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31 Heun FunctionsProperties

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

Let α=-n, n=0,1,2,, and qn,m, m=0,1,,n, be the eigenvalues of the tridiagonal matrix

31.5.1 [0aγ00P1-Q1R100P2-Q2Rn-100Pn-Qn],

where Pj,Qj,Rj are again defined as in §31.3(i). Then

31.5.2 Hpn,m(a,qn,m;-n,β,γ,δ;z)=H(a,qn,m;-n,β,γ,δ;z)

is a polynomial of degree n, and hence a solution of (31.2.1) that is analytic at all three finite singularities 0,1,a. These solutions are the Heun polynomials. Some properties are included as special cases of properties given in §31.15 below.