Math 213a: Complex analysis Problem Set #9 (19 November 2003): Hypergeometric functions

Recall that the hypergeometric function F (a, b; c; z) (a.k.a. _{2}F_{1}(a, b; c; z)) is defined by the power series

1 + \frac{ab}{c}z + \frac{a(a + 1)b(b + 1)}{c(c + 1)} \frac{z^{2}}{2!} + \frac{a(a + 1)(a + 2)b(b + 1)(b + 2)}{c(c + 1)(c + 2)} \frac{z^{3}}{3!} + ⋯

(|z| < 1, c ≠ 0, −1, −2, …), and satisfies the hypergeometric differential equation

(z − z^{2})w′′ + (c − (a + b + 1)z)w′ − abw = 0.

1. Prove that if Re(c) > Re(b) > 0 then

F (a, b; c; z) = \frac{1}{B(b, c − b)} ∫_{0}^{1} t^{b−1}(1 − t)^{c−b−1}(1 − zt)^{−a} dt,

and deduce the value of F (a, b; c; 1) = lim_{z→1−} F (a, b; c; z).

2. Prove the identity F (a, b; c; z) = (1−z)^{−a}F (a, c−b; c; z/(z −1)). What is the relation between F (a, b; c; z) and F (c − a, c − b; c; z)? [Note that (a, b, c) and (c − a, c − b, c) yield the same angles π|1 − c|, π|c − a − b|, π|a − b| at z = 0, 1, ∞.]

3. Suppose 2c = a + b + 1.

i) Prove that F (a, b; c; z) = F (a/2, b/2; c; 4z(1 − z)), and use this to evaluate F (a, b; c; 1/2).

ii) Show that (unless a, b, c ∈ {0, −1, −2, …}) F (a, b; c; z) and F (a, b; c; 1−z) constitute a basis of the solutions of the same hypergeometric differential equation.

4. Assume further that a+b = 0, so c = 1/2. Evaluate F (a, −a; 1/2; z) in closed form, and verify directly that if 0 < a < 1 then

F (a, −a; 1/2; z)/F (a, −a; 1/2; 1 − z)

conformally maps the upper half-plane to a spherical triangle on the Riemann sphere.

5.^{∗} In the first problem set we encountered the finite subgroup G ⊂ PGL_{2}(C) generated by the fractional linear transformations w ↦ (1+w)/(1−w) and w ↦ iw. Find an explicit rational function z(w) such that z(w) = z(w′) if and only if w′ is in the orbit G(w), normalized so that if w = g(w) for some nontrivial g ∈ G then z(w) ∈ {0, 1, ∞}. Express (a branch of) the inverse function w(z) explicitly in terms of hypergeometric functions. Be sure to check that the first few coefficients of the power-series expansion of your formula do agree with the desired inverse function.

It is known that G ≅ S_{4}, and that there are two further exceptional discrete subgroups of PGL_{2}(C): the “tetrahedral” and “icosahedral” groups, isomorphic with A_{4} and A_{5}. Be thankful that I didn’t ask you to do the icosahedral version of this exercise!

6. Fix distinct z_{1}, …, z_{n} ∈ P^{1}(C) and α_{j}, β_{j} ∈ C (j = 1, …, n) such that α_{j} − β_{j} ∉ Z. We seek differential equations w′′ + pw′ + qw = 0 whose only singularities are regular singular points at the z_{j} with exponents α_{j}, β_{j}. When n = 3 we saw, following Riemann, that such an equation exists if and only if ∑_{j=1}^{3} α_{j} + β_{j} = 1, and then the equation is uniquely determined. Find the analogous condition on the α_{j}, β_{j} when n > 3, and show that under this condition the equation is determined up to n − 3 parameters.

This problem set is due Wednesday, November 26, at the beginning of class.