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15 Hypergeometric FunctionProperties

§15.11 Riemann’s Differential Equation

Contents

§15.11(i) Equations with Three Singularities

The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). The most general form is given by

15.11.1 d2wdz2+(1-a1-a2z-α+1-b1-b2z-β+1-c1-c2z-γ)dwdz+((α-β)(α-γ)a1a2z-α+(β-α)(β-γ)b1b2z-β+(γ-α)(γ-β)c1c2z-γ)w(z-α)(z-β)(z-γ)=0,

with

15.11.2 a1+a2+b1+b2+c1+c2=1.

Here {a1,a2}, {b1,b2}, {c1,c2} are the exponent pairs at the points α, β, γ, respectively. Cases in which there are fewer than three singularities are included automatically by allowing the choice {0,1} for exponent pairs. Also, if any of α, β, γ, is at infinity, then we take the corresponding limit in (15.11.1).

The complete set of solutions of (15.11.1) is denoted by Riemann’s P-symbol:

15.11.3 w=P{αβγa1b1c1za2b2c2}.

In particular,

15.11.4 w=P{0100az1-cc-a-bb}

denotes the set of solutions of (15.10.1).

§15.11(ii) Transformation Formulas

A conformal mapping of the extended complex plane onto itself has the form

15.11.5 t=(κz+λ)/(μz+ν),

where κ, λ, μ, ν are real or complex constants such that κν-λμ=1. These constants can be chosen to map any two sets of three distinct points {α,β,γ} and {α~,β~,γ~} onto each other. Symbolically:

15.11.6 P{αβγa1b1c1za2b2c2}=P{α~β~γ~a1b1c1ta2b2c2}.

The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by

15.11.7 P{αβγa1b1c1za2b2c2}=(z-αz-γ)a1(z-βz-γ)b1P{0100a1+b1+c1(z-α)(β-γ)(z-γ)(β-α)a2-a1b2-b1a1+b1+c2}.

We also have

15.11.8 zλ(1-z)μP{01a1b1c1za2b2c2}=P{01a1+λb1+μc1-λ-μza2+λb2+μc2-λ-μ},

for arbitrary λ and μ.