Integral Representation Of Confluent Hypergeometric Function
Integral Representation Of Confluent Hypergeometric Function : Theorem : If γ>α>0 , then the function F(α;γ;x) can be expressed as 1 Γ(γ)/Γ(γ)Γ(γ-α) ∫ eˣᵗ t^(α-1)(1-t)^(γ-α-1) dt. 0 Proof : We know that B(α+n,γ-α)/B(α,γ-α) = Γ(α+n)Γ(γ-α)/Γ(γ+n) / Γ(α)Γ(γ-α)/Γ(γ) = Γ(α+n)/Γ(α) / Γ(γ+n)/Γ(γ) But Γ(α+n)/Γ(α) = (α)ₙ and Γ(γ+n)/Γ(γ) = (γ)ₙ Therefore, (α)ₙ/(γ)ₙ = B(α+n,γ-α)/Β(α,γ-α) 1 = Γ(γ)/Γ(α)Γ(γ-α) ∫ t^(α+n-1) (1-t)^(γ-α-1) dt 0 ∞ Now F(α;γ;x) = Σ (α)ₙxⁿ/n!(γ)ₙ n=0 = Γ(γ)/Γ(α)Γ(γ-α) ∞ 1 Σ xⁿ/n! ∫ t^(α-1) tⁿ(1-t)^(γ-α-1) dt n=0 0 Interchanging the order of summa