Solution

Set

u = \frac{1 - z}{2} .

This Möbius change sends

z = 1, - 1, \infty

u = 0, 1, \infty

, respectively, without changing the corresponding exponents. The hypergeometric P-symbol has exponents

00 1 - C 10 C - A - B \infty A B .

Matching it with

(0, 1/2)

at both finite singularities and

(- n, n)

at infinity gives

C = \frac{1}{2}, A = n, B = - n .

The exponent-zero solution normalized to one at

u = 0

is therefore

w_{1} (z) = F (n, - n; \frac{1}{2}; \frac{1 - z}{2}) .

The second standard local solution at

u = 0

u^{1 - C} F (A - C + 1, B - C + 1; 2 - C; u) .

After absorbing the constant

2^{- 1/2}

into its normalization, this becomes

w_{2} (z) = (1 - z)^{1/2} F (- n + \frac{1}{2}, n + \frac{1}{2}; \frac{3}{2}; \frac{1 - z}{2}) .

Their exponents at

z = 1

are

0

and

1/2

, so they are linearly independent.

Solved by gpt-5.6-sol high.

 Ancestors (11)