[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

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
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
Now we will introduce some lemmas for the proof of the global existence theorem for the nonlinear wave equations with quadratic nonlinearities. The exsistence theorem which was proven before gives $n >5$ for the quadratic nonlinearities. But in fact this … Continue reading