When written in terms of $\vartheta$-functions, the classical Frobenius-Stickelberger pseudo-addition formula takes a very simple form. Generalizations of this functional identity are studied, where the functions involved are derivatives (including derivatives with respect to the modular parameter) of the elliptic trilogarithm function introduced by Beilinson and Levin. A differential identity satisfied by this function is also derived. These generalized Frobenius-Stickelberger identities play a fundamental role in the development of elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde equations of associativity, with the simplest case reducing to the above mentioned differential identity.
A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, introduced by Veselov, of a V-system. Rational and trigonometric limits are studied together with examples of elliptic V-systems based on various Weyl groups. Jacobi group orbit spaces are studied: these c…
We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities of arbitrary rank. Second, we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp(2iπ/s), where s is any integer. Third, we systematize the local identities by deriving four local ``master identities'' analogous to the master identities for th…
Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at $p$ equally shifted points on the real axis were recently found. These identities played a crucial role in discovering linear superposition solutions of a large number of important nonlinear equations. We derive four master identities, from which the identities discussed earlier are derivable as special cases. Master identities are also obtained which lead to cyclic identities with alternating signs. We discuss an extension of our results to pure imaginary and complex shifts as well as to the ratio o…
We state and discuss numerous mathematical identities involving Jacobi elliptic functions sn(x,m), cn(x,m), dn(x,m), where m is the elliptic modulus parameter. In all identities, the arguments of the Jacobi functions are separated by either 2K(m)/p or 4K(m)/p, where p is an integer and K(m) is the complete elliptic integral of the first kind. Each p-point identity of rank r involves a cyclic homogeneous polynomial of degree r (in Jacobi elliptic functions with p equally spaced arguments) related to other cyclic homogeneous polynomials of degree r-2 or smaller. Identities corresponding to small…
We claim that some non-trivial theta-function identities at higher genus can stand behind the Poisson commutativity of the Hamiltonians of elliptic integrable systems, which are made from the theta-functions on Jacobians of the Seiberg-Witten curves. For the case of three-particle systems the genus-2 identities are found and presented in the paper. The connection with the Macdonald identities is established. The genus-2 theta-function identities provide the direct way to construct the Poisson structure in terms of the coordinates on the Jacobian of the spectral curve and the elements of its pe…
We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show thair linear independence for the numbers of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.
We propose and prove a set of identities for ${\rm GL}_M$ elliptic $R$-matrix (in the fundamental representation). In the scalar case ($M=1$) these are elliptic function identities derived by S.N.M. Ruijsenaars as necessary and sufficient conditions for his kernel identity underlying construction of integral solutions to quantum spinless Ruijsenaars-Schneider model. In this respect the result of the present paper can be considered as the first step towards constructing solutions of quantum eigenvalue problem for the anisotropic spin Ruijsenaars model.
We establish a bilinear framework for elliptic soliton solutions which are composed by the Lamé-type plane wave factors. $τ$ functions in Hirota's form are derived and vertex operators that generate such $τ$ functions are presented. Bilinear identities are constructed and an algorithm to calculate residues and bilinear equations is formulated. These are investigated in detail for the KdV equation and sketched for the KP hierarchy. Degenerations by the periods of elliptic functions are investigated, giving rise to the bilinear framework associated with trigonometric/hyperbolic and rational func…
We prove a recently conjectured star-star relation, which plays the role of an integrability condition for a class of 2D Ising-type models with multicomponent continuous spin variables. Namely, we reduce this relation to an identity for elliptic gamma functions, previously obtained by Rains.
In recent work on superconformal quantum field theories, Razamat arrived at elliptic kernel identities of a new and striking character: They relate solely to the root systems $A_2$ and $A_3$ and have no coupling type parameters \cite{Ra18}. The pertinent 2- and 3-variable Hamiltonians are analytic difference operators and the kernel functions are built from the elliptic gamma function. Razamat presented compelling evidence for the validity of these identities, and checked them to a certain order in a power series expansion. This paper is mainly concerned with analytical proofs of these identit…
Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at $p$ equally shifted points were recently found. The purpose of this paper is to re-express these cyclic identities in terms of ratios of Jacobi theta functions, since many physicists prefer using Jacobi theta functions rather than Jacobi elliptic functions.
The theory of quadrature domains for harmonic functions and the Hele-Shaw problem of the fluid dynamics are related subjects of the complex variables and mathematical physics. We present results generalizing the above subjects for elliptic PDEs with variable coefficients emerging in a class of the free-boundary problems for viscous flows in non-homogeneous media. Such flows posses an infinite number of conservation laws, whose special cases may be viewed as quadrature identities for solutions of variable-coefficient elliptic PDEs. If such PDEs are gauge equivalent to the Laplace equation (gaug…
We consider generalizations of the $BC$-type relativistic Calogero-Moser-Sutherland models, comprising of the rational, trigonometric, hyperbolic, and elliptic cases, due to Koornwinder and van Diejen, and construct an explicit eigenfunction for these generalizations. In special cases, we find the various kernel function identities, and also a Chalykh-Feigin-Sergeev-Veselov type deformation of these operators and their corresponding kernel functions, which generalize the known kernel functions for the Koornwinder-van Diejen models.
