Multiple elliptic integrals related to the generalized Clebsch–Gordan (CG) integral are of importance in many areas physics and special functions theory. Zhou has introduced applied Legendre function-based techniques prove symbolic evaluations for CG form involving twofold threefold products complete expressions, this includes Zhou's remarkable proof an open problem due Wan. The foregoing consi...

We study the elliptic C n and BC n Ruijsenaars-Schneider models which is elliptic generalization of system given in [1]. The Lax pairs for these models are constructed by Hamiltonian reduction technology. We show that the spectral curves can be parameter-ized by the involutive integrals of motion for these models. Taking nonrelativistic limit and scaling limit, we verify that they lead to the s...

We prove analogues for elliptic interpolation functions of Macdonald’s version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic “hypergeometric” sums as skew interpolation functions. One such analogue has an interpretation as a “vanishing integral”, generalizing a result of [9]; the structure of this analogue gives suff...

This book is about elliptic curves and modular functions, two topics that are intimately related in both accidental and essential ways. As emphasized by André Weil in his magisterial historical introduction to contemporary number theory [W], the arithmetic study of elliptic curves is, in spite of the clear reference to the integral calculus in the adjective elliptic, in many respects antecedent...

This book is about elliptic curves and modular functions, two topics that are intimately related in both accidental and essential ways. As emphasized by André Weil in his magisterial historical introduction to contemporary number theory [W], the arithmetic study of elliptic curves is, in spite of the clear reference to the integral calculus in the adjective elliptic, in many respects antecedent...

In recent work on multivariate elliptic 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 E7. 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 t...

We recall the form factors f (j) N,N corresponding to the λ-extension C(N,N ;λ) of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential equations which exhibit both a “Russian-doll” nesting, and a decomposition of the linear differential operators as a direct sum of operators (equivalent to symmetric powers of the differen...