generalized hypergeometric functions should be implemented

[dandrake]

Sage should have support for generalized hypergeometric functions. They are an important class of special functions. They show up everywhere from differential equations to binomial coefficient identities.

CC: @burcin @sagetrac-fstan @kcrisman @sagetrac-ktkohl @benjaminfjones @sagetrac-dsm @eviatarbach @mezzarobba

Component: symbolics

Keywords: hypergeometric, special

Author: Fredrik Johansson, Eviatar Bach, Ralf Stephan

Branch: b2f8dc5

Reviewer: Ralf Stephan, Nils Bruin, Volker Braun

Issue created by migration from https://trac.sagemath.org/ticket/2516

[fredrik-johansson]

Attachment: hyper.patch.gz

tentative implementation of generalized hypergeometric functions

[fredrik-johansson]

Author: fredrik.johansson

[fredrik-johansson]

comment:1

I've attached essentially the code I wrote during with Sage Days 24 with some additions. Features include:

• Representation of PFQs as symbolic functions (with automatic simplification) and in always-unevaluated form

• Numerical evaluation (wrapping mpmath)

• Evaluation in terms of elementary functions in some simple cases

Important missing features include:

• Rewriting symbolic sums as PFQs

• Evaluation in terms of polygamma functions

• Various transformations permitting closed-form evaluation in special cases

The patch includes symbolic versions of Bessel functions, which can be expanded as PFQs. This obviously needs to be merged with the existing Bessel function classes.

The main remaining issue is to handle multivariate symbolic functions properly (the current approach in Function_PFQ is something of a hack). Also, in the present patch, there are two PFQ classes. This is partially due to limitations/awkwardness of the symbolic function class, but regardless of whether necessary, it may be desirable to have a separate "backend" representation for hypergeometric functions that doesn't need to be concerned with automatic evaluation, numerical conversions, etc. I'm not sure what's the best way to organize it all.

[kcrisman]

comment:3

See also #9908, which is at least related (we need to translate Maxima hg functions).

Also, I think that we already have a lot of Bessel functions implemented, though they are not yet symbolic.

[kcrisman]

comment:5

Changing component. But this is a really nice start, hopefully can be combined with some of the currently existing bessel stuff etc.

[sagetrac-dsm]

comment:7

Okay, I'm going to see what can be done to get this merged.

[sagetrac-dsm]

comment:8

Having played around with this some, I think it'll be pretty straightforward to get in.. after we have Burcin's ticket allowing numerical_approx to take more general kwargs. If it weren't for the need to maintain backward compatibility with the algorithm kwarg for the old bessel functions, life would be easier.

I have a half-functional wrapper which monkeypatches things to sort of get them to work, but once that patch gets in it becomes much simpler. So I think this should stay in a holding pattern until we get that one in.

[kcrisman]

comment:9

If it weren't for the need to maintain backward compatibility with the algorithm kwarg for the old bessel functions, life would be easier.

Since #4102 is nearly finished, and has at least sort of dealt with this issue, perhaps one can just ignore that part of the Bessel issue. Instead, one could just add the _expand_hyper methods to the stuff there. I do like the idea here and it would be a shame for it to bit rot further...

[eviatarbach]

comment:12

Here is an initial version of an updated patch. It needs some work because it's failing a few tests, but I just wanted to post for comments.