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)ₙxⁿ⁻¹/(n-1)!(γ)ₙ

                                  n=1

On putting k=n-1, the right hand side becomes 

             ∞

             Σ  (α)ₖ₊₁(β)ₖ₊₁ xᵏ/k!(γ)ₖ₊₁

            k=0

We know that 

       (α)ₖ₊₁ = α(α+1)....(α+k)

Also (α+1)ₖ = (α+1)(α+2)...(α+k)

Thus   (α)ₖ₊₁ = α(α+1)ₖ

and hence 

     

                           

d/dx [F(α,β;γ;x)]

                      ∞

                   = Σ α(α+1)ₖβ(β+1)ₖ xᵏ/k!γ(γ+1)ₖ

                     k=0

                            ∞

            =   αβ/γ  [Σ  (α+1)ₖ(β+1)ₖ xᵏ

                          k=0

            = αβF(α+1,β+1;γ+1;x)/γ    ....(2)

Repeated application of equation(2) leads to the formula 

     dᵐ/dxᵐ [F(α,β;γ;x)] =                                                           (α)ₘ(β)ₘF(α+m,β+m;γ+m;x)/(γ)ₘ

    m= 1,2.....                          ..........(3)

From now on , to simplify the notation , we write

F(α,β;γ;x) ≡F , F(α±1,β;γ,x)≡F(α±1)

F(α,β±1;γ;x)≡F(β±1) , F(α,β;γ±1;x)≡F(γ±1) . Then the function F(α±1),F(β±1) and F(γ±1) are said to be contiguous to F . The function F and any two functions contiguous to F are connected by recurrence relations whose coefficients are linear functions of the variable x . There are altogether fifteen(15) recurrence relations involving F and its contiguous functions . We cite a few of the relations given below .

(γ-α-β)F + α(1-x)F(α+1)-(γ-β)F(β-1)=0 ......(4)

(γ-α-1)F+αF(α+1)-(γ-1)F(γ-1)=0  ......(5)

γ(1-x)F-γF(α-1)+(γ-β)xF(γ+1)=0 .......(6)

 which can be verified by direct substitution of the series (9). For example,

  substituting (9) into (4) , we obtain

(γ-α-β)F +α(1-x)F(α+1)-(γ-β)F(β-1)
                ∞
= (γ-α-β) Σ (α)ₙ(β)ₙ xⁿ/n!(γ)ₙ
              n=0
                 ∞
   + α(1-x) Σ (α+1)ₙ(β)ₙ xⁿ/n!(γ)ₙ
                n=0
              ∞
  - (γ-β) Σ (α)ₙ(β-1)ₙxⁿ/n!(γ)ₙ
              n=0
               ∞
=(γ-α-β) Σ (α)ₙ(β)ₙxⁿ/n!(γ)ₙ
              n=0

        ∞
 + α Σ (α+1)ₙ(β)ₙxⁿ / n!(γ)ₙ
       n=0
            ∞
  -(γ-β) Σ (α)ₙ(β-1)ₙxⁿ/n!(γ)ₙ
           n=0
    ∞
-α Σ (α+1)ₙ(β)ₙxⁿ⁺¹/n!(γ)ₙ
   n=0
                 ∞
= (γ-α-β)+Σ (α)ₙ(β)ₙxⁿ/n!(γ)ₙ
               n=1
           ∞
 +α+α Σ (α+1)ₙ(β)ₙxⁿ/n!(γ)ₙ

          n=1
                    ∞
-(γ-β)-(γ-β) Σ (α)ₙ(β-1)ₙxⁿ/n!(γ)ₙ
                   n=1
     ∞
 -α Σ (α+1)ₙ₋₁(β)ₙ₋₁xⁿ/(n-1)!(γ)ₙ₋₁
    n=1
   ∞
= Σ(α)ₙ(β)ₙ₋₁/n!(γ)ₙ  [(γ-α-β)(β+n-1)+(α+n)
  n=1             (β+n-1)-(γ-β)(β-1)-(γ+n-1)]xⁿ = 0

Similarly the other two can be verified . Three other formulae are an immediate consequence of equation(4)-(6) and symmetry condition

(γ-α-β)F + β(1-x)F(β+1)-(γ-α)F(α -1)=0 ...(7)

(γ-β-1)F+βF(β+1)-(γ-1)F(γ-1)=0   .......(8)

γ(1-x)F-γF(β-1)+(γ-α)xF(γ+1)=0 .........(9)

  The rest of the recurrence relation can be obtained from equation(4)-(9) by eliminating a common contiguous function from an approximate pair of formulae . For example , combining equation (5) and (8), or (6) and (9) , we obtain

 (α-β) F - αF(α+1)+βF(β+1)=0        .........(10)

(α-β)(1-x)F + (γ-α)F(α-1) - (γ-β)F(β-1) = 0 , as so on                                                     ..........(11)