# [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

# [M2 Seminar I] Week 6 : 끝이 보이기 시작하다

2013년 5월 30일 (목) Fourier series가 포함된 계산들을 하다보니, 개념은 간단한 것인데도 불구하고 수식이 너무 복잡해져 버렸다 OTL. 내일이 세미나인데 빨리 다른 표현을 찾지 않으면… 아래는 작업하던거 일부분 복사+붙여넣기: (이후삭제) 오늘도 계산의 소용돌이 속에서 별이 바람에 스치운다..   2013년 5월 … Continue reading

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