[Calculation 3] Euler’s Transformation Formula

c3

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 1] Fundamentals of Hypergeometric Functions

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