[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