§31.3 Basic Solutions

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. If the other exponent is not a positive integer, that is, if $\gamma \ne 0,-1,-2,\mathrm{\dots }$, then from §2.7(i) it follows that $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ exists, is analytic in the disk , and has the Maclaurin expansion

where $c_0=1$,

with

Similarly, if $\gamma \ne 1,2,3,\mathrm{\dots }$, then the solution of (31.2.1) that corresponds to the exponent $1-\gamma$ at $z=0$ is

When $\gamma \in \mathbb{Z}$, linearly independent solutions can be constructed as in §2.7(i). In general, one of them has a logarithmic singularity at $z=0$.

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 $1-\delta$ at $z=1$ are respectively,

Solutions of (31.2.1) corresponding to the exponents $0$ and $1-ϵ$ at $z=a$ are respectively,

Solutions of (31.2.1) corresponding to the exponents $\alpha$ and $\beta$ at $z=\mathrm{\infty }$ are respectively,

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 $192/8=24$ equivalent expressions for each of the 8. For example, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is equal to

which arises from the homography $\tilde{z}=z/a$, and to

which arises from $\tilde{z}=z/(z-1)$, 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).