# [Calculation 13] A Simple Formula Related to Digamma Functions

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

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

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

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

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 $w\in\mathbb R$: \tag{1} (1-w)^{-a} {}_2 F_1 \left( a,b;c; – \frac{w}{1-w}\right) … Continue reading

# [Calculation 8] Gauss’s Quadratic Trasformation

Theorem. (Gauss’s Quadratic Transformation) $$\tag{1} (1+z)^{-2a} {}_2 F_1 \left(a,b; 2b; \frac{4z}{(1+z)^2}\right) = {}_2 F_1 \left(a, 1+\frac{1}{2}-b;b+\frac{1}{2};z^2\right).$$   Proof. The proof is almost similar to that of [2]. Note that the left hand side of (1) can be expanded in … Continue reading

# [Calculation 7] Bailey’s Formulas for Hypergeometric Functions

Theorem. (Bailey) The followings are valid: \begin{align*} &\text{(i) } (1-z)^{-a} {}_2 F_1 \left( a,b;c; – \frac{z}{1-z}\right) = {}_2 F_1 (a,c-b;c;z),\quad|z|<1,\;\text{Re}z< \frac{1}{2},\\ &\text{(ii) } {}_2 F_1 \left( a,b; \frac{a+b+1}{2}; \frac{1}{2}\right) = \frac{\Gamma(\frac{1}{2}) \Gamma(\frac{1+a+b}{2})}{\Gamma(\frac{1+a}{2})\Gamma(\frac{1+b}{2})},\\ &\text{(iii) } {}_2 F_1 \left(a,1-a;c;\frac{1}{2}\right) = \frac{\Gamma(\frac{1}{2}c)\Gamma(\frac{c+1}{2})}{\Gamma(\frac{c+a}{2})\Gamma(\frac{1+c-a}{2})}. \end{align*} … Continue reading

# [Calculation 6] Kummer’s Theorem

Theorem. (Kummer’s Theorem) $${}_2 F_1 (a,b;1+a-b;-1) = \frac{\Gamma(1+a-b)\Gamma\left(1+\frac{1}{2}a\right)}{\Gamma(1+a)\Gamma\left(1+\frac{1}{2}a-b\right)}$$   To prove Kummer’s theorem, we introduce the following lemma, which is called Kummer’s quadratic transformation:   Lemma. (Kummer’s Quadratic Transformation) \tag{1} {}_2 F_1 (a,b;1+a-b;z) = (1-z)^{-a} {}_2 F_1 \left( … 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

# [Calculation 4] Saalschütz’s Theorem

Theorem. (Saalschütz’s Theorem) $${}_3 F_2 (a,b,-n;c,1+a+b-c-n;1) = \frac{(c-a)_n (c-b)_n}{(c)_n (c-a-b)_n}$$ holds for $n\in \mathbb N_0$.   Proof. We recall the Euler Transformation Formula: $$(1-z)^{a+b-c} {}_2 F_1 (a,b;c;z) = {}_2 F_1 (c-a,c-b;c;z).$$ Equating the coefficients of $z^n$ from both … Continue reading

# [Calculation 3] Euler’s Transformation Formula

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

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

# [Calculation 1] Fundamentals of Hypergeometric Functions

The classical hypergeometric function ${}_{2}F_1$ is defined by $${}_{2}F_1(a,b;c;z) = \sum_{k=0}^\infty \frac{(a)_k(b)_k}{(c)_k} \frac{z^k}{k!}$$ where $(\cdot)_k$ is Pochhammer symbol, that is, $$(q)_k = \frac{\Gamma(q+k)}{\Gamma(q)}$$ provided that $q+k$ is not a negative integer, with the convention \$1/{\pm\infty} = … Continue reading