By studying Cameron’s operator in terms of determinants, two kinds “integer” sequences incomplete numbers were introduced. One was the sequence restricted numbers, including s-step Fibonacci sequences. Another associated Lamé higher order. classical Trudi’s formula and inverse relation, more expressions able to be obtained. These relations identities can extended those negative integers or rati...

Stirling numbers of the second kind and Bell numbers for graphs were defined by Duncan and Peele in 2009. In a previous paper, one of us, jointly with Nyul, extended the known results for these special numbers by giving new identities, and provided a list of explicit expressions for Stirling numbers of the second kind and Bell numbers for particular graphs. In this work we introduce q-Stirling ...

We study the q-dependent susceptibility χ(q) of a series of quasiperiodic Ising models on the square lattice. Several different kinds of aperiodic sequences of couplings are studied, including the Fibonacci and silver-mean sequences. Some identities and theorems are generalized and simpler derivations are presented. We find that the q-dependent susceptibilities are periodic, with the commensura...

This paper is devoted to the study of eigen-sequences for some important operators acting on sequences. Using functional equations involving generating functions, we completely solve the problem of characterizing the fixed sequences for the Generalized Binomial operator. We give some applications to integer sequences. In particular we show how we can generate fixed sequences for Generalized Bin...

In this paper we introduce a family of infinite words that generalize the Fibonacci word and we study their combinatorial properties. We associate with this family of words a family of curves that are like the Fibonacci word fractal and reveal some fractal features. Finally, we describe an infinite family of polyominoes stems from the generalized Fibonacci words and we study some of their geome...

In this paper we find closed forms of the generating function ∞ ∑ k=0 U r nx , for powers of any non-degenerate second-order recurrence sequence, Un+1 = aUn+bUn−1, a +4b 6= 0, completing a study began by Carlitz [1] and Riordan [4] in 1962. Moreover, we generalize a theorem of Horadam [3] on partial sums involving such sequences. Also, we find closed forms for weighted (by binomial coefficients...

The Fibonacci polynomial has been generalized in many ways,some by preserving the initial conditions,and others by preserving the recurrence relation.In this article,we study new generalization {Mn}(x ), with initial conditions M0(x ) = 2 and M1(x ) = m (x ) + k (x ), which is generated by the recurrence relation Mn+1(x ) = k (x )Mn (x )+Mn−1(x ) for n ≥ 2, where k (x ), m (x ) are polynomials ...

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or po...

We study the largest values of the rth row of Stern’s diatomic array. In particular, we prove some conjectures of Lansing. Our main tool is the connection between the Stern sequence, alternating binary expansions, and continuants. This allows us to reduce the problem of ordering the elements of the Stern sequence to the problem of ordering continuants. We describe an operation that increases th...

An inverse-conjugate composition of a positive integer m is an ordered partition of m whose conjugate coincides with its reversal. In this paper we consider inverseconjugate compositions in which the part sizes do not exceed a given integer k. It is proved that the number of such inverse-conjugate compositions of 2n − 1 is equal to 2F n , where F (k) n is a Fibonacci k-step number. We also give...