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

Jacobi Triple Product Formula Cor

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 17] Jacobi Triple Product Formula

Jacobi Triple Product Formula

We introduce the Jacobi Triple Product Formula [1] here. Actually it can be easily obtained from the Ramanujan ${}_1\psi_1$ Summation Formula [2] with some proper coefficients. Here we denote $f$ as the Ramanujan Theta Function which is defined in [3] … Continue reading

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

Ramanujan 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 15] Introduction to the Ramanujan Theta Functions

Ramanujan theta function

In this post, we introduce the Ramanujan theta functions $f(a,b)$, which generalize the form of the Jacobi theta functions. Here we define the Ramanujan theta function, and introduce some elementary properties. First, we define the Ramanujan theta function as $$ … Continue reading

[Calculation 14] q-Series and the q-Binomial Theorem

q-series and q-binomial theorem

In this post, we introduce q-Series and the q-Binomial theorem. For any complex number $a$, we write $$ (a;q)_k = (1-a)(1-aq)(1-aq^2) \cdots (1-aq^{k-1}) $$ where $|q|<1$. Also we write $$ (a;q)_\infty = \prod_{k=0}^\infty (1-aq^k). $$ With these notations, we will … Continue reading