differential-difference equations for hypergeometric function

The hypergeometric function satisfies several equations whichrelate derivatives with respect to the argument $z$ toshifting the parameters $a,b,c,d$ by unity (Here, theprime 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 serieswhich defines the hypergeometric equation. By eliminating the derivativesbetween these equations, one obtains the contiguity relations for thehypergeometric function. By differentiating them once more and takingsuitable linear combinations, one may obtain the differential equationof the hypergeometric function.