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31 Heun FunctionsProperties

§31.10 Integral Equations and Representations

Contents

§31.10(i) Type I

If w(z) is a solution of Heun’s equation, then another solution W(z) (possibly a multiple of w(z)) can be represented as

31.10.1 W(z)=C𝒦(z,t)w(t)ρ(t)dt

for a suitable contour C. The weight function is given by

31.10.2 ρ(t)=tγ-1(t-1)δ-1(t-a)ϵ-1,

and the kernel 𝒦(z,t) is a solution of the partial differential equation

31.10.3 (𝒟z-𝒟t)𝒦=0,

where 𝒟z is Heun’s operator in the variable z:

31.10.4 𝒟z=z(z-1)(z-a)(2/z2)+(γ(z-1)(z-a)+δz(z-a)+ϵz(z-1))(/z)+αβz.

The contour C must be such that

31.10.5 p(t)(𝒦tw(t)-𝒦dw(t)dt)|C=0,

where

31.10.6 p(t)=tγ(t-1)δ(t-a)ϵ.

Kernel Functions

Set

31.10.7 cosθ =(zta)1/2,
sinθcosϕ =i((z-a)(t-a)a(1-a))1/2,
sinθsinϕ =((z-1)(t-1)1-a)1/2.

The kernel 𝒦 must satisfy

31.10.8 sin2θ(2𝒦θ2+((1-2γ)tanθ+2(δ+ϵ-12)cotθ)𝒦θ-4αβ𝒦)+2𝒦ϕ2+((1-2δ)cotϕ-(1-2ϵ)tanϕ)𝒦ϕ=0.

The solutions of (31.10.8) are given in terms of the Riemann P-symbol (see §15.11(i)) as

31.10.9 𝒦(θ,ϕ)=P{01012-δ-σαcos2θ1-γ12-ϵ+σβ}P{0100-12+δ+σcos2ϕ1-ϵ1-δ-12+ϵ-σ},

where σ is a separation constant. For integral equations satisfied by the Heun polynomial Hpn,m(z) we have σ=12-δ-j, j=0,1,,n.

For suitable choices of the branches of the P-symbols in (31.10.9) and the contour C, we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution).

Example 1

Let

31.10.10 𝒦(z,t)=(zt-a)12-δ-σF12(12-δ-σ+α,12-δ-σ+βγ;zta)F12(-12+δ+σ,-12+ϵ-σδ;a(z-1)(t-1)(a-1)(zt-a)),

where γ>0, δ>0, and C be the Pochhammer double-loop contour about 0 and 1 (as in §31.9(i)). Then the integral equation (31.10.1) is satisfied by w(z)=wm(z) and W(z)=κmwm(z), where wm(z)=(0,1)Hfm(a,qm;α,β,γ,δ;z) and κm is the corresponding eigenvalue.

Example 2

Fuchs–Frobenius solutions Wm(z)=κ~mz-αH(1/a,qm;α,α-γ+1,α-β+1,δ;1/z) are represented in terms of Heun functions wm(z)=(0,1)Hfm(a,qm;α,β,γ,δ;z) by (31.10.1) with W(z)=Wm(z), w(z)=wm(z), and with kernel chosen from

31.10.11 𝒦(z,t)=(zt-a)12-δ-σ(zt/a)-12+δ+σ-α×F12(12-δ-σ+α,32-δ-σ+α-γα-β+1;azt)×P{0100-12+δ+σ(z-a)(t-a)(1-a)(zt-a)1-ϵ1-δ-12+ϵ-σ}.

Here κ~m is a normalization constant and C is the contour of Example 1.

§31.10(ii) Type II

If w(z) is a solution of Heun’s equation, then another solution W(z) (possibly a multiple of w(z)) can be represented as

31.10.12 W(z)=C1C2𝒦(z;s,t)w(s)w(t)ρ(s,t)dsdt

for suitable contours C1, C2. The weight function is

31.10.13 ρ(s,t)=(s-t)(st)γ-1((1-s)(1-t))δ-1((1-(s/a))(1-(t/a)))ϵ-1,

and the kernel 𝒦(z;s,t) is a solution of the partial differential equation

31.10.14 ((t-z)𝒟s+(z-s)𝒟t+(s-t)𝒟z)𝒦=0,

where 𝒟z is given by (31.10.4). The contours C1, C2 must be chosen so that

31.10.15 p(t)(𝒦tw(t)-𝒦dw(t)dt)|C1 =0,
and
31.10.16 p(s)(𝒦sw(s)-𝒦dw(s)ds)|C2 =0,

where p(t) is given by (31.10.6).

Kernel Functions

Set

31.10.17 u =(stz)1/2a,
v =((s-1)(t-1)(z-1)1-a)1/2,
w =i((s-a)(t-a)(z-a)a(1-a))1/2.

The kernel 𝒦 must satisfy

31.10.18 2𝒦u2+2𝒦v2+2𝒦w2+2γ-1u𝒦u+2δ-1v𝒦v+2ϵ-1w𝒦w=0.

This equation can be solved in terms of cylinder functions 𝒞ν(z)10.2(ii)):

31.10.19 𝒦(u,v,w)=u1-γv1-δw1-ϵ𝒞1-γ(uσ1)𝒞1-δ(vσ2)𝒞1-ϵ(iwσ1+σ2),

where σ1 and σ2 are separation constants.

Transformation of Independent Variable

A further change of variables, to spherical coordinates,

31.10.20 u =rcosθ,
v =rsinθsinϕ,
w =rsinθcosϕ,

leads to the kernel equation

31.10.21 2𝒦r2+2(γ+δ+ϵ)-1r𝒦r+1r22𝒦θ2+(2(δ+ϵ)-1)cotθ-(2γ-1)tanθr2𝒦θ+1r2sin2θ2𝒦ϕ2+(2δ-1)cotϕ-(2ϵ-1)tanϕr2sin2θ𝒦ϕ=0.

This equation can be solved in terms of hypergeometric functions (§15.11(i)):

31.10.22 𝒦(r,θ,ϕ)=rmsin2pθP{0100acos2θ12(3-γ)cb}P{0100acos2ϕ1-ϵ1-δb},

with

31.10.23 m2+2(α+β)m-σ1 =0,
p2+(α+β-γ-12)p-14σ2 =0,
a+b =2(α+β+p)-1,
ab =p2-p(1-α-β)-14σ1,
c =γ-12-2(α+β+p),
a+b =δ+ϵ-1,
ab =-14σ2,

and σ1 and σ2 are separation constants.

For integral equations for special confluent Heun functions (§31.12) see Kazakov and Slavyanov (1996).