论文标题
用二次微分方程猜测
Guessing With Quadratic Differential Equations
论文作者
论文摘要
通过自动猜测，我们表示通过序列的生成函数满足的多项式系数找到线性微分方程的过程，其中只有几个第一个术语是已知的。三十年来，已在计算机代数中使用了自动猜测，以证明在证据之前的直觉过程中的猜测和训练范式的价值，这在解决方案的艺术中传播（Polya，1978）。在用于执行猜测的突出包装中，可以引用Salvy和Zimmermann的Maple Gfun包装； Mallinger的Mathematica生成手套包；以及Kauers，Jaroschek和Johansson的Sage Ore_algebra包。 我们提出了一种方法，该方法通过允许目标微分方程为最多两个，从而扩展了自动猜测。因此，它使我们能够捕获更多的生成功能，而不仅仅是全体功能。相应的复发方程与Bernoulli，Euler和Bell数的已知方程相似。结果，我们的软件为上/下数字的生成函数（https://oeis.org/a000111）找到了正确的复发和微分方程，zeta函数在正整数上的评估，lambert W功能的taylor系数，等等。我们的枫木实施（$ delta2guess $）是FPS软件包的一部分
By holonomic guessing, we denote the process of finding a linear differential equation with polynomial coefficients satisfied by the generating function of a sequence, for which only a few first terms are known. Holonomic guessing has been used in computer algebra for over three decades to demonstrate the value of the guess-and-prove paradigm in intuition processes preceding proofs, as propagated in The Art of Solving (Polya, 1978). Among the prominent packages used to perform guessing, one can cite the Maple Gfun package of Salvy and Zimmermann; the Mathematica GeneratingFunctions package of Mallinger; and the Sage ore_algebra package of Kauers, Jaroschek, and Johansson. We propose an approach that extends holonomic guessing by allowing the targeted differential equations to be of degree at most two. Consequently, it enables us to capture more generating functions than just holonomic functions. The corresponding recurrence equations are similar to known equations for the Bernoulli, Euler, and Bell numbers. As a result, our software finds the correct recurrence and differential equations for the generating functions of the up/down numbers (https://oeis.org/A000111), the evaluations of the zeta function at positive even integers, the Taylor coefficients of the Lambert W function, and many more. Our Maple implementation ($delta2guess$) is part of the FPS package which can be downloaded at http://www.mathematik.uni-kassel.de/~bteguia/FPS_webpage/FPS.htm
