# §31.3 Basic Solutions

## §31.3(i) Fuchs–Frobenius Solutions at

denotes the solution of (31.2.1) that corresponds to the exponent 0 at and assumes the value 1 there. If the other exponent is not a positive integer, that is, if , then from §2.7(i) it follows that exists, is analytic in the disk , and has the Maclaurin expansion

where ,

with

When , linearly independent solutions can be constructed as in §2.7(i). In general, one of them has a logarithmic singularity at .

## §31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

With similar restrictions to those given in §31.3(i), the following results apply. Solutions of (31.2.1) corresponding to the exponents 0 and at are respectively,

31.3.7

## §31.3(iii) Equivalent Expressions

Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. Each is related to the solution (31.3.1) by one of the automorphisms of §31.2(v). There are 192 automorphisms in all, so there are equivalent expressions for each of the 8. For example, is equal to

which arises from the homography , and to

which arises from , and also to 21 further expressions. The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).