hyperg_U basically fails with b=1, a non-integer; because
gsl_sf_poch_e(1+a-b,-a,&r1); is throwing a domain error when given
gamma(0)/gamma(a).
Checking on and using b=1 after a-integer is checked is illustrated
below in Octave. I also put in recursion to evaluate b>=2.
I checked the b=1 expression against Maple; for a few values x<0,a<0,b=1
and x<0,a<0,b>=2 integer.
--------------
Unfortunately the routine in Octave to call hyperg_U is only set up for
real returns, which was okay for versions <1.14 . Sad to say I am the
one who implemented the hyperg_U interface, and will probably have to go
back :-( . Integrating these functions into Octave was not pleasant;
but perhaps somebody made it easier. I did translate the active parts
of hyperg_U into octave though; so it can be used in that way.
Ray
#
#
# Test function to evaluate b=1 for gsl 1.14 hyperg_U x<0
#
function anss=hyperg_U_negx_1(a,b,x)
int_a=(floor(a)==a);
int_b=(floor(b)==b);
#neg, int, a is already taken care of so use it
if (int_a && a<=0)
anss=hyperg_U(a,b,x);
elseif (int_b && (b==1))
#from the new NIST DLMF 13.2.41
anss=gamma(1-a)*exp(x)*(hyperg_U(1-a,1,-x)/gamma(a)-hyperg_1F1(1-a,1,-x)*exp((1-a)*pi*I));
elseif (b>=2)
#DLMF 13.3.10
anss=((b-1-a)*hyperg_U_negx_1(a,b-1,x) +
hyperg_U_negx_1(a-1,b-1,x))/x;
else
anss=hyperg_U(a,b,x);
endif
#
endfunction
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