Binet Formulas Research Articles

On the Properties of Iterated Binomial Transforms for the Padovan and Perrin Matrix Sequences

In this study, we apply "r" times the binomial transform to the Padovan and Perrin matrix sequences. Also, the Binet formulas, summations, generating functions of these transforms are found using recurrence relations. Finally, we give the relationships of between iterated binomial transforms for Padovan and Perrin matrix sequences.

Some Combinatorial Properties of the k-Fibonacci and the k-Lucas Quaternions

Abstract In this paper, we define the k-Fibonacci and the k-Lucas quaternions. We investigate the generating functions and Binet formulas for these quaternions. In addition, we derive some sums formulas and identities such as Cassini’s identity.

On Dual Fibonacci Octonions

In this study, we define the dual Fibonacci octonions and consider the dual Fibonacci quaternions. We investigate some fundamental algebraic properties of the Fibonacci octonions and quaternions. In addition we give the Binet formulas and the generating functions of them.

Kneading determinants of infinite order linear recurrences

Infinite order linear recurrences are studied via kneading matrices and kneading determinants. The concepts of kneading matrix and kneading determinant of an infinite order linear recurrence, introduced in this work, are defined in a purely linear algebraic context. These concepts extend the classical notions of Frobenius companion matrix to infinite order linear recurrences and to the associated discriminant of finite order linear recurrences. Asymptotic Binet formulas are deduced for general classes of infinite order linear recurrences as a consequence of the analytical properties of the generating functions obtained for the solutions of these infinite order linear recurrences.

A New Approach To Fibonacci, Lucas Numbers and Dual Vectors

Dual Fibonacci and dual Lucas numbers are defined with dual Fibonacci and Lucas quaternions in Nurkan and G¨uven [18]. In this study, we work on these dual Fibonacci and dual Lucas numbers. We obtain the properties e.g. D’ocagnes, Cassini, Catalan, negadual Fibonacci identities, Binet formulas and relations of them. We also define new vectors which are called dual Fibonacci vectors. We give properties of these vectors to exert in geometry of dual space.

Dual Fibonacci Quaternions

In this study, we define the dual Fibonacci quaternion and the dual Lucas quaternion. We derive the relations between the dual Fibonacci and the dual Lucas quaternion which connected the Fibonacci and the Lucas numbers. Furthermore, we give the Binet and Cassini formulas for these quaternions.

New generalizations of Fibonacci and Lucas sequences

We consider the sequences { } and { } which are generated by the recurrence relations ( ) and ( ) with the initial conditions and where a and b are any non – zero real numbers. We obtain generating functions, Binet formulas for these two sequences and give generalizations of some well – known identities. Mathematics Subject Classification: 11B39

Generalization of Binet's Gamma function formulas

Several representations for the logarithm of the Gamma function exist in the literature. There are four important expansions which bear the name of Binet. Hermite generalized Binet's first formula to the logarithm of the Gamma function with shifted argument. The generalization of Binet's second formula is apparently not known; however, it follows easily from another result of Hermite. The aim of this paper is to give possible generalizations of the third and fourth Binet formulas.

Split Fibonacci Quaternions

Starting from ideas given by Horadam in [5] , in this paper, we will define the split Fibonacci quaternion, the split Lucas quaternion and the split generalized Fibonacci quaternion. We used the well-known identities related to the Fibonacci and Lucas numbers to obtain the relations between the split Fibonacci, split Lucas and the split generalized Fibonacci quaternions. Moreover, we give Binet formulas and Cassini identities for these quaternions.

Binomial Transforms of the Padovan and Perrin Matrix Sequences

We apply the binomial transforms to Padovan and Perrin matrix sequences. Also, the Binet formulas, summations, and generating functions of these transforms are found by recurrence relations. Finally, we illustrate the relations between these transforms by deriving new formulas.

On Fibonacci Quaternions

In this paper, we investigate the Fibonacci and Lucas quaternions. We give the generating functions and Binet formulas for these quaternions. Moreover, we derive some sums formulas for them.

Multi Parameters Golden Ratio and Some Applications

The present paper is devoted to the generalized multi parameters golden ratio. Variety of features like two-dimensional continued fractions, and conjectures on geometrical properties concerning to this subject are also presented. Wider generalization of Binet, Pell and Gazale formulas and wider generalizations of symmetric hyperbolic Fibonacci and Lucas functions presented by Stakhov and Rozin are also achieved. Geometrical applications such as applications in angle trisection and easy drawing of every regular polygons are developed. As a special case, some famous identities like Cassini’s, Askey’s are derived and presented, and also a new class of multi parameters hyperbolic functions and their properties are introduced, finally a generalized Q-matrix called Gn-matrix of order n being a generating matrix for the generalized Fibonacci numbers of order n and its inverse are created. The corresponding code matrix will prevent the attack to the data based on previous matrix.

