differential-difference equations for hypergeometric function

The hypergeometric functionDlmfDlmfDlmfMathworldPlanetmath satisfies several equations which relate derivatives with respect to the argument z to shifting the parameters a,b,c,d by unity (Here, the prime denotes derivative with respect to z.):

zF(a,b;c;z)+aF(a,b;c;z) =F(a+1,b;c;z)
zF(a,b;c;z)+bF(a,b;c;z) =F(a,b+1;c;z)
zF(a,b;c;z)+(c-1)F(a,b;c;z) =F(a,b;c-1;z)
(1-z)zF(a,b;c;z) =(c-a)F(a-1,b;c;z)+(a-c+bz)F(a,b;c;z)
(1-z)zF(a,b;c;z) =(c-b)F(a,b-1;c;z)+(b-c+az)F(a,b;c;z)
(1-z)zF(a,b;c;z) =z(c-a)(c-b)F(a,b;c+1;z)+zc(a+b-c)F(a,b;c;z)

These equations may readily be verified by differentiating the series which defines the hypergeometric equation. By eliminating the derivatives between these equations, one obtains the contiguity relationsMathworldPlanetmath for the hypergeometric function. By differentiating them once more and taking suitable linear combinationsMathworldPlanetmath, one may obtain the differential equationMathworldPlanetmath of the hypergeometric function.

Title differential-difference equations for hypergeometric function
Canonical name DifferentialdifferenceEquationsForHypergeometricFunction
Date of creation 2013-03-22 17:36:18
Last modified on 2013-03-22 17:36:18
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Theorem
Classification msc 33C05