Differentiating The Gauss Summation Formula To Produce Infinitely Many Series For 1/π

In this paper, firstly we use the theory of series of complex function terms to prove some important properties and theorems of the complex infinite products. Secondly, we introduce the definition of the Gamma function with the help of the infinite products theory. At the same time, we also produce Weierstrass’s definition for the Gamma function and the Euler reflection formula. Then using logarithmic derivative, we will have the Digamma and Trigamma function. By Abelian summation theorem, we prove the Gauss theorem and get some special figures. Next, using the absolute uniform convergence of the complex series of function terms and several algebraic skills, we will have the well-known Gauss summation formula for the classical hypergeometric series. Finally, with some properties of the Gamma function and the Gauss summation formula, we use asymptotic analysis and absolute uniform convergence to prove the commutative law of the summation and differentiation. By differentiating the Gauss summation formula with respect to the parameter a, we get some series identities. Afterwards, we use corresponding parameter transformation and the Euler reflection formula to get some four-parameter series expansion formulas, which can produce infinitely many series for 1/π.