differential-difference equations for hypergeometric function

The hypergeometric function 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 $z$.):

$z{F}^{\prime }(a,b;c;z)+aF(a,b;c;z)$ $=F(a+1,b;c;z)$

$z{F}^{\prime }(a,b;c;z)+bF(a,b;c;z)$ $=F(a,b+1;c;z)$

$z{F}^{\prime }(a,b;c;z)+(c-1)F(a,b;c;z)$ $=F(a,b;c-1;z)$

$(1-z)z{F}^{\prime }(a,b;c;z)$ $=(c-a)F(a-1,b;c;z)+(a-c+bz)F(a,b;c;z)$

$(1-z)z{F}^{\prime }(a,b;c;z)$ $=(c-b)F(a,b-1;c;z)+(b-c+az)F(a,b;c;z)$

$(1-z)z{F}^{\prime }(a,b;c;z)$ $=z(c-a)(c-b)F(a,b;c+1;z)+zc(a+b-c)F(a,b;c;z)$

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 relations for the hypergeometric function. By differentiating them once more and taking suitable linear combinations, one may obtain the differential equation 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