In this paper, we evaluate several families of Toeplitz-Hessenberg matrices whose entries are generalized Fibonacci numbers. In particular, we find simple formulas for several determinants whose entries are translates of the Chebyshev polynomials of the second kind. Equivalently, these determinant formulas may also be rewritten as identities involving sums of products of generalized Fibonacci numbers and multinomial coefficients. Combinatorial proofs which make use of sign-reversing involutions and the definition of a determinant as a signed sum over the symmetric group S n are given for our formulas in several particular cases, including those involving the Chebyshev polynomials.

The main object of the present paper is to reveal connections between Mersenne numbers $M_n=2^n-1$ and generalized Fibonacci (i.e., Horadam) numbers $w_n$ defined by a second order linear recurrence $w_n=pw_{n-1}+qw_{n-2}$, $n\geq 2$, with $w_0=a$ and $w_1=b$, where $a$, $b$, $p>0$ and $q\ne0$ are integers. This is achieved by relating the respective (ordinary and exponential) generating functions to each other. Several explicit examples involving Fibonacci, Lucas, Pell, Jacobsthal and balancing numbers are stated to highlight the results.

In this paper we study some families of Toeplitz-Hessenberg determinants the entries of which are Fibonacci-Narayana (or Narayana's cows) numbers. This leads to discover some identities for these numbers. In particular, we establish connection between Fibonacci-Narayana numbers with Fibonacci and tribonacci numbers. We also present new formulas for Fibonacci-Narayana numbers via recurrent determinants of four-diagonal matrix.

No abstract

We study a certain family of infinite series with reciprocal Catalan numbers. We first evaluate two special candidates of the family in closed form, where we also present some Catalan–Fibonacci relations. Then, we focus on the general properties of the family and prove explicit formulas, including two types of integral representations.