We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter a when they are convergent. Sometimes, even if they do not converge, they are valid if we consider these identities as limits.

In this paper we prove some Ramanujan type formulas for 1/π but without using the theory of modular forms. Instead we use the WZ-method created by H. Wilf and D. Zeilberger and find some hypergeometric functions in two variables which are second components of WZ-pairs than can be certified using Zeilberger's EKHAD package. These certificates have an additional property which allows us to get generalized Ramanujan's type series which are routinely proven by computer. We call these second hypergeometric components of the WZ-pairs generators. Finding generators seems a hard task but using a kind of experimental research (explained below), we have succeeded in finding some of them. Unfortunately we have not found yet generators for the most impressive Ramanujan's formulas. We also prove some interesting binomial sums for the constant 1/π 2 . Finally we rewrite many of the obtained series using pochhammer symbols and study the rate of convergence.

The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and of the second author for Euler's constant γ and its alternating analog ln(4/π), and on the other hand the infinite products of the first author for e, of the second author for π, and of Ser for e γ . We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch's transcendent of Hadjicostas's double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch's function, including Hasse's series. We also use Ramanujan's polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations between certain dilogarithm values.

We propose a new kind of Ramanujan-type formula for 1/π 2 and conjecture that it is related to the theory of modular functions.

Using the WZ-method we find some of the easiest Ramanujan's formulae and also some new interesting Ramanujan-like sums.