I need to make a relatively quick calculation of confluent Heun function (It is HeunC in Maplesoft Maple notation) values with different parameters in different points. I understand that it is possible to solve a Heun equation numerically, but that way seems not to be effective.
Of course, it is possible to use, e.g. Maple, for numerical procedure realization I need this in my study, but applying some series expansion or hypergeometric functions is more preferable.
So, I would appreciate simple (because I'm not a pure mathematician) information about series expansions of the Heun confluent function, its expression by hypergeometrics or any other known functions which are available to realize the numerical calculation.
See
www.supermathematica.com;
www.supermathematica.org;
www.supermatematica.ro;
www.supermatematicaonline.blogspot.ro
As far as I know, the HeunC function in Maple is said to have some problems.
There are some interesting formulas for the calculation in the attached link.
http://arxiv.org/pdf/math-ph/0404014v2.pdf
Thank you, Dr. Homeier
I think the reference and its' references will be useful.
Actually, the main problem for me is really slow calculation Maple HeunC function because it uses numerical integration outside the series expansion convergence region (i.e., where |z| > 1). Probably, it is impossible and there is no coordinate mapping can turn the outside area into the inside region due to function and Heun equation singularities. I continue the search.
If you have a series expansion valid for |z| 1, e.g. Levin-type algorithms, Borel, ...
A series expansion is provided in https://arxiv.org/pdf/0904.0245.pdf
Is it not possible to transform a confluent Heun function into a hypergeometric equation? The expansion of the latter is well established in literature.
Dear Amaechi,
Please read the https://arxiv.org/pdf/1606.00896.pdf, eq. 64.
Best regards
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