A generalized recurrence for Bell polynomials: An alternate approach to Spivey and Gould–Quaintance formulas

AbstractLetting Bn(x) the n-th Bell polynomial, it is well known that Bn admit specific integer coordinates in the two following bases {xi}i=0,…,n and {xBi(x)}i=0,…,n−1 according, respectively, to Stirling numbers and binomial coefficients. Our aim is to prove that, for r+s=n, the sequence {xjBk(x)}j=0,…,rk=0,…,s is a family of bases of the Q-vectorial space formed by polynomials of Q[X] for which Bn admits a Binomial Recurrence Coefficient

On the self-convolution of generalized Fibonacci numbers

We focus on a family of equalities pioneered by Zhang and generalized by Zao and Wang and hence by Mansour which involves self convolution of generalized Fibonacci numbers. We show that all these formulas are nicely stated in only one equation involving a bivariate ordinary generating function and we give also a formula for the coefficients appearing in that context. As a consequence, we give the general forms for the equalities of Zhang, Zao-Wang and Mansour

Tiling approach to obtain identities for generalized Fibonacci and Lucas numbers

In Proofs that Really Count [2], Benjamin and Quinn have used “square and domino tiling” interpretation to provide tiling proofs of many Fibonacci and Lucas formulas. We explore this approach in order to provide tiling proofs of some generalized Fibonacci and Lucas identities. Keywords: Generalized Fibonacci and Lucas numbers; Tiling proofs

Continued fractions for bi-periodic Fibonacci sequence

In this paper, we study generalized continued fractions for the expression of bi-periodic Fibonacci ratios