We derive new integral representations for objects arising in the classical theory of elliptic functions: the Eisenstein series $E_s$, and Weierstrass' $\wp$ and $ζ$ functions. The derivations proceed from the Laplace-Mellin transformation for multipoles, and an elementary lemma on the summation of 2D geometric series. In addition, we present new results concerning the analytic continuation of the Eisenstein series as an entire function in $s$, and the value of the conditionally convergent series, denoted by $\widetilde{E}_2$ below, as a function of summation over increasingly large rectangles with arbitrary fixed aspect ratio.
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.
In this paper, we establish several asymptotical bounds for the complete elliptic integrals of the second kind $\mathcal{E}(r)$, and improve the well-known conjecture $\mathcal{E}(r)>π[(1+(1-r^2)^{3/4})/2]^{2/3}/2$ for all $r\in(0,1)$ proposed by 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.
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 $ζ(3)$.
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.
Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind). Numerical check values, consistency checks, and relations to Legendre's integrals and Bulirsch's integrals are included.
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.
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 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…
Symmetric elliptic integrals, which have been used as replacements for Legendre's integrals in recent integral tables and computer codes, are homogeneous functions of three or four variables. When some of the variables are much larger than the others, asymptotic approximations with error bounds are presented. In most cases they are derived from a uniform approximation to the integrand. As an application the symmetric elliptic integrals of the first, second, and third kinds are proved to be linearly independent with respect to coefficients that are rational functions.
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.
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.
In math.QA/0309252, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.
In this article we prove a new elliptic hypergeometric integral identity. It previously appeared (as a conjecture) in articles by Rains, and Spiridonov and Vartanov. Moreover it gives a different proof of an identity in another article by Rains. We also give some basic hypergeometric and classical limits of this identity. The classical limit gives identities (some known, some new) between generalizations of the Selberg integral.
For $r\in(0,1)$, the function $\K(r)=\int_0^{π/2}(1-r^2\sin^2t)^{-1/2}dt$ is known as the complete elliptic integral of the first kind. In this paper, we prove the absolute monotonicity of two functions involving $\K(r)$. As a consequence, we improve Alzer and Richards' result.
We prove a simple relation for a special case of Carlson's elliptic integral $R_D$. The findings are applied to derive explicit formulae for the asymptotics of certain moments of the angular central Gaussian distribution in terms of the second moment.
By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect to the elliptic modulus and integral representations of several of the series in terms of the inverse Mellin transforms related to the Riemann zeta function. The relation with the corresponding case of the Voronoi summation formula is exhibited. The involved series are expressed in closed form in terms of complete elliptic integrals of the first and second ki…
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".
The elliptic integral and its various generalizations are playing very important and basic role in different branches of modern mathematics. It is well known that they cannot be represented by the elementary transcendental functions. Therefore, there is a need for sharp computable bounds for the family of integrals. In this paper, by virtue of two new tools, we study monotonicity and convexity of certain combinations of the complete elliptic integrals of the first kind, and obtain new sharp bounds and inequalities for them. In particular, we prove that the function $\mathcal{K}\left( \sqrt{% x…
Eric Rains conjectured several quadratic transformations between multivariate elliptic hypergeometric functions in "Elliptic Littlewood Identities", with the integrand multiplied by interpolation functions. In this article two of these conjectures are proven in the case where the interpolation functions are constant, and we obtain a third conjecture as a corollary. Two other equations for elliptic Selberg integrals with 10 parameters, two of which multiplying to pq/t, are given, as they are needed in the proof. The proofs consist essentially of a calculation which strings together many ellipti…
In the article, the complete elliptic integrals of the first and second kinds are bounded by using the power series expansions of some functions, the celebrated Wallis' inequality, and an integral inequality due to R. P. Agarwal, P. Cerone, S. S. Dragomir and F. Qi.
In this article we give a new transformation between elliptic hypergeometric beta integrals, which gives rise to a Weyl group symmetry of type F_4. The transformation is a generalization of a series transformation discovered by Langer, Schlosser, and Warnaar. Moreover we consider various limits of this transformation to basic hypergeometric functions obtained by letting p tend to 0.
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 paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss hypergeometric functions, of arbitrary high degree. The Gauss hypergeometric functions can be identified as elliptic integrals on the genus 1 curves $y=x^3-x$ or $y=x^3-1$. Especially interesting are algebraic transformations of the hypergeometric functions into themselves; these transformations come from isogenies of the respective elliptic curves.
We bound the j -invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that the modular curve Xsplit (p3) has no non-trivial rational point if p is a sufficiently large prime number. Assuming the GRH, one can replace p3 by p2 .
