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§31.16 Mathematical Applications


§31.16(i) Uniformization Problem for Heun’s Equation

The main part of Smirnov (1996) consists of V. I. Smirnov’s 1918 M. Sc. thesis “Inversion problem for a second-order linear differential equation with four singular points”. It describes the monodromy group of Heun’s equation for specific values of the accessory parameter.

§31.16(ii) Heun Polynomial Products

Expansions of Heun polynomial products in terms of Jacobi polynomial (§18.3) products are derived in Kalnins and Miller (1991a, b, 1993) from the viewpoint of interrelation between two bases in a Hilbert space:

31.16.1 Hpn,m(x)Hpn,m(y)=j=0nAjsin2jθPn-j(γ+δ+2j-1,ϵ-1)×(cos2θ)Pj(δ-1,γ-1)(cos2ϕ),

where n=0,1,, m=0,1,,n, and

31.16.2 x =sin2θcos2ϕ,
y =sin2θsin2ϕ.

The coefficients Aj satisfy the relations:

31.16.3 Q0A0+R0A1=0,
31.16.4 PjAj-1+QjAj+RjAj+1=0,


31.16.5 Pj =(ϵ-j+n)j(β+j-1)(γ+δ+j-2)(γ+δ+2j-3)(γ+δ+2j-2),
31.16.6 Qj =-aj(j+γ+δ-1)-q+(j-n)(j+β)(j+γ)(j+γ+δ-1)(2j+γ+δ)(2j+γ+δ-1)+(j+n+γ+δ-1)j(j+δ-1)(j-β+γ+δ-1)(2j+γ+δ-1)(2j+γ+δ-2),
31.16.7 Rj =(n-j)(j+n+γ+δ)(j+γ)(j+δ)(γ+δ+2j)(γ+δ+2j+1).

By specifying either θ or ϕ in (31.16.1) and (31.16.2) we obtain expansions in terms of one variable.