超几何型和非完整幂级数的符号计算
Symbolic computation of hypergeometric type and non-holonomic power series
摘要：
对于给定的正整数$m$，如果比率$a_{n+m}/a_n$是特征为零的域$K$上的有理函数，则项$a_n$是$m$ -倍超几何的.建立了完整递归方程的结构，即对任意正整数$m$，在$K$上具有$m$ -重超几何项解的多项式系数线性齐次递归方程.因此，我们描述了一个算法，比如说$mfoldHyper$，它扩展了van Hoeij的算法（1998），该算法计算完整递归方程的超几何$(m=1)$项解的子空间的基到更一般的$m$ -折叠超几何项的情况。通过考虑系数为$m$ -重超几何项的Laurent-Puietrix级数的线性组合，推广了Koepf（1992）提出的超几何型幂级数的概念.因此，由于$mfoldHyper$，我们推导出一个完整的程序来计算这些幂级数;事实上，事实证明，每一个线性组合的幂级数与$m$倍超几何项系数，为许多值的$m$，检测。另一方面，我们研究了非完整函数幂级数的表示算法。该算法遵循相同的步骤Koepf的算法，但不是寻求完整的微分方程，二次微分方程的计算和柯西乘积规则是用来推导递推方程的幂级数系数。该算法定义了一个正规函数，该正规函数在具有足够初值的情况下，可得到许多非完整函数的幂级数的范式。因此，非平凡的身份自动证明使用这种方法。本文附带了计算机代数系统（CAS）Maxima 5.44.0和Maple 2019中的实现。
英文摘要：
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.
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