Elementary Properties Of The Hypergeometric Function
Elementary Properties Of The Hyper geometric Function : In this section we consider some properties of the hyper geometric function which are immediate consequences of its definition by the series ∞ y₁= Σ (α)ₙ(β)ₙxⁿ/n!(γ)ₙ = F(α,β,;γ,x) n=0 γ≠0,-1,-2...... (9) (i) We observe that the terms of the series do not change if the parameters α and β are permuted (interchanged). Hence we obtain the symmetry property F(α,β;γ;x) = F(β,α;γ;x) .........(1) (ii) d/dx[ F(α,β;γ;x) ]= αβ F(α+1,β+1;γ+1;x)/γ Proof : From equation(9), we have ∞ F(α,β;γ;x) = 1+Σ (α)ₙ(β)ₙxⁿ/n!(γ)ₙ n=1 ∞ Now d/dx [F(α,β;γ;x)]=Σ (α)ₙ(β)ₙn.xⁿ⁻¹/n!(γ)ₙ n=1 ∞ = Σ (α)ₙ(β)