[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

[Calculation 13] A Simple Formula Related to Digamma Functions

digamma function

Theorem. The following holds: $$ \pi \;{}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; 1-x\right) = \log \left( \frac{16}{x}\right) {}_2 F_1 \left( \frac{1}{2}, \frac{1}{2};1;x\right) – 4 \sum_{k=1}^\infty \frac{(\frac{1}{2})_k^2}{(k!)^2} \sum_{j=1}^k \frac{x^k}{(2j-1)(2j)}. $$   Proof. First, we recall the Corollary in [2] with $a=b=-\frac{1}{2}$ … Continue reading

[Calculation 12] A Basic Formula for Hypergeometric Functions

Hypergeometric Function

Theorem. Let $n \notin \mathbb Z$. Then we have \begin{align}\tag{1} &{}_2 F_1 \left( a+n+1, b+n+1; a+b+n+2; 1-z\right)\\ &\, \qquad\qquad\qquad=\frac{\Gamma(a+b+n+2)\Gamma(-n)}{\Gamma(a+1)\Gamma(b+1)}\; {}_2 F_1 (a+n+1,b+n+1;n+1;z)\\ &\, \qquad\qquad\qquad\qquad+\frac{\Gamma(a+b+n+2)\Gamma(n)z^{-n}}{\Gamma(a+n+1)\Gamma(b+n+1)} \;{}_2 F_1 (a+1,b+1;-n+1;z). \end{align}   Proof. We consider the following ODE, which is called hypergemoetric differential … Continue reading

[Calculation 11] Simple Examples from the Dixon Theorem

Dixon Theorem

Here we introduce some examples from the Dixon Theorem. By setting suitable coefficients, we can obtain simple formulas of infinite series, which are related to the Gamma functions.   Example. From the Dixon Theorem, we have \begin{align*} &\text{(i) } 1 … Continue reading

[Calculation 10] Dixon Theorem

Dixon Theorem

Here we note Dixon’s theorem, which gives some special values of ${}_3 F_2$, since the proof is almost automatic by using Gauss and Kummer’s formulas which we’ve shown before. Theorem. (Dixon’s Theorem) $$ {}_3 F_2 (a,b,c;1+a-b,1+a-c;1) = \frac{\Gamma(1+\frac{a}{2})\Gamma(1+a-b)\Gamma(1+a-c)\Gamma(1+\frac{a}{2}-b-c)}{\Gamma(1+a)\Gamma(1+\frac{a}{2}-b)\Gamma(1+\frac{a}{2}-c)\Gamma(1+a-b-c)} $$   … Continue reading

[Calculation 9] Simple Corollaries from Gauss and Bailey Formula

s9

Corollary 1. For $\frac{1}{2} < z < 2$, $$ {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-\frac{1}{z}\right) = \sqrt{z} {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-z\right). $$   Proof. We recall Bailey’s Formula ((i) in [2]) for $w\in\mathbb R$: \begin{equation}\tag{1} (1-w)^{-a} {}_2 F_1 \left( a,b;c; – \frac{w}{1-w}\right) … Continue reading

[Calculation 5] Gauss’s Summation Formula

c5

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

[Calculation 3] Euler’s Transformation Formula

c3

Theorem (Euler’s Transformation Formula) $${}_2 F_1 (a,b;c;z) = (1-z)^{c-a-b} {}_2 F_1 (c-a,c-b;c;z)$$   Proof. Applying Pfaff’s Transformation Formula twice, we obtain \begin{eqnarray*} {}_2 F_1(a,b;c;z) &=& (1-z)^{-a} {}_2 F_1 \left(a,c-b;c; \frac{z}{z-1} \right)\\ &=& (1-z)^{-a} \left(1-\frac{z}{z-1}\right)^{b-c} {}_2 F_1 \left(c-a,c-b;c;\frac{\frac{z}{z-1}}{\frac{z}{z-1} -1} \right)\\ &=& … Continue reading

[Calculation 2] Pfaff’s Transformation Formula

Hypergeometric Function

Theorem. (Pfaff’s Transformation Formula) $${}_2 F_1(a,b;c;z) = (1-z)^{-a} {}_2 F_1 \left(a,c-b;c; \frac{z}{z-1} \right)$$   Proof. We remember the Euler Integral Representation for the hypergeometric function: $$ {}_2 F_1 (a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 t^{b-1}(1-t)^{c-b-1} (1-tz)^{-a}\,dt. $$ Substitution $t=1-s$ yields \begin{eqnarray*} {}_2 … Continue reading