A new form of the generalized complete elliptic integrals

Abstract

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 transformation is also given.

An inequality for generalized complete elliptic integral

“…Applying the asymptotic formula ([10], equality (2)) and expression [10] where and denote a classical hypergeometric function and a beta function, respectively, we obtain Putting and , we complete the proof. □…”

Complete p-elliptic integrals and a computation formula of $$\pi _p$$ π p for $$p=4$$ p = 4

“…It is natural to try to relate the generalized trigonometric function to the complete elliptic integrals. Following [13], we define the complete p-elliptic integrals of the first kind 1 p for x ∈ [0, π p /2]. Using I p and J p , we can write…”

“…We are interested in finding a formula as (1.4) of π p for p = 2. Recently, the author [13] dealt with the case p = 3. We take sequences a n+1 = a n + 2b n 3 , b n+1 = 3 (a 2 n + a n b n + b 2 n )b n 3 , n = 0, 1, 2, .…”

Complete (p,q)-elliptic integrals with application to a family of means

“…Moreover, K p,q (k) for p = q has been already studied in [9]. In this paper we will apply the complete (p, q)-elliptic integral K p,q (k) to study a family of means defined by Bhatia and Li [2] and to give an alternative proof of their theorem.…”