1,243 research outputs found
An Identity for Generalized Bernoulli Polynomials
Recognizing the great importance of Bernoulli numbers and Bernoulli polynomials in various branches of mathematics, the present paper develops two results dealing with these objects. The first one proposes an identity for the generalized Bernoulli polynomials, which leads to further generalizations for several relations involving classical Bernoulli numbers and Bernoulli polynomials. In particular, it generalizes a recent identity suggested by Gessel. The second result allows the deduction of similar identities for Fibonacci, Lucas, and Chebyshev polynomials, as well as for generalized Euler polynomials, Genocchi polynomials, and generalized numbers of Stirling
Combinatorial sums associated with balancing and Lucas-balancing polynomials
The aim of the paper is to use some identities involving binomial coefficients to derive new combinatorial identities for balancing and Lucas-balancing polynomials. Evaluating these identities at specific points, we can also establish some combinatorial expressions for Fibonacci and Lucas numbers
Combinatorial Identities for Incomplete Tribonacci Polynomials
The incomplete tribonacci polynomials, denoted by T_n^{(s)}(x), generalize
the usual tribonacci polynomials T_n(x) and were introduced in [10], where
several algebraic identities were shown. In this paper, we provide a
combinatorial interpretation for T_n^{(s)}(x) in terms of weighted linear
tilings involving three types of tiles. This allows one not only to supply
combinatorial proofs of the identities for T_n^{(s)}(x) appearing in [10] but
also to derive additional identities. In the final section, we provide a
formula for the ordinary generating function of the sequence T_n^{(s)}(x) for a
fixed s, which was requested in [10]. Our derivation is combinatorial in nature
and makes use of an identity relating T_n^{(s)}(x) to T_n(x)
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
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