Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important requirement to be satisfied by the function under consideration. The targeted functions mainly summarize that of meromorphic functions. However, expressions like $\tan(z)$, $z/(\exp(z)-1)$, $\sec(z)$, etc. are not holonomic, therefore their power series are inaccessible by non-pattern matching implementations like the current Maple \texttt{convert/FormalPowerSeries}. From the mathematical dictionaries, one can observe that most of the known closed-form formulas of non-holonomic power series involve another sequence whose evaluation depends on some finite summations. In the case of $\tan(z)$ and $\sec(z)$ the corresponding sequences are the Bernoulli and Euler numbers, respectively. Thus providing a symbolic approach that yields complete representations when linear summations for power series coefficients of non-holonomic functions appear, might be seen as a step forward towards the representation of non-holonomic power series. By adapting the method of ansatz with undetermined coefficients, we build an algorithm that computes least-order quadratic differential equations with polynomial coefficients for a large class of non-holonomic functions. A differential equation resulting from this procedure is converted into a recurrence equation by applying the Cauchy product formula and rewriting powers into polynomials and derivatives into shifts. Finally, using enough initial values we are able to give normal form representations to characterize several non-holonomic power series and prove non-trivial identities. We discuss this algorithm and its implementation for Maple 2022.
翻译:全息函数在计算机代数中扮演着不可或缺的角色, 因为它们允许应用许多象征性算法 。 在寻找电源序列公式的所有算法尝试中, holonomic 属性仍然是审议中的函数所要满足的最重要要求。 目标函数主要概括了 molortic 函数。 然而, $tan( z) 、 $z/ (\ ex( z)-1) 美元、 $\\ sec( z) 等表达方式不是 Holonomic, 因此它们的电源序列无法被非模式的变数匹配执行( 如当前 Maple \ tett{ conver/ FormalpoolateSeries ) 。 从数学字典中可以观察到, 大多数已知的非超额公式公式公式的函数包含另一个序列, 其评价取决于一定的比值 。 在 $\ z ( ) 和 $\\\ 20 ( z) 中, 对应的序列是 足够多的变数和 Eul 数字。 因此, 提供了一种完整的表达式方法, 当我们用一个变数序列的直线性变数的变数的变数函数的变数函数的变数, 。