In this paper, we will firstly define a new generalization of numbers (p, q) and then derive the appropriate Binet's formula and generating functions concerning (p,q)-Fibonacci numbers, (p,q)- Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Also, some useful generating functions are provided for the products of (p,q)-numbers with bivariate complex Fibonacci and Lucas polynomials.
The aim of this paper is to define new families of combinatorial numbers and
polynomials associated with Peters polynomials. These families are also a
modification of the special numbers and polynomials in [11]. Some fundamental
properties of these polynomials and numbers are given. Moreover, a
combinatorial identity, which calculates the Fibonacci numbers with the aid
of binomial coefficients and which was proved by Lucas in 1876, is proved by
different method with the help of these combinatorial numbers. Consequently,
by using the same method, we give a new recurrence formula for the Fibonacci
numbers and Lucas numbers. Finally, relations between these combinatorial
numbers and polynomials with their generating functions and other well-known
special polynomials and numbers are given.
In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.
In this paper, we introduce a new operator in order to derive some new
symmetric properties of Fibonacci numbers and Chebychev polynomials of first
and second kind. By making use of the new operator defined in this paper, we
give some new generating functions for Fibonacci numbers and Chebychev
polynomials of first and second kinds.
AbstractNew results about some sums s n(k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Ý, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $$ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $$ are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s n(k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.
Abstract
A permutation of length n may be represented, equivalently, by a sequence a1a2 • • • an satisfying 0 < ai < i for all z, which is called an inversion sequence. In analogy to the usual case for permutations, the pattern avoidance question is addressed for inversion sequences. In particular, explicit formulas and/or generating functions are derived which count the inversion sequences of a given length that avoid a single pattern of length three. Among the sequences encountered are the Fibonacci numbers, the Schröder numbers, and entry A200753 in OEIS. We make use of both algebraic and combinatorial methods to establish our results. An explicit Injection is given between two of the avoidance classes, and in three cases, the kernel method is used to solve a functional equation satisfied by the generating function enumerating the class in question.
In this paper we consider the generalized Fibonacci numbers Fn,m and the
generalized Lucas numbers Ln,m. Also, we introduce new sequences of numbers
An,m, Bn,m, Cn,m and Dn,m. Further, we find the generating functions and
some recurrence relations for these sequences of numbers.
AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.
In this paper, we derive some new symmetric properties of k-Fibonacci
numbers by making use of symmetrizing operator. We also give some new
generating functions for the products of some special numbers such as
k-Fibonacci numbers, k-Pell numbers, Jacobsthal numbers, Fibonacci
polynomials and Chebyshev polynomials.
We demonstrate here a remarkably simple method for deriving a large number of identities involving the Fibonacci numbers, Lucas numbers and binomial coefficients. As will be shown, this is based on the utilisation of some straightforward properties of the golden ratio in conjunction with a result concerning irrational numbers. Indeed, for the simpler cases at least, the derivations could be understood by able high-school students. In particular, we avoid the use of exponential generating functions, matrix methods, Binet's formula, involved combinatorial arguments or lengthy algebraic manipulations.