[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

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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 8] Gauss’s Quadratic Trasformation

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Theorem. (Gauss’s Quadratic Transformation) \begin{equation}\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). \end{equation}   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 5] Gauss’s Summation Formula

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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