Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series
Bertrand Teguia Tabuguia, Wolfram Koepf · Programming and Computer Software · 2022
A term an is m-fold hypergeometric, for a given positive integer m, if the ratio $${{a}_{{n + m}}}{\text{/}}{{a}_{n}}$$ is a rational function over a field $$\mathbb{K}$$ of characteristic zero. We establish the structure of holonomic recurrence equations, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have m-fold hypergeometric term solutions over $$\mathbb{K}$$ , for any positive integer m. Consequently, we describe a new algorithm, say mfoldHyper, that extends the algorithms by Petkovšek (1992) and van Hoeij (1998) which compute a basis of hypergeometric (m = 1) term solutions of holonomic recurrence equations to the more general case of m-fold hypergeometric terms.