We obtain an explicit formula in terms of the partitions of the positive integer n to express the nth coefficient of the formal series expansion of the reciprocal of a given function. A brief survey shows that our arithmetic proof differs from others, some obtained already in the XIX century. Examples are given to establish explicit formulas for Bernoulli, Euler, and Fibonacci numbers.
Reciprocal Function Series Coefficients with Integer Partitions
Talamini, Vittorino
2018-01-01
Abstract
We obtain an explicit formula in terms of the partitions of the positive integer n to express the nth coefficient of the formal series expansion of the reciprocal of a given function. A brief survey shows that our arithmetic proof differs from others, some obtained already in the XIX century. Examples are given to establish explicit formulas for Bernoulli, Euler, and Fibonacci numbers.File in questo prodotto:
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Reciprocal Series_MJoM_confibdispari_rev.pdf
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