# §15.10 Hypergeometric Differential Equation

## §15.10(i) Fundamental Solutions

This is the hypergeometric differential equation. It has regular singularities at , with corresponding exponent pairs , , , respectively. When none of the exponent pairs differ by an integer, that is, when none of , , is an integer, we have the following pairs , of fundamental solutions. They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity.

### ¶ Singularity

(a) If equals , and , then fundamental solutions in the neighborhood of are given by (15.10.2) with the interpretation (15.2.5) for .

(b) If equals , and , then fundamental solutions in the neighborhood of are given by and

Moreover, in (15.10.9) and (15.10.10) the symbols and are interchangeable.

(c) If the parameter in the differential equation equals , then fundamental solutions in the neighborhood of are given by times those in (a) and (b), with and replaced throughout by and , respectively.

(d) If equals , or , then fundamental solutions in the neighborhood of are given by those in (a), (b), and (c) with replaced by .

(e) Finally, if equals , or , then fundamental solutions in the neighborhood of are given by times those in (a), (b), and (c) with and replaced by and , respectively.

## §15.10(ii) Kummer’s 24 Solutions and Connection Formulas

The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions.

The connection formulas for the principal branches of Kummer’s solutions are: