We establish two recurrence relations for some Clausen’s hypergeometric functions with unit argument. solve them to give the explicit formulas. Additionally, we use moments of Ramanujan’s generalized elliptic integrals obtain these relations.

By studying the monotonicity properties of K(r), E(r) and some combinations elementary functions special functions, new inequalities for complete elliptic integrals first second kinds are established. where K(r) =f0 ?/2

We remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be satisfied by a squarefree polynomial D.x/ whose square root generates a quadratic function field with non-trivial unit. We detail the genus 1 case. 2000 Mathema...

We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise an intrinsic regularization of such singular integrals in terms the underlying conformal geometry. Applied products surfaces, this scheme establishes analytic theory for over configuration spaces, including Feynman graph arising from two dimensional chiral quantum ...