# [Calculation 18] Some Useful Formulas from the Jacobi Triple Product

In this post, we introduce some useful formulas, which can be proved by the Jacobi Triple Product [2]. As before, we always denote $f$ as the Ramanujan Theta Function, which is defined in [3].   Corollary. If $|q|<1$ then, \begin{align*} … Continue reading

# [Calculation 16] Ramanujan ‘s 1ψ1 (1-psi-1) Summation Formula

In this post, we will introduce one of the famous formulas discovered by Ramanujan, which is called Ramanujan’s ${}_1\psi_1$ Summation Formula. It was first introduced by Hardy, and he called it as “a remarkable formula with many parameters”. The first … Continue reading

# [Calculation 5] Gauss’s Summation Formula

Theorem. (Gauss’s Summation Formula) For $\text{Re}c>\text{Re}b>0$, $${}_2 F_1 (a,b;c;1) = \frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$$ holds.   Proof. We remember that the Euler Integral Representation for the hypergeometric function is $$_2 F_1 (a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 \frac{t^{b-1} (1-t)^{c-b-1}}{(1-tz)^a}\,dt.$$ Taking … Continue reading