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31 Heun FunctionsProperties

§31.4 Solutions Analytic at Two Singularities: Heun Functions

For an infinite set of discrete values qm, m=0,1,2,, of the accessory parameter q, the function H(a,q;α,β,γ,δ;z) is analytic at z=1, and hence also throughout the disk |z|<a. To emphasize this property this set of functions is denoted by

31.4.1 (0,1)Hfm(a,qm;α,β,γ,δ;z),
m=0,1,2,.

The eigenvalues qm satisfy the continued-fraction equation

31.4.2 q=aγP1Q1+q-R1P2Q2+q-R2P3Q3+q-,

in which Pj,Qj,Rj are as in §31.3(i).

More generally,

31.4.3 (s1,s2)Hfm(a,qm;α,β,γ,δ;z),
m=0,1,2,,

with (s1,s2){0,1,a,}, denotes a set of solutions of (31.2.1), each of which is analytic at s1 and s2. The set qm depends on the choice of s1 and s2.

The solutions (31.4.3) are called the Heun functions. See Ronveaux (1995, pp. 39–41).