and its multiparameter generalizations to integrals on the $A_n$ and $C_n$ root systems. We prove also some new unit circle multiple elliptic beta integrals, which are well defined for $|q|=1$, and their $p\to 0$ degenerations.
We start from an interpretation of the $BC_2$-symmetric "Type I" (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation, and give an extension to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding elliptic beta integral and transformation formula in a new way, by proving both sides satisfy the same difference equations, and that the difference equations satisfy a Galois-theoretical condition that ensures uniqueness of simultaneous solution.
In this paper we treat certain elliptic and hyper-elliptic integrals in a unified way. We introduce a new basis of these integrals coming from certain basis $ϕ_n(x)$ of polynomials and show that the transition matrix between this basis and the traditional monomial basis is certain upper triangular band matrix. This allows us to obtain explicit formulas for the considered integrals. Our approach, specified to elliptic case, is more effective than known recursive procedures for elliptic integrals. We also show that basic integrals enjoy symmetry coming from the action of the dihedral group $D_{M…
We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations.
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。
We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of the Szegö polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered.
In this paper, authors study the generalized complete $(p,q)$-elliptic integrals of the first and the second kind as an application of generalized trigonometric functions with two parameters, and establish the Turán type inequalities of these functions.
We present an elliptic version of Selberg's integral formula.
We present an elliptic version of Selberg's integral formula.
In this paper, we give a two dimensional analogue of the Euler-MacLaurin summation formula. By using this formula, we obtain an integral representation of Weil's elliptic functions which was introduced in the book "Elliptic functions according to Eisenstein and Kronecker".
We consider the moments of products of complete elliptic integrals of the first and second kinds. In particular, we derive new results using elementary means, aided by computer experimentation and a theorem of W. Zudilin. Diverse related evaluations, and two conjectures, are also given.
The complete $p$-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function $\sin_pθ$ and its half-period $π_p$. It is shown, only for $p=4$, that the generalized $p$-elliptic integrals yield a computation formula of $π_p$ in terms of the arithmetic-geometric mean. This is a $π_p$-version of the celebrated formula of $π$, independently proved by Salamin and Brent in 1976.
By using the generalized Abel-Plana formula, we derive a summation formula for the series over the zeros of a combination of the associated Legendre functions with respect to the degree. The summation formula for the series over the zeros of the combination of the Bessel functions, previously discussed in the literature, is obtained as a limiting case. As an application we evaluate the Wightman function for a scalar field with general curvature coupling parameter in the region between concentric spherical shells on background of constant negative curvature space. For the Dirichlet boundary con…
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