About the Project
31 Heun FunctionsApplications

§31.16 Mathematical Applications

Contents
  1. §31.16(i) Uniformization Problem for Heun’s Equation
  2. §31.16(ii) Heun Polynomial Products

§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=0nAjsin2jθPnj(γ+δ+2j1,ϵ1)(cos(2θ))×Pj(δ1,γ1)(cos(2ϕ)),

where n=0,1,, m=0,1,,n, and x,y are implicitly defined by

31.16.2 xy =asin2θcos2ϕ,
(x1)(y1) =(1a)sin2θsin2ϕ,
(xa)(ya) =a(a1)cos2θ.

The coefficients Aj satisfy the relations:

31.16.3 A0=n!(γ+δ)n𝐻𝑝n,m(1),Q0A0+R0A1=0,
31.16.4 PjAj1+QjAj+RjAj+1=0,
j=1,2,,n,

where

31.16.5 Pj =(ϵj+n)j(β+j1)(γ+δ+j2)(γ+δ+2j3)(γ+δ+2j2),
31.16.6 Qj =aj(j+γ+δ1)q+(jn)(j+β)(j+γ)(j+γ+δ1)(2j+γ+δ)(2j+γ+δ1)+(j+n+γ+δ1)j(j+δ1)(jβ+γ+δ1)(2j+γ+δ1)(2j+γ+δ2),
31.16.7 Rj =(nj)(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.