用二次微分方程进行猜测
Guessing With Quadratic Differential Equations
摘要：
通过完整猜测，我们表示的过程中找到一个线性微分方程的多项式系数满足的生成函数的序列，其中只有少数第一项是已知的。完整猜想已经在计算机代数中使用了三十多年，以证明在证明之前的直觉过程中猜测和证明范式的价值，正如在The Art of Solving（Polya，1978）中所传播的那样。在用来进行猜测的著名软件包中，可以引用Salvy和Zimmermann的Maple Gfun软件包; Mallinger的Mathematica GeneratingFunctions软件包;以及Kauers，Jaroschek和Johansson的Sage ore_algebra软件包。我们提出了一种方法，通过允许目标微分方程的度最多为2，扩展完整猜测。因此，它使我们能够捕获更多的生成函数，而不仅仅是完整函数。相应的递归方程类似于伯努利数、欧拉数和贝尔数的已知方程。因此，我们的软件可以找到上/下数字的生成函数的正确递归和微分方程（https：//oeis.org/A000111），zeta函数在正偶数处的计算，Lambert W函数的泰勒系数等等。我们的Maple实现（$delta2guess$）是FPS包的一部分，可以从http://www.mathematik.uni-kassel.de/~bteguia/FPS_webpage/FPS.htm下载
英文摘要：
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
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