Algebraic properties of Manin matrices 1

We study a class of matrices with noncommutative entries, which were first considered by Yu.I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative endomorphisms” of a polynomial algebra. More explicitly their defining conditions read: (1) elements in the same column commute; (2) commutators of the cross terms are equal: [ M i j , M k l ] = [ M k j , M i l ] (e.g. [ M 11 , M 22 ] = [ M 21 , M 12 ] ). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true, that is, Manin matrices are the most general class of matrices such that linear algebra holds true for them. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new. In particular we provide complete proofs that an inverse to a Manin matrix is again a Manin matrix and for the Schur formula for the determinant of a block matrix; we generalize the noncommutative Cauchy–Binet formulas discovered recently arXiv:0809.3516, which includes the classical Capelli and related identities. We also discuss many other properties, such as the Cramer formula for the inverse matrix, the Cayley–Hamilton theorem, Newton and MacMahon–Wronski identities, Plücker relations, Sylvester's theorem, the Lagrange–Desnanot–Lewis Carroll formula, the Weinstein–Aronszajn formula, some multiplicativity properties for the determinant, relations with quasideterminants, calculation of the determinant via Gauss decomposition, conjugation to the second normal (Frobenius) form, and so on and so forth. Finally several examples and open question are discussed. We refer to [A. Chervov, G. Falqui, Manin matrices and Talalaev's formula, J. Phys. A 41 (2008) 194006; V. Rubtsov, A. Silantiev, D. Talalaev, Manin matrices, elliptic commuting families and characteristic polynomial of quantum gl n elliptic Gaudin model, in press] for some applications in the realm of quantum integrable systems.

The generalized Pell ( p, i)-numbers and their Binet formulas, combinatorial representations, sums

The theory of generalized Pell p-numbers was introduced by Stakhov and then have been studied by several authors. In this paper, we consider the usual Pell numbers and as similar to the Fibonacci p-numbers, we give fair generalization of the Pell numbers, which we call the generalized Pell ( p, i)- numbers for 0 ⩽ i ⩽ p. First we give relationships between the generalized Pell ( p, i)-numbers and give the generating matrices for these numbers. Also we derive the generalized Binet formulas, sums, combinatorial representations and generating function of the generalized Pell p-numbers. Also using matrix methods, we derive an explicit formula for the sums of the generalized Fibonacci p-numbers. Finally, we derive relationships between generalized Pell ( p, i)-numbers and their sums and permanents of certain matrices.

On the m-extension of the Fibonacci and Lucas p-numbers

In this article, we define the m-extension of the Fibonacci and Lucas p-numbers ( p ⩾ 0 is integer and m > 0 is real number) from which, specifying p and m, classic Fibonacci and Lucas numbers ( p = 1, m = 1), Pell and Pell–Lucas numbers ( p = 1, m = 2), Fibonacci and Lucas p-numbers ( m = 1), Fibonacci m-numbers ( p = 1), Pell and Pell–Lucas p-numbers ( m = 2) are obtained. Afterwards, we obtain the continuous functions for the m-extension of the Fibonacci and Lucas p-numbers using the generalized Binet formulas. Also we introduce in the article a new class of mathematical constants – the Golden ( p, m)-Proportions, which are a wide generalization of the classical golden mean, the golden p-proportions and the golden m-proportions. The article is of fundamental interest for theoretical physics where Fibonacci numbers and the golden mean are used widely.

Generating functions for the powers of Fibonacci sequences

In this note, based on the Binet formulas and the power-reducing techniques, closed forms of generating functions for the powers of Fibonacci sequences are presented. The corresponding results are extended to some other famous sequences as well.

The continuous functions for the Fibonacci and Lucas p-numbers

Abstract The new continuous functions for the Fibonacci and Lucas p-numbers using Binet formulas are introduced. The article is of a fundamental interest for Fibonacci numbers theory and theoretical physics.

Theory of Binet formulas for Fibonacci and Lucas p-numbers

Abstract Modern natural science requires the development of new mathematical apparatus. The generalized Fibonacci numbers or Fibonacci p-numbers (p = 0, 1, 2, 3, …), which appear in the “diagonal sums” of Pascal’s triangle and are assigned in the recurrent form, are a new mathematical discovery. The purpose of the present article is to derive analytical formulas for the Fibonacci p-numbers. We show that these formulas are similar to the Binet formulas for the classical Fibonacci numbers. Moreover, in this article, there is derived one more class of the recurrent sequences, which is defined to be a generalization of the Lucas numbers (Lucas p-numbers).