Related papers: Weighted Tribonacci sums
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.
In this paper, we consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.
In this paper, we study the weighted sums of multiple t-values and of multiple t-star values at even arguments. Some general weighted sum formulas are given, where the weight coefficients are given by (symmetric) polynomials of the…
We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the $q$-Fibonacci polynomials appearing in Schur's work. The proofs of most of…
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 present a different combinatorial interpretations of Lucas and Gibonacci numbers. Using these interpretations we prove several new identities, and simplify the proofs of several known identities. Some open problems are discussed towards…
We develop new closed form representations of sums of (n + {\alpha})th shifted harmonic numbers and reciprocal binomial coefficients in terms of {\alpha}th shifted harmonic numbers. Some interesting new consequences and illustrative…
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and…
We derive a formula for the evaluation of weighted generalized Fibonacci sums of the type $S_k^n (w,r) = \sum_{j = 0}^k {w^j j^r G_j{}^n }$. Several explicit evaluations are presented as examples.
We derive generalizations of a couple of inverse tangent summation identities involving Fibonacci and Lucas numbers. As byproducts we establish many new inverse tangent identities involving the Fibonacci and Lucas numbers.
In this paper, we investigated properties of Tribonacci-Lucas polynomials which generalized Tribonacci-Lucas numbers. From this generalization, we also obtain some new algebraic properties on these numbers and polynomials as Binet formula,…
We develop closed form expressions for various finite binomial Fibonacci and Lucas sums depending on the modulo 5 nature of the upper summation limit. Our expressions are inferred from some trigonometric identities.
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…
Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.
In this study, we apply the binomial transforms to Tribonacci and Tribonacci-Lucas sequences. Also, the Binet formulas, summations, generating functions of these transforms are found using recurrence relations. Finally, we illustrate the…
We give convolution identities without binomial coefficients for Tetranacci numbers and convolution identities with binomial coefficients for Tetranacci and Tetranacci-type numbers.