Heun's Biconfluent equation,
can be transformed into another version of itself, that is, an equation with one regular and one irregular singularity respectively located at 0 and through transformations of the form
where are new variables, and . Under this transformation, the HeunB parameters transform according to -> , -> , -> and -> . These transformations form a group and imply on a number of identities, among which you have
A relation between HeunB and the confluent 1F1 hypergeometric function is
When, in HeunB(,,,,z), , with a positive integer, the th coefficient in the series expansion is a polynomial in of order . If is a root of that polynomial, that th coefficient and the subsequent ones are zero. The series then truncates and HeunB reduces to a polynomial. For example, this is the necessary condition for a polynomial form
Considering the first non-trivial case, for , the function is
So the coefficient of in the series expansion is
solving for , requesting from solve to return using RootOf, you have
substituting in we have
When the function admits a polynomial form, as is the case of by construction, to obtain the actual polynomial of degree (in this case ) use