I have some code that supports basic simplifications
on the incomplete gamma function. I'll send it to
anybody that might like to work on it (a 9K zip file).
The code has some support for working with recursion relations
and integral representations. Understand that the code isn't
suitable for doing real work---I don't remember testing this
code all that much.
(%i1) load("incomplete-gamma.lisp")$
(%i2) incomplete_gamma(a,0);
(%o2) gamma(a)
(%i3) incomplete_gamma(1/2,x);
(%o3) sqrt(%pi)*(1-erf(sqrt(x)))
(%i4) incomplete_gamma(5,x);
(%o4) 24*(x^4/24+x^3/6+x^2/2+x+1)*%e^(-x)
(%i5) diff(incomplete_gamma(a,x),x);
(%o5) -x^(a-1)*%e^(-x)
(%i6) tex(incomplete_gamma(a,x));
$$\Gamma\left(a,x\right)$$
(%o6) false
(%i7) load("integral-representation")$
(%i8) INTEGRAL_REP(incomplete_gamma(a,x));
(%o8) integrate(xo^(a-1)*%e^(-xo),xo,x,inf)
(%i9)
Barton