Tribonacci and Tribonacci-Lucas Sedenions

The sedenions form a 16-dimensional Cayley-Dickson algebra. In this paper, we introduce the Tribonacci and Tribonacci-Lucas sedenions. Furthermore, we present some properties of these sedenions and derive relationships between them.Comment: 17 pages, 1 figur

Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet polynomials, Humbert-type numbers and polynomials, chain and anti-chain polynomials, rank polynomials of the lattices, length of any alphabet of words, partitions, and other graph polynomials. By applying the Euler transform and the Lambert series to these generating functions, many new identities and relations are derived. By using differential equations of these generating functions, some new recurrence relations for these polynomials are found. Moreover, general Binet's type formulas for these polynomials are given. Finally, some new classes of polynomials and their corresponding certain family of special numbers are investigated with the help of these generating functions.Comment: 29 page

A short proof for a determinantal formula for generalized Fibonacci numbers

The aim of this note is to provide a short proof for a recent determinantal formula of generalized Fibonacci numbers

Fibonacci–Lucas–Pell–Jacobsthal relations

In this paper, we prove several identities involving linear combinations of convolutions of the generalized Fibonacci and Lucas sequences. Our results apply more generally to broader classes of second-order linearly recurrent sequences with constant coefficients. As a consequence, we obtain as special cases many identities relating exactly four sequences amongst the Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal, and Jacobsthal–Lucas number sequences. We make use of algebraic arguments to establish our results, frequently employing the Binet-like formulas and generating functions of the corresponding sequences. Finally, our identities above may be extended so that they include only terms whose subscripts belong to a given arithmetic progression of the non-negative integers

Fibonacci–Lucas–Pell–Jacobsthal relations

In this paper, we prove several identities involving linear combinations of convolutions of the generalized Fibonacci and Lucas sequences. Our results apply more generally to broader classes of second-order linearly recurrent sequences with constant coefficients. As a consequence, we obtain as special cases many identities relating exactly four sequences amongst the Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal, and Jacobsthal–Lucas number sequences. We make use of algebraic arguments to establish our results, frequently employing the Binet-like formulas and generating functions of the corresponding sequences. Finally, our identities above may be extended so that they include only terms whose subscripts belong to a given arithmetic progression of the non-negative integers

Representation of special polynomials by the cycle indicator

Certain classes of special functions have been shown to have Cycle Indicator representations as well as recurrence relations. Most remarkably, it is shown how the Cycle Indicator can be used to unify or generalize special functions. Methods of unifying special functions are elaborated on. It follows that classical special functions with simple logarithms of their generating functions can be classified in this way. Also, there are counter-examples where the Cycle Indicator doesn\u27t represent the special functions given; This thesis represents a study of a journal article called Cycle Indicators and Special Functions by Leetsch C. Hsu and Peter Jau-Shyong Shiue in the Annals of Combinatorics (cf. Hsu and Shiue, [10])