#5221 gamma_incomplete_lower(a, z) with a rational order a and a float z is not evaluated

[David Scherfgen]

Bug report written by Claude.

For an order that is neither an integer nor a half integer, gamma_incomplete_lower with a float or bigfloat argument returns gamma(a) - gamma_incomplete(a, z) with gamma(a) left exact, instead of a number. The upper function evaluates, and so does gamma_incomplete_lower for a half-integer or an integer order.

(%i1) display2d : false$
(%i2) gamma_incomplete_lower(1/3, 8.0);
(%o2) gamma(1/3)-7.799182611869946e-5
(%i3) gamma_incomplete_lower(1/3, -8.0);
(%o3) gamma(1/3)+719.2849028282662*%i+412.60039373722566
(%i4) gamma_incomplete_lower(1/3, 8.0*%i);
(%o4) gamma(1/3)+0.10901538887260959*%i+0.221572520567445
(%i5) gamma_incomplete_lower(1/3, 8.0b0*%i);
(%o5) gamma(1/3)+1.090153888726096b-1*%i+2.215725205674452b-1
(%i6) gamma_incomplete_lower(0.5, 8.0*%i);
(%o6) 0.22561853882998203*%i+2.040645378736624
(%i7) gamma_incomplete(1/3, 8.0*%i);
(%o7) -(0.10901538887260959*%i)-0.221572520567445
(%i8) float(gamma_incomplete_lower(1/3, 8*%i));
(%o8) 0.10901538887260959*%i+2.900511055275193

Expected: a float in each of (%o2) to (%o5), as in (%o8): 2.6788605428816297 for gamma_incomplete_lower(1/3, 8.0).

Where: the simplifier of gamma_incomplete_lower in src/gamma.lisp, whose numerical clause subtracts the numerically evaluated gamma_incomplete(a, z) from gamma(a) built from the exact a, so that gamma(a) is not evaluated when a is exact and only z is a float.

A related omission: gamma_incomplete_lower has no conjugate-function property, so rectform(gamma_incomplete_lower(1/3, %i*x^3)) returns realpart and imagpart noun forms, where rectform(gamma_incomplete(1/3, %i*x^3)) uses the mirror symmetry of the function.