### Abstract

A term *a*_{n} 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.

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## Notes

Mostly \(\mathbb{K}: = \mathbb{Q}({{\alpha }_{1}}, \ldots ,{{\alpha }_{N}})\) is the field of rational functions in several variables

Originally called hypergeometric type but we avoid this calling since we are redefining this terminology.

Throughout this paper we mainly give representations about

*z*_{0}= 0 since the case of arbitrary*z*_{0}deduces easily.A Maple command to use package names as Maple procedures.

The tangent function is usually not encoded as \(\sin (z){\text{/cos}}(z)\), and this fact is also ignore by the implemented differentiation.

For each fixed

*m*corresponding to an*m*-fold hypergeometric term we have a basis, and the basis of all*m*-fold hypergeometric term solutions is the collection of these bases.The brackets around m,j means optional arguments

Originally the type was used to denote the value of \(m\) for an \(m\)‑fold hypergeometric term coefficient.

Integer shift used in Petkovšek’s algorithm, see also Lemma 8

This is to make sure that cancellation of common factors is avoided.

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## ACKNOWLEDGMENTS

We thank Anna Ryabenko and Sergei Abramov who gave the opportunity to take part in the 4^{th} international conference “Computer Algebra”, Moscow, June 28-29, 2021, and handled our submission to the Russian journal of Programming and Computer Software.

We also thank Jürgen or Gerhard from Maplesoft for the opportunity to implement our algorithms for the Maple 2022 release.

We would like to thank the reviewers for their important help. This gave us some insights that could be incorporated into the final version.

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Teguia Tabuguia, B., Koepf, W. Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series.
*Program Comput Soft* **48**, 125–146 (2022). https://doi.org/10.1134/S0361768822020104

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DOI: https://doi.org/10.1134/S0361768822020104