On Convolved Generalized Fibonacci and Lucas Polynomials

We define the convolved h(x)-Fibonacci polynomials as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the h(x)-Fibonacci and h(x)-Lucas polynomials. Moreover we obtain the convolved h(x)-Fibonacci polynomials form a family of Hessenberg matrices

Enumeration of kk-Fibonacci Paths using Infinite Weighted Automata

In this paper, we introduce a new family of generalized colored Motzkin paths, where horizontal steps are colored by means of $F_{k,l}$ colors, where $F_{k,l}$ is the $l$th $k$-Fibonacci number. We study the enumeration of this family according to the length. For this, we use infinite weighted automata.Comment: arXiv admin note: substantial text overlap with arXiv:1310.244

On the average number of elements in a finite field with order or index in a prescribed residue class

For any prime p we consider the density of elements in the multiplicative group of the finite field F_p having order, respectively index, congruent to a(mod d). We compute these densities on average, where the average is taken over all finite fields of prime order. Some connections between the two densities are established. It is also shown how to compute these densities with high numerical accuracy.Comment: 25 pages, 4 tables. A conjecture made in the previous version is now resolved. Tables are also improved, thanks to a C++ program written by Yves Gallo

The formal series Witt transform

Given a formal power series f(z) we define, for any positive integer r, its rth Witt transform, W_f^{(r)}, by rW_f^{(r)}(z)=sum_{d|r}mu(d)f(z^d)^{r/d}, where mu is the Moebius function. The Witt transform generalizes the necklace polynomials M(a,n) that occur in the cyclotomic identity 1-ay=prod (1-y^n)^{M(a,n)}, where the product is over all positive integers. Several properties of the Witt transform are established. Some examples relevant to number theory are considered.Comment: 18 pages, small improvements in contents and presentation, to appear in Discrete Mathematic

Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet polynomials, Humbert-type numbers and polynomials, chain and anti-chain polynomials, rank polynomials of the lattices, length of any alphabet of words, partitions, and other graph polynomials. By applying the Euler transform and the Lambert series to these generating functions, many new identities and relations are derived. By using differential equations of these generating functions, some new recurrence relations for these polynomials are found. Moreover, general Binet's type formulas for these polynomials are given. Finally, some new classes of polynomials and their corresponding certain family of special numbers are investigated with the help of these generating functions.Comment: 29 page

Fibonacci and Lucas Differential Equations

The second-order linear hypergeometric differential equation and the hypergeometric function play a central role in many areas of mathematics and physics. The purpose of this paper is to obtain differential equations and the hypergeometric forms of the Fibonacci and the Lucas polynomials. We also write again these polynomials by means of Olver’s hypergeometric functions. In addition, we present some relations between these polynomials and the other well-known functions