Integral Formula For The Hypergeometric Series
Integral Formula For The Hypergeometric Series : Theorem : The hypergeometric function F(α,β;γ;x) can be represented by [Γ(γ)/Γ(β)Γ(γ-β)] 1 ∫(1-t)^(γ-β-1) t^(β-1) (1-xt)^-α dt 0 Proof : We know that (β)ₙ = β(β+1)...(β+n-1) = Γ(β+n)/Γ(β) Similarly (γ)ₙ = Γ(γ+n)/Γ(γ) Again B(m,n) = Γ(m)Γ(n)/Γ(m+n) Consider, B(β+n,γ-β) / B(β,γ-β) = [Γ(β+n)Γ(γ-β)/Γ(γ+n)]/[Γ(β)Γ(γ-β)/Γ(γ) = [Γ(β+n)/Γ(β)] / [Γ(γ+n)/Γ(γ)] Thus (β)ₙ/(γ)ₙ = B(β+n, γ-β) / B(β, γ-β) From the definition of Beta function , we have 1 B(p,q) = ∫ tᵖ⁻¹ (1-t)^(q-1) dt , t<1. 0 Therefore, (β)ₙ/(γ)ₙ = 1/B(β, γ-β) 1 ∫ t^(β+n-1) (1-t)^(γ-β-1) dt 0