A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently, we describe an algorithm, say $mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis of the subspace of hypergeometric $(m=1)$ term solutions of holonomic recurrence equations to the more general case of $m$-fold hypergeometric terms. We generalize the concept of hypergeometric type power series introduced by Koepf (1992), by considering linear combinations of Laurent-Puiseux series whose coefficients are $m$-fold hypergeometric terms. Thus thanks to $mfoldHyper$, we deduce a complete procedure to compute these power series; indeed, it turns out that every linear combination of power series with $m$-fold hypergeometric term coefficients, for finitely many values of $m$, is detected. On the other hand, we investigate an algorithm to represent power series of non-holonomic functions. The algorithm follows the same steps of Koepf's algorithm, but instead of seeking holonomic differential equations, quadratic differential equations are computed and the Cauchy product rule is used to deduce recurrence equations for the power series coefficients. This algorithm defines a normal function that yields together with enough initial values normal forms for many power series of non-holonomic functions. Therefore, non-trivial identities are automatically proved using this approach. This paper is accompanied by implementations in the Computer Algebra Systems (CAS) Maxima 5.44.0 and Maple 2019.

中文翻译：

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超几何类型和非完整幂级数的符号计算

对于给定的正整数$ m $，如果比率$ a_ {n + m} / a_n $是特征值为零的字段$ K $的有理函数，则项$ a_n $是$ m $倍超几何。我们建立完整的递归方程的结构，即具有多项式系数的线性和齐次递归方程，对于任何正整数$ m $，其具有超过$ K $的$ m $倍超几何项解。因此，我们描述了一种算法，例如$ mfoldHyper $，该算法将van Hoeij（1998）的算法扩展到了更常见的$ m情况，该算法计算完整几何递归方程的超几何$（m = 1）$项解的子空间的基础$倍超几何项。我们归纳了Koepf（1992）提出的超几何型幂级数的概念，通过考虑Laurent-Puiseux级数的线性组合，其系数为$ m $-倍超几何项。因此，由于有了$ mfoldHyper $，我们得出了一个完整的过程来计算这些幂级数。实际上，事实证明，对于$ m $的有限许多值，具有$ m $倍超几何项系数的幂级数的线性组合都可以检测到。另一方面，我们研究了一种表示非完整函数幂级数的算法。该算法遵循与Koepf算法相同的步骤，但是不是寻找完整的微分方程，而是计算二次微分方程，并使用柯西乘积规则推导幂级数系数的递推方程。该算法定义了一个正常函数，该函数可以为许多非完整函数的幂级数生成足够的初始值和正常形式的形式。因此，使用这种方法会自动证明非平凡的身份。本文随附计算机代数系统（CAS）Maxima 5.44.0和Maple 2019中的实现。