This equation is obtained from Kummer’s equation (13.2.1) via the substitutions , , and . It has a regular singularity at the origin with indices , and an irregular singularity at infinity of rank one.
Standard solutions are:
except that does not exist when .
converge for all .
In general and are many-valued functions of with branch points at and . The principal branches correspond to the principal branches of the functions and on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i).
Although does not exist when , many formulas containing continue to apply in their limiting form. For example, if , then
If , where , then
Except when , each branch of the functions and is entire in and . Also, unless specified otherwise and are assumed to have their principal values.
In cases when , where is a nonnegative integer,
In all other cases
For with use (13.14.31).
Except when (polynomial cases),
where is an arbitrary small positive constant. Also,
A fundamental pair of solutions that is numerically satisfactory in the sector near the origin is
When is not an integer