The formulas in this section are given in Svartholm (1939) and Erdélyi (1942b, 1944).

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 $w(z)$ be any Fuchs–Frobenius solution of Heun’s equation. Expand

where (§15.11(i))

with

The Fuchs-Frobenius solutions at $\mathrm{\infty }$ are

Taking $P_j=P_j^5$ or $P_j=P_j^6$ the coefficients $c_j$ satisfy the equations

where we take $c_0=1$ and where

$\lambda$, $\mu$ must also satisfy the condition

Here

or

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 $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with

and (31.11.1) converges to (31.3.10) outside the ellipse $ℰ$ in the $z$-plane with foci at 0, 1, and passing through the third finite singularity at $z=a$.

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 $z=a$ and has exponent $\alpha$ at $\mathrm{\infty }$. The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities $z=0$ and $z=1$. In this case the accessory parameter $q$ is a root of the continued-fraction equation

The case $\alpha =-n$ for nonnegative integer $n$ corresponds to the Heun polynomial ${\mathrm{𝐻𝑝}}_{n,m}\left(z\right)$.

The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse $ℰ$.

Here one of the following four pairs of conditions is satisfied:

In each case $P_j^6$ 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 $ℰ$. Every Heun function can be represented by a series of Type II.

Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions.