This SAGE code is written by Tolga Birkandan as an appendix to the article:

"Physical Examples of the Heun-to-Hypergeometric Reduction", arXiv:1401.0449 [math-ph].

See Table 2 of R. S. Maier, "The 192 Solutions of the Heun Equation", Math. Comp. 76 (2007), 811-843, arXiv:math/0408317 (line 5 & line 9 respectively) for the identities.

You can either download and install SAGE: http://www.sagemath.org/

or use it via SAGE Cloud: https://cloud.sagemath.com/

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Please see the code for further comments and the usage.