On some fibinomial identities

Abstract

Fibinomial identity is identity that combine Fibonacci numbers and binomial or multinomial coefficients.
In this paper, for obtaining new fibinomial identities we consider determinants and permanents for some families of lower Toeplitz–Hessenberg matrices $H_n=(h_{ij})$,
where $h_{ij}=0$ for all $j>i+1$, $h_{ij}=a_{i-j+1}$, and $a_{i,i+1}=2$, having various translates of the Fibonacci numbers $F_n$ for the nonzero entries.

These determinant and permanent formulas may also be rewritten as identities involving sums of products of Fibonacci numbers and multinomial coefficients.
For example, for $n\geq1$, the following formula holds
$$
\sum_{s_1+2s_2+\cdots+ns_n=n}(-1)^{s_1+\cdots+s_n}{s_1+\cdots+s_n\choose s_1,\ldots, s_n}\left(\frac{F_2}{2}\right)^{s_1}\left(\frac{F_4}{2}\right)^{s_2}\cdots\left(\frac{F_{2n}}{2}\right)^{s_n}=
\frac{1-4^n}{3\cdot 2^n},
$$
where ${s_1+\cdots+s_n\choose s_1,\ldots, s_n}=\frac{(s_1+\cdots+s_n)!}{s_1!\cdots s_n!}$ is multinomial coefficient, and the summation is over non\-negative integers
$s_j$ satisfying Diophantine equation $s_1 +2s_2 +\cdots +ns_n=n$.

Also, we establish connection formulas between Jacobsthal, Pell, Pell-Lucas numbers and Fibonacci numbers using Toeplitz-Hessenberg determinants.