Special functions Hypergeometric functions Theorem

Kummer's 24 solutions

Builds on Hypergeometric differential equation Hypergeometric function Pfaff and Euler transformations Möbius transformation

Statement

Let $a, b, c \in \C$ be such that none of $c$ , $c - a - b$ , $a - b$ is an integer, and consider the hypergeometric differential equation with these parameters. Let ${}_2F_1$ be the hypergeometric function.

The following six functions are solutions, each on a suitable simply connected domain: $\begin{aligned} w_1 &= {}_2F_1(a, b; c; z), & w_2 &= z^{1-c} \, {}_2F_1(a - c + 1, b - c + 1; 2 - c; z), \\ w_3 &= {}_2F_1(a, b; a + b - c + 1; 1 - z), & w_4 &= (1 - z)^{c-a-b} \, {}_2F_1(c - a, c - b; c - a - b + 1; 1 - z), \\ w_5 &= z^{-a} \, {}_2F_1(a, a - c + 1; a - b + 1; 1/z), & w_6 &= z^{-b} \, {}_2F_1(b, b - c + 1; b - a + 1; 1/z). \end{aligned}$
Applying the Pfaff and Euler transformations to each $w_i$ gives four expressions $z^{\lambda} (1 - z)^{\mu} \, {}_2F_1(a', b'; c'; \xi)$ . In each, $\xi$ is one of $z$ , $1 - z$ , $1/z$ , $1/(1 - z)$ , $z/(z - 1)$ , $(z - 1)/z$ . This gives $24$ expressions of solutions, falling into six groups of four equal functions.

Remarks

The six variables are the images of $z$ under the six Möbius transformations that permute $\set{0, 1, \infty}$ . Each such map carries the equation to a hypergeometric equation with permuted local exponents. The solutions $w_1, \ldots, w_6$ are the pairs of local solutions at $0$ , $1$ and $\infty$ , each with a pure power behaviour there. The hypothesis that $c$ , $c - a - b$ and $a - b$ are not integers makes the two exponents at each singular point differ by a non-integer, so each pair is a basis. Without it, some of the six formulas are undefined or coincide, and logarithmic solutions appear.

Example: the group of $w_1$ consists of ${}_2F_1(a, b; c; z) = (1 - z)^{-a} \, {}_2F_1\left(a, c - b; c; \tfrac{z}{z - 1}\right) = (1 - z)^{-b} \, {}_2F_1\left(c - a, b; c; \tfrac{z}{z - 1}\right) = (1 - z)^{c - a - b} \, {}_2F_1(c - a, c - b; c; z).$

Since the solution space is two-dimensional, any three of the $w_i$ are linearly dependent. For instance, on $\C \setminus \big((-\infty, 0] \cup [1, \infty)\big)$ , $w_1 = \frac{\Gamma(c) \Gamma(c - a - b)}{\Gamma(c - a) \Gamma(c - b)} \, w_3 + \frac{\Gamma(c) \Gamma(a + b - c)}{\Gamma(a) \Gamma(b)} \, w_4,$ and the first coefficient is the value given by Gauss's hypergeometric theorem.

Going deeper

For a student

The list is the tool for continuing ${}_2F_1$ from one singular point to another. Linear relations between solutions at different points, the connection formulas, follow from Gauss's theorem and give the monodromy explicitly.

Reading path

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Hypergeometric differential equation concept
Hypergeometric function concept
Möbius transformation concept
Pfaff and Euler transformations statement
Kummer's 24 solutions statement

Tag	`7QFW771QMA`
URI	`m2://7QFW771QMA`
Kind	statement, theorem
Writing	W0 stub
MSC	33C05, 34M03
Status	This copy of the site has no build data. Run `m2 status 7QFW771QMA`.
About	Hypergeometric differential equation
Uses	Hypergeometric differential equation, Hypergeometric function, Pfaff and Euler transformations, Möbius transformation