Three closed forms for convolved Fibonacci numbers

Year 2020, Volume: 3 Issue: 4, 185 - 195, 30.12.2020

Feng Qi

Abstract

In the paper, by virtue of the Fa\`a di Bruno formula and several properties of the Bell polynomials of the second kind, the author computes higher order derivatives of the generating function of convolved Fibonacci numbers and, consequently, derives three closed forms for convolved Fibonacci numbers in terms of the falling and rising factorials, the Lah numbers, the signed Stirling numbers of the first kind, and the golden ratio.

In the paper, by virtue of the Fa\`a di Bruno formula and several properties of the Bell polynomials of the second kind, the author computes higher order derivatives of the generating function of convolved Fibonacci numbers and, consequently, derives three closed forms for convolved Fibonacci numbers in terms of the falling and rising factorials, the Lah numbers, the signed Stirling numbers of the first kind, and the golden ratio.

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Keywords

closed form, convolved Fibonacci number, Fa\`a di Bruno formula, Bell polynomial of the second kind, higher order derivative, generating function, falling factorial, rising factorial, Lah number, Stirling number of the first kind, golden ratio

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