Pfaff-Saalschütz Theorem/Examples/3F2(0.5,0.5,-2;1.5,-1.5;1)

Example of Use of Pfaff-Saalschütz Theorem

 * $\ds \map { {}_3 \operatorname F_2} { { {\dfrac 1 2, \dfrac 1 2, -2} \atop {\dfrac 3 2, -\dfrac 3 2} } \, \middle \vert \, 1} = \dfrac {64} {45}$

Proof
From Pfaff-Saalschütz Theorem:


 * $\ds \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1} = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } { c^{\overline n} \paren {c - a - b}^{\overline n} }$

where:
 * ${}_3 \operatorname F_2$ is the generalized hypergeometric function
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$.

We have:

and:

Therefore:
 * $\ds \map { {}_3 \operatorname F_2} { { {\dfrac 1 2, \dfrac 1 2, -2} \atop {\dfrac 3 2, -\dfrac 3 2} } \, \middle \vert \, 1} = \dfrac {64} {45}$