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31 Heun FunctionsProperties

§31.8 Solutions via Quadratures

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

31.8.1 β-α =m0+12,
γ =-m1+12,
δ =-m2+12,
ϵ =-m3+12,
m0,m1,m2,m3=0,1,2,,

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 m=(m0,m1,m2,m3) and λ=-4q. Then

31.8.2 w±(m;λ;z)=Ψg,N(λ,z)exp(±iν(λ)2z0ztm1(t-1)m2(t-a)m3dtΨg,N(λ,t)t(t-1)(t-a))

are two independent solutions of (31.2.1). Here Ψg,N(λ,z) is a polynomial of degree g in λ and of degree N=m0+m1+m2+m3 in z, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. The degree g is given by

31.8.3 g=12max(2max0k3mk,1+N-(1+(-1)N)(12+min0k3mk)).

The variables λ and ν are two coordinates of the associated hyperelliptic (spectral) curve Γ:ν2=j=12g+1(λ-λj). (This ν is unrelated to the ν in §31.6.) Lastly, λj, j=1,2,,2g+1, are the zeros of the Wronskian of w+(m;λ;z) and w-(m;λ;z).

By automorphisms from §31.2(v), similar solutions also exist for m0,m1,m2,m3, and Ψg,N(λ,z) may become a rational function in z. For instance,

31.8.4 Ψ1,2 =z2+λz+a,
ν2 =(λ+a+1)(λ2-4a),
m=(1,1,0,0),

and

31.8.5 Ψ1,-1 =(z2+(λ+3a+3)z+a)/z3,
ν2 =(λ+4a+4)((λ+3a+3)2-4a),
m=(1,-2,0,0).

For m=(m0,0,0,0), these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. The curve Γ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for mj. When λ=-4q 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).