Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i). There are four standard forms, as follows:
This has regular singularities at and 1, and an irregular singularity of rank 1 at .
This has irregular singularities at and , each of rank 1.
This has a regular singularity at , and an irregular singularity at of rank 2.
This has one singularity, an irregular singularity of rank 3 at .