13.14.1 | |||
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:
13.14.2 | |||
13.14.3 | |||
except that does not exist when .
Conversely,
13.14.4 | |||
13.14.5 | |||
The series
13.14.6 | |||
, | |||
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
13.14.7 | |||
If , where , then
13.14.8 | |||
, | |||
or
13.14.9 | |||
. | |||
13.14.10 | |||
Except when , each branch of the functions and is entire in and . Also, unless specified otherwise and are assumed to have their principal values.
13.14.14 | |||
. | |||
In cases when , where is a nonnegative integer,
13.14.15 | |||
In all other cases
13.14.16 | |||
, , | |||
13.14.17 | |||
13.14.18 | |||
, , | |||
13.14.19 | |||
For with use (13.14.31).
Except when (polynomial cases),
13.14.20 | |||
, | |||
where is an arbitrary small positive constant. Also,
13.14.21 | |||
. | |||
Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are
13.14.22 | |||
, | |||
, | |||
13.14.23 | |||
, | |||
. | |||
A fundamental pair of solutions that is numerically satisfactory in the sector near the origin is
13.14.24 | |||
, | |||
. | |||
13.14.25 | |||
13.14.26 | |||
13.14.27 | |||
13.14.28 | |||
13.14.29 | |||
13.14.30 | |||
13.14.31 | |||
13.14.32 | |||
When is not an integer
13.14.33 | |||
13.14.34 | |||
13.14.35 | |||