In recent work (math.QA/0309252) on multivariate hypergeometric integrals, the author generalized a conjectural integral formula of van Diejen and Spiridonov to a ten parameter integral provably invariant under an action of the Weyl group E_7. In the present note, we consider the action of the affine Weyl group, or more precisely, the recurrences satisfied by special cases of the integral. These are of two flavors: linear recurrences that hold only up to dimension 6, and three families of bilinear recurrences that hold in arbitrary dimension, subject to a condition on the parameters. As a coro…
Motivated by the work of Alzer and Richards \cite{ar}, here authors study the monotonicity and convexity properties of the function $$Δ_{p,q} (r) = \frac{{E_{p,q}(r) - \left( {r'} \right)^p K_{p,q}(r) }}{r^p } - \frac{{E'_{p,q}(r) - r^p K'_{p,q}(r) }}{{\left( {r'} \right)^p }},$$ where $K_{p,q}$ and $E_{p,q}$ denote the complete $(p,q)$- elliptic integrals of the first and the second kind, respectively.
The classical Selberg integral contains a power of the Vandermonde determinant. When that power is a square, it is easy to prove Selberg's identity by interpreting it as a determinant of one-variable integrals. We give similar proofs of summation and transformation formulas for continuous and discrete elliptic Selberg integrals. In the continuous case, the same proof was previously given by Noumi. Special cases of these identities have found applications in combinatorics.
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.
The complete elliptic integrals are generalized by using the generalized trigonometric functions with two parameters. It is shown that a particular relation holds for the generalized integrals. Moreover, as an application of the integrals, an alternative proof of a result for a family of means by Bhatia and Li, which involves the logarithmic mean and the arithmetic-geometric mean, is given.
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.
In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p->0, for given behavior of the parameters as p->0. This article is therefore the multivariate version of our earlier paper "Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions". The integrand of the elliptic Selberg integral is the measure for the BC_n symmetric biorthogonal functions introduced by the second author, so we also consider the limits of the associated bilinear form. We al…
We show the modular properties of the multiple 'elliptic' gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's transformation, and that of the elliptic gamma function was provided by Felder and Varchenko. In this paper we deal with the multiple sine functions, since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite product. We also derive integral representations of the …
We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Euler's and Selberg's beta integrals, elliptic analogue of the Euler-Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for …
General elliptic hypergeometric functions are defined by elliptic hypergeometric integrals. They comprise the elliptic beta integral, elliptic analogues of the Euler-Gauss hypergeometric function and Selberg integral, as well as elliptic extensions of many other plain hypergeometric and $q$-hypergeometric constructions. In particular, the Bailey chain technique, used for proving Rogers-Ramanujan type identities, has been generalized to integrals. At the elliptic level it yields a solution of the Yang-Baxter equation as an integral operator with an elliptic hypergeometric kernel. We give a brie…
In this paper, we prove that the double inequality \begin{equation*} 1+αr'^2<\frac{\mathcal{K}_{a}(r)}{\sin(πa)\log(e^{R(a)/2}/r')}<1+βr'^2 \end{equation*} holds for all $a\in (0, 1/2]$ and $r\in (0, 1)$ if and only if $α\leq π/[R(a)\sin(πa)]-1$ and $β\geq a(1-a)$, where $r'=\sqrt{1-r^2}$, $\mathcal{K}_{a}(r)$ is the generalized elliptic integral of the first kind and $R(x)$ is the Ramanujan constant function. Besides, as the key tool, the series expression for the Ramanujan constant function $R(x)$ is given.
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…
We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, we also consider their "Type II" integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint integral operators; a different proof gives rise to pairs of adjoint difference operators. These allow us to construct a family of biorthogonal abelian functions generalizing the Koornwinder polynomials, …
We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $BC_n$ root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in terms of the lens elliptic gamma function, a generalisation of the elliptic gamma function that depends on an additional integer variable, as well as a complex variable and two elliptic nomes. As an application of our results, we prove an equality between $S^1\times S^3/\mathbb{Z}_r$ supersymmetric indices, for a pair of four-dimensional $\mathcal{N}=1$ supers…
We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transfo…
We investigate the elliptic umbilic canonical integral with an approach based on a series expansion of its initial distribution shifted to the caustic points. An absolutely convergent integral representation for the elliptic umbilic is obtained. Using it, we find the elliptic umbilic particular values in terms of 2F2 hypergeometric functions. We also derive an integral over the product of Gaussian and two Airy functions in terms of Bessel functions of fractional orders. Some other corollaries including 3F2 hypergeometric function special values and the Airy polynomials relations are also discu…
