For half-odd-integer values of the exponent parameters:

the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows.

Denote and . Then

are two independent solutions of (31.2.1). Here is a polynomial of degree in and of degree in , that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. The degree is given by

The variables and are two coordinates of the associated hyperelliptic (spectral) curve . (This is unrelated to the in §31.6.) Lastly, , , are the zeros of the Wronskian of and .

By automorphisms from §31.2(v), similar solutions also exist for , and may become a rational function in . For instance,

and

For , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. The curve reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for . When approaches the ends of the gaps, the solution (31.8.2) becomes the corresponding Heun polynomial. For more details see Smirnov (2002).

The solutions in this section are finite-term Liouvillean solutions which can be constructed via Kovacic’s algorithm; see §31.14(ii).