关于非完整一元幂级数的表示
On the representation of non-holonomic univariate power series
摘要：
完整函数在计算机代数中起着至关重要的作用，因为它们允许许多符号算法的应用。在所有试图寻找幂级数公式的算法中，完整性仍然是所考虑的函数所要满足的最重要的要求。目标函数主要是对亚纯函数的目标函数的总结。然而，诸如$\tan(z)$、$z/(\exp(z)-1)$、$\sec(z)$等的表达式，特别是完整函数的倒数、导数和合成，一般不是完整的。因此，它们的幂级数是完整力学框架所不能达到的。从数学词典中可以看到，大多数已知的非完整幂级数的闭式公式都涉及另一个序列，其求值依赖于某些有限求和。在$\tan(z)$和$\sec(z)$的情况下，相应的序列分别是伯努利数和欧拉数。因此，提供一个符号的方法，产生完整的表示时，非完整函数的幂级数系数的线性求和出现，可能会被视为一个步骤，迈向非完整的幂级数的表示。通过采用待定系数的反演算法，我们建立了一个计算一类非完整函数的多项式系数的最小阶二次微分方程的算法。通过应用柯西乘积公式和将幂改写为多项式，将导数改写为移位，将由此产生的微分方程转换为递归方程。最后，利用足够的初值，我们能够给出几个非完整幂级数的范式表示，并证明了非平凡恒等式。我们讨论了该算法及其在Maple 2022上的实现。
英文摘要：
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., particularly, reciprocals, quotients and compositions of holonomic functions, are generally not holonomic. Therefore their power series are inaccessible by the holonomic framework. 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.
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