Mersenne Lucas numbers and complete homogeneous symmetric functions
21 Jan 2021-Vol. 24, Iss: 02, pp 127-139
TL;DR: In this paper, Binet's formula, generating function and symmetric function of Mersenne Lucas numbers were given for products of (p, q)-numbers with Mersennes Lucas numbers at positive and negative indice.
Abstract: In this paper, we first introduce new definition of Mersenne Lucas numbers sequence as, for n > 2, mn = 3mn−1 − 2mn−2 with the initial conditions m0 = 2 and m1 = 3. Considering this sequence, we give Binet’s formula, generating function and symmetric function of Mersenne Lucas numbers. By using the Binet’s formula we obtain some well-known identities such as Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. After that, we give some new generating functions for products of (p,q)-numbers with Mersenne Lucas numbers at positive and negative indice.
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01 Mar 2021
TL;DR: In this article, a new definition of k-Mersenne-Lucas numbers was introduced and properties of these numbers were investigated, and some identities and established connection formulas between them through the use of Binet's formula.
Abstract: In this paper, we will introduce a new definition of k-Mersenne–Lucas numbers and investigate some properties. Then, we obtain some identities and established connection formulas between k-Mersenne–Lucas numbers and k-Mersenne numbers through the use of Binet’s formula.
6 citations
Cites methods from "Mersenne Lucas numbers and complete..."
...[7] The Mersenne–Lucas numbers with negative index are given by m−n = 1 2n mn, for all n ∈ N....
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TL;DR: In this article, a new concept of bivariate Mersenne Lucas polynomials is introduced and the recurrence relation of them is given, and Binet's formula, generating function, Catalan's identity and Cassini's identity for this type of polynomial are obtained.
Abstract: The main aim of this paper is to introduce new concept of bivariate Mersenne Lucas polynomials { m n ( x , y ) } n = 0 ∞ , we first give the recurrence relation of them. We then obtain Binet’s formula, generating function, Catalan’s identity and Cassini’s identity for this type of polynomials. After that, we give the symmetric function, explicit formula and d’Ocagne’s identity of bivariate Mersenne and bivariate Mersenne Lucas polynomials. By using the Binet’s formula we obtain some well-known identities of these bivariate polynomials. Also, some summation formulas of bivariate Mersenne and bivariate Mersenne Lucas polynomials are investigated.
4 citations
25 Jul 2022
TL;DR: The closed form formulae for these octonions are given and some well-known identities like Cassini's identity, d’Ocagne’s identity, Catalan identity, Vajda's identity and generating functions of them are obtained.
Abstract: This paper aims to introduce the k -Mersenne and k -Mersenne-Lucas octonions. We give the closed form formulae for these octonions and obtain some well-known identities like Cassini’s identity, d’Ocagne’s identity, Catalan identity, Vajda’s identity and generating functions of them. As a consequence k = 1 yields all the above properties for Mersenne and Mersenne-Lucas octonions.
1 citations
TL;DR: In this paper , the symmetric and generating functions for odd and even terms of second-order linear recurrence sequences were studied and a new family of generating functions of odd terms of (p,q)-Fibonacci-like numbers and polynomials were derived.
Abstract: In this paper, we study the symmetric and the generating functions for odd
and even terms of the second-order linear recurrence sequences. we introduce
a operator in order to derive a new family of generating functions of odd
and even terms of Mersenne numbers, Mersenne Lucas numbers, (p,q)-
Fibonacci-like numbers, k-Pell polynomials and k-Pell Lucas polynomials. By
making use of the operator defined in this paper, we give some new
generating functions of the products of (p,q)-Fibonacci-like numbers
with odd and even terms of certain numbers and polynomials.
TL;DR: In this article , a generalized family of numbers and polynomials of one or more variables attached to the formal composition f . ( g ) of generating functions f , g and h .
Abstract: A bstract . In this paper, we introduce a generalized family of numbers and polynomials of one or more variables attached to the formal composition f . ( g ◦ h ) of generating functions f , g and h . We give explicit formula and apply the obtained result to two special families of polynomials; the first concerns generalization of some polynomials applied to the theory of hyperbolic differential equations recently introduced and studied by M. Mihoubi and M. Sahari. The second concerns two variables Laguerre-based generalized Hermite-Euler polynomials introduced and should be updated to studied recently by N. U. Khan et al..
References
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TL;DR: New generalizations are given for Pell and Pell–Lucas numbers sequences which have elements of elements of { p n ( s, t ) } n ∈ N and { q n( s , t) } n∈ N .
37 citations
TL;DR: In this article, a generalization for the symmetry between complete symmetric functions and elementary symmetric function is given, and a large variety of identities involving integer partitions and multinomial coefficients can be generated using this generalization.
Abstract: A generalization for the symmetry between complete symmetric functions and elementary symmetric functions is given. As corollaries we derive the inverse of a triangular Toeplitz matrix and the expression of the Toeplitz-Hessenberg determinant. A very large variety of identities involving integer partