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31 Heun FunctionsProperties

§31.3 Basic Solutions


§31.3(i) Fuchs–Frobenius Solutions at z=0

H(a,q;α,β,γ,δ;z) denotes the solution of (31.2.1) that corresponds to the exponent 0 at z=0 and assumes the value 1 there. If the other exponent is not a positive integer, that is, if γ0,-1,-2,, then from §2.7(i) it follows that H(a,q;α,β,γ,δ;z) exists, is analytic in the disk |z|<1, and has the Maclaurin expansion

31.3.1 H(a,q;α,β,γ,δ;z)=j=0cjzj,

where c0=1,

31.3.2 aγc1-qc0=0,
31.3.3 Rjcj+1-(Qj+q)cj+Pjcj-1=0,


31.3.4 Pj =(j-1+α)(j-1+β),
Qj =j((j-1+γ)(1+a)+aδ+ϵ),
Rj =a(j+1)(j+γ).

Similarly, if γ1,2,3,, then the solution of (31.2.1) that corresponds to the exponent 1-γ at z=0 is

31.3.5 z1-γH(a,(aδ+ϵ)(1-γ)+q;α+1-γ,β+1-γ,2-γ,δ;z).

When γ, linearly independent solutions can be constructed as in §2.7(i). In general, one of them has a logarithmic singularity at z=0.

§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

With similar restrictions to those given in §31.3(i), the following results apply. Solutions of (31.2.1) corresponding to the exponents 0 and 1-δ at z=1 are respectively,

31.3.6 H(1-a,αβ-q;α,β,δ,γ;1-z),
31.3.7 (1-z)1-δH(1-a,((1-a)γ+ϵ)(1-δ)+αβ-q;α+1-δ,β+1-δ,2-δ,γ;1-z).

Solutions of (31.2.1) corresponding to the exponents 0 and 1-ϵ at z=a are respectively,

31.3.8 H(aa-1,αβa-qa-1;α,β,ϵ,δ;a-za-1),
31.3.9 (a-za-1)1-ϵH(aa-1,(a(δ+γ)-γ)(1-ϵ)a-1+αβa-qa-1;α+1-ϵ,β+1-ϵ,2-ϵ,δ;a-za-1).

Solutions of (31.2.1) corresponding to the exponents α and β at z= are respectively,

31.3.10 z-αH(1a,α(β-ϵ)+αa(β-δ)-qa;α,α-γ+1,α-β+1,δ;1z),
31.3.11 z-βH(1a,β(α-ϵ)+βa(α-δ)-qa;β,β-γ+1,β-α+1,δ;1z).

§31.3(iii) Equivalent Expressions

Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. Each is related to the solution (31.3.1) by one of the automorphisms of §31.2(v). There are 192 automorphisms in all, so there are 192/8=24 equivalent expressions for each of the 8. For example, H(a,q;α,β,γ,δ;z) is equal to

31.3.12 H(1/a,q/a;α,β,γ,α+β+1-γ-δ;z/a),

which arises from the homography z~=z/a, and to

31.3.13 (1-z)-αH(aa-1,-q-aαγa-1;α,α+1-δ,γ,α+1-β;zz-1),

which arises from z~=z/(z-1), and also to 21 further expressions. The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).