Hypergeometric Equation And Hypergeometric Function
Hypergeometric Equation And Hypergeometric Function : The linear equation x(1-x)y" + [γ-(1+α+β)x]y'-αβy = 0, .....(1) Where α,β and γ are constants which can take various real or complex values is called the Hypergeometric equation or the Gauss equation . The equation (1) can be put to the form y" + [γ-(1+α+β)x]y'/x(1-x) - αβy/x(1-x) = 0 or y"+(1+x+x²+...)[γ-(1+α+β)x]y'/x -αβ(1+x+x²+...)y/x = 0 or y" + p(x)y' + q(x)y = 0, Since xp(x) and x²q(x) can be expressed as power series , x=0 is a regular singularity . Hence the above equation can be solved by Frobenius method . We write ∞ y = Σ cₙ xⁿ⁺ʳ , where c₀≠0, n=0 so that y' = Σcₙ(n+r)xⁿ⁺ʳ⁻¹ and y" = Σcₙ(n+r)(n+r-1)xⁿ⁺ʳ⁻² Substituting these expressions into the equation(1), we have Σcₙ(n+r)(n+r-1+γ)xⁿ⁺ʳ⁻¹ - Σcₙ [(n+r)(n+r+