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31 Heun FunctionsProperties

§31.9 Orthogonality

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§31.9(i) Single Orthogonality

With

31.9.1 wm(z)=(0,1)Hfm(a,qm;α,β,γ,δ;z),

we have

31.9.2 ζ(1+,0+,1-,0-)tγ-1(1-t)δ-1(t-a)ϵ-1wm(t)wk(t)dt=δm,kθm.

Here ζ is an arbitrary point in the interval (0,1). The integration path begins at z=ζ, encircles z=1 once in the positive sense, followed by z=0 once in the positive sense, and so on, returning finally to z=ζ. The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning.

The normalization constant θm is given by

31.9.3 θm=(1-e2πiγ)(1-e2πiδ)ζγ(1-ζ)δ(ζ-a)ϵf0(q,ζ)f1(q,ζ)q𝒲{f0(q,ζ),f1(q,ζ)}|q=qm,

where

31.9.4 f0(qm,z) =H(a,qm;α,β,γ,δ;z),
f1(qm,z) =H(1-a,αβ-qm;α,β,δ,γ;1-z),

and 𝒲 denotes the Wronskian (§1.13(i)). The right-hand side may be evaluated at any convenient value, or limiting value, of ζ in (0,1) since it is independent of ζ.

For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64).

§31.9(ii) Double Orthogonality

Heun polynomials wj=Hpnj,mj, j=1,2, satisfy

31.9.5 12ρ(s,t)w1(s)w1(t)w2(s)w2(t)dsdt=0,
|n1-n2|+|m1-m2|0,

where

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

and the integration paths 1, 2 are Pochhammer double-loop contours encircling distinct pairs of singularities {0,1}, {0,a}, {1,a}.

For further information, including normalization constants, see Sleeman (1966a). For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2). For other generalizations see Arscott (1964b, pp. 206–207 and 241).