This problem is a brief introduction to Gauss’s hypergeo- metric equationwhere α, β, and γ...

This problem is a brief introduction to Gauss’s hypergeo- metric equation

where α, β, and γ are constants. This famous equation has wide-ranging applications in mathematics and physics. (a) Show that x = 0 is a regular singular point of Eq, with exponents 0 and 1 − γ. (b) If γ is not zero or a negative integer, it follows (why?) that Eq has a power series solution

with c0 ≠ 0. Show that the recurrence relation for this series is

for n ≧ 0. (c) Conclude that with c0 = 1 the series in part (b) is

where αn = α(α + 1)(α + 2) ? (α + n − 1) for n ≧ 1, and (βn and γn are defined similarly. (d) The series in (36) is known as the hypergeometric series and is commonly denoted by F(α, β, γ, x). Show that