Three closed forms for convolved Fibonacci numbers

Abstract

<p>In the paper, by virtue of the Faa 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.</p>

Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials

“…Remark 3.6. The motivations in the papers [5,6,8,11,13,14,15,16,17,23,24,28,29,31,32,33,35,36,37,46,43,44,45,47,48,49] are same as the one in this paper.…”

Explicit Expressions Related to Degenerate Cauchy Numbers and Their Generating Function

“…Remark 5. The motivations in the papers [5,6,9,10,11,12,13,14,15,16,20,21,24,25,26,27,30,31,32,38,35,36,37,40,41,42] are same as the one in this paper.…”

“…In this paper, by virtue of the Faà di Bruno formula (9) and two identities (10) and (11) for the Bell polynomials of the second kind B n,k (x 1 , x 2 , . .…”

Some identities related to Eulerian polynomials and involving the Stirling numbers

“…Remark 10. The motivation of this paper is same as the one in [10,11,16,19,20,21,22,23,24,25,27,29,31,32,33,34,37,38,39,40,42,43,44,45,46,48] and closely related references therein.…”