# Confluent hypergeometric functions in Sage

The confluent hypergeometric functions are solutions to Kummer's differential equation, $$z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0.$$ Two linearly independent solutions are functions denoted as $$M$$ and $$U$$, which I implemented symbolically in Sage ($$U$$ was already implemented, but only numerically).

Here are some things it can do:

sage: (hypergeometric_M(1, 1, 1) +
....:  hypergeometric_U(1, 2, 1)).simplify_hypergeometric()
e + 1
sage: hypergeometric_U(1, 3, x).simplify_hypergeometric()
(x + 1)/x^2
sage: hypergeometric_M(1, 3/2, 1).simplify_hypergeometric()
1/2*sqrt(pi)*e*erf(1)
sage: hypergeometric_U(2, 2, x).series(x == 3, 100).subs(x=1).n()
0.403652637676806
sage: hypergeometric_U(2, 2, 1).n()
0.403652637676806

As far as I can tell, no open-source computer algebra system has this level of simplification of confluent hypergeometric functions; Maxima is used here, but it is not wrapped like this in Maxima. I hope this will prove useful for those working with hypergeometric functions or differential equations in Sage.

It's fairly trivial to implement the Whittaker functions in a similar way; the reason I haven't yet is because I didn't want to make the patch larger and stall review, and because the Maxima conversions are a bit trickier.

The newly implemented dynamic attributes for symbolic expressions have proved enormously useful for this ticket, as well as for the generalized hypergeometric functions and for Volker Braun's new implementation of piecewise functions. I may write a new section for the Developer's Guide which explains symbolic functions in detail, including the dynamic attributes.

I uploaded a patch at #14896; however, since it makes use of the generalized hypergeometric framework of #2516, that patch has to be merged first. Any help with review would be greatly appreciated!