Related papers: Lucas-Euler relations using balancing and Lucas-ba…
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph…
We show that certain weighted Fibonacci and Lucas series can always be expressed as linear combinations of polylogarithms. In some special cases we evaluate the series in terms of Bernoulli polynomials, making use of the connection between…
While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down…
We derive some Fibonacci and Lucas identities which contain inverse binomial coefficients. Extension of the results to the general Horadam sequence is possible, in some cases.
We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associated with it. Also, we present…
The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of…
Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.
In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius-Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of…
Based on a variant of Sury's polynomial identity we derive new expressions for various finite Fibonacci (Lucas) sums. We extend the results to Fibonacci and Chebyshev polynomials, and also to Horadam sequences. In addition to deriving sum…
In this note we prove combinatorially some new formulas connecting poly-Bernoulli numbers with negative indices to Eulerian numbers.
In this paper, we introduce relations between binomial sums involving (generalized) Fibonacci and Lucas numbers, and different kinds of binomial coefficients. We also present some relations between sums with two and three binomial…
Based on well-known properties of Fibonacci and Lucas numbers and polynomials we give a self-contained approach to some bivariate analogs.
In this paper, by using some families of special numbers and polynomials with their generating functions, we give various properties of these numbers and polynomials. These numbers are related to the well-known numbers and polynomials,…
We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.
In this study, we define the dual Fibonacci quaternion and the dual Lucas quternion. We derive the relations between the dual Fibonacci and the dual Lucas quaternion which connected the Fibonacci and the Lucas numbers. Furthermore, we give…
This paper is concerned with developing some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. All the connection coefficients involve hypergeometric functions of the type $_2F_{1}(z)$, for certain…
In this paper, we give some interesting identities of higher-order Bernoulli, Frobenius-Euler and Euler polynomials arising from umbral calculus. From our method of this paper, we can derive many interesting identities of special…
In the present paper, we obtain new interesting relations and identities of the Apostol-Bernoulli polynomials of higher order, which are derived using a Bernoulli polynomial basis. Finally, by utilizing our method, we also derive formulas…
By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio, $\alpha=(1+\sqrt 5)/2$ and its inverse, $\beta=-1/\alpha=(1-\sqrt 5)/2$, a multitude of Fibonacci and Lucas identities have been developed in the…
By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and…