The series of Type I (§31.11(iii)) are useful since they represent the functions in large domains. Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i).
For other expansions see §31.16(ii).
Let be any Fuchs–Frobenius solution of Heun’s equation. Expand
The Fuchs-Frobenius solutions at are
Taking or the coefficients satisfy the equations
where we take and where
, must also satisfy the condition
Then condition (31.11.9) is satisfied.
Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. For instance, choose (31.11.10). Then the Fuchs–Frobenius solution at belonging to the exponent has the expansion (31.11.1) with
and (31.11.1) converges to (31.3.10) outside the ellipse in the -plane with foci at 0, 1, and passing through the third finite singularity at .
Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function.
For example, consider the Heun function which is analytic at and has exponent at . The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities and . In this case the accessory parameter is a root of the continued-fraction equation
The case for nonnegative integer corresponds to the Heun polynomial .
Here one of the following four pairs of conditions is satisfied:
In each case can be expressed in terms of a Jacobi polynomial (§18.3). Such series diverge for Fuchs–Frobenius solutions. For Heun functions (§31.4) they are convergent inside the ellipse . Every Heun function can be represented by a series of Type II.