求和公式、生成函数和多项式除法

Q4 Mathematics
E. Berkove, Michael A. Brilleslyper
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引用次数: 0

摘要

摘要我们描述了一种一般的方法,它可以找到系数由线性递推关系产生的幂级数的部分和的闭合形式。这些闭合形式允许我们导出大量涉及斐波那契数和其他相关序列的恒等式。尽管该方法的动机是多项式长除法问题,但它自然地符合标准的生成函数框架。我们还描述了一种计算两个生成函数的Hadamard乘积的生成函数的显式方法,这是一种类似于点积的幂级数的构造。这允许将该方法用于系数的递推关系最初未知的许多示例。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Summation Formulas, Generating Functions, and Polynomial Division
Summary We describe a general method that finds closed forms for partial sums of power series whose coefficients arise from linear recurrence relations. These closed forms allow one to derive a vast collection of identities involving the Fibonacci numbers and other related sequences. Although motivated by a polynomial long division problem, the method fits naturally into a standard generating function framework. We also describe an explicit way to calculate the generating function of the Hadamard product of two generating functions, a construction on power series which resembles the dot product. This allows one to use the method for many examples where the recurrence relation for the coefficients is not initially known.
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来源期刊
Mathematics Magazine
Mathematics Magazine Mathematics-Mathematics (all)
CiteScore
0.20
自引率
0.00%
发文量
68
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