Landen formulas, which connect Jacobi elliptic functions with different modulus parameters, were first obtained over two hundred years ago by making a suitable quadratic transformation of variables in elliptic integrals. We obtain and discuss significant generalizations of the celebrated Landen formulas. Our approach is based on some recently obtained periodic solutions of physically interesting nonlinear differential equations and numerous remarkable new cyclic identities involving Jacobi elliptic functions.
We consider the relativistic generalization of the quantum $A_{N-1}$ Calogero-Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh- Feigin-Veselov-Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible appli…
Landen transformation formulas, which connect Jacobi elliptic functions with different modulus parameters, were first obtained over two hundred years ago by changing integration variables in elliptic integrals.We rediscover known results as well as obtain more generalized Landen formulas from a very different perspective, by making use of the recently obtained periodic solutions of physically interesting nonlinear differential equations and numerous remarkable new cyclic identities involving Jacobi elliptic functions. We find that several of our Landen transformations have a rather different a…
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…
Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by virtue of some remarkable new identities satisfied by the elliptic functions.
We derive and study Yangian Ward identities for the infinite class of four-point ladder integrals and their Basso-Dixon generalisations. These symmetry equations follow from interpreting the respective Feynman integrals as correlation functions in the bi-scalar fishnet theory. Alternatively, the presented identities can be understood as anomaly equations for a momentum space conformal symmetry. The Ward identities take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. We employ a manifestly conformal tensor reduction in ord…
This article presents a family of nonlinear differential identities for the spatially periodic function $u_s(x)$, which is essentially the Jacobian elliptic function $\cn^2(z;m(s))$ with one non-trivial parameter $s$. More precisely, we show that this function $u_s$ fulfills equations of the form {equation*} \big(u_s^{(α)}u_s^{(β)}\big)(x)=\sum_{n=0}^{2+α+β}b_{α,β}(n)u_s^{(n)}(x)+c_{α,β}, {equation*} for any $s>0$ and for all $α,β\in\N_0$. We give explicit expressions for the coefficients $b_{α,β}(n)$ and $c_{α,β}$ for given $s$. Moreover, we show that for any $s$ satisfying $\sinh(π/(2s))\geq…
The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d-3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functio…
We introduce stochastic Interaction-Round-a-Face (IRF) models that are related to representations of the elliptic quantum group $E_{τ,η}(sl_2)$. For stochasic IRF models in a quadrant, we evaluate averages for a broad family of observables that can be viewed as higher analogs of $q$-moments of the height function for the stochastic (higher spin) six vertex models. In a certain limit, the stochastic IRF models degenerate to (1+1)d interacting particle systems that we call dynamic ASEP and SSEP; their jump rates depend on local values of the height function. For the step initial condition, we ev…
We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same $_2F_1$ hypergeometric function with different rational pullbacks. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus $_2F_1$ hypergeometric function …
We derive new, localized geometric integral identities for solutions to the $3D$ compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The identities are coercive in the first derivatives of the specific vorticity and the second derivatives of the entropy, and the error terms exhibit remarkable regularity and null structures. Our framework allows one to simultaneously unleash the full power of the geometric vectorfield method for both the wave- and transport- parts of the flow on compact regions. In particular, the integral identities yield localiz…
We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses. The differential equation is obtained by viewing the Feynman integral as a period of a variation of a mixed Hodge structure, where the variation is with respect to the external momentum squared. The fibre is the complement of an elliptic curve. From the fact that the first cohomology group of this elliptic curve is two-dimensional we obtain a second-order differential equation. This is an improvement compared to the usual way of deriving differential equations: Integration-by-…
We present a general method for analytically factorizing the n-fold form factor integrals $f^{(n)}_{N,N}(t)$ for the correlation functions of the Ising model on the diagonal in terms of the hypergeometric functions $_2F_1([1/2,N+1/2];[N+1];t)$ which appear in the form factor $f^{(1)}_{N,N}(t)$. New quadratic recursion and quartic identities are obtained for the form factors for n=2,3. For n= 2,3,4 explicit results are given for the form factors. These factorizations are proved for all N for n= 2,3. These results yield the emergence of palindromic polynomials canonically associated with ellipti…
In earlier work we studied features of non-holomorphic modular functions associated with Feynman graphs for a conformal scalar field theory on a two-dimensional torus with zero external momenta at all vertices. Such functions, which we will refer to as modular graph functions, arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We here introduce a class of single-valued elliptic multiple polylogarithms, which are defined as elliptic functions associated with Feynman graphs with vanishing external momenta at all but two vertices. These functions depend o…
