Related papers: More Fibonacci-Bernoulli relations with and withou…
Balancing and Lucas-balancing numbers are solutions of a Diophantine equation and satisfy a second order homogeneous recurrence relation. Interestingly, these numbers can be seen as numerators and denominators in the steady state…
In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and…
We consider the numbers $\mathcal{B}_{r,s} = (\mathbf{B}+1)^r \mathbf{B}^s$ (in umbral notation $\mathbf{B}^n = \mathbf{B}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the…
We find particular relations which we call "Bernoulli-type" in some noncommutative polynomial ring with a single nontrivial relation. More precisely, our ring is isomorphic to the universal enveloping algebra of a two-dimensional…
In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii) Geometry of Polynomials Level Sets, and iii) Shape…
In 2016, Y\"uce and Torunbalc\i\ Ayd\i n \cite{Yuc-Tor} defined dual Fibonacci quaternions. In this paper, we defined the dual third-order Jacobsthal quaternions and dual third-order Jacobsthal-Lucas quaternions. Also, we investigated the…
In this paper we present two families of Fibonacci-Lucas identities, with the Sury's identity being the best known representative of one of the families. While these results can be proved by means of the basic identity relating Fibonacci…
In this paper, we study degenerate ordered Bell polynomials with the viewpoint of Carlitz's degenerate Bernoulli and Euler polynomials and derive by using umbral calculus some properties and new identities for the degenerate ordered Bell…
This contribution presents all possible solutions to the Diophantine equations $F_k=L_mL_n$ and $L_k=F_mF_n$. To be clear, Fibonacci numbers that are the product of two arbitrary Lucas numbers and Lucas numbers that are the product of two…
We study formal power series which can be interpreted as interpolations of Fibonacci and Lucas polynomials with even (or odd) indices.
In this paper, we study relations among several types of Eulerian polynomials from a combinatorial viewpoint. We establish an identity between the restricted Eulerian polynomials of types $A$ and $B$. As an application, we present a…
In this paper, we derive some new and interesting idebtities for Bernoulli, Euler and Hermite polynomials associated with Chebyshev polynomials.
The balancing numbers $B_n$ ($n=0,1,\cdots$) are solutions of the binary recurrence $B_n=6B_{n-1}-B_{n-2}$ ($n\ge 2$) with $B_0=0$ and $B_1=1$. In this paper we show several relations about the sums of product of two balancing numbers of…
For an arbitrary homogeneous linear recurrence sequence of order d with constant coefficients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coefficients of these recurrences are given…
We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and…
In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler…
In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and…
In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials…
Many mathematicians have been studying various degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our main focus here is a new type of degenerate poly-Euler polynomials and numbers. This…
I recent years, many mathematicians studied various degenerate version of some spcial polynomials of which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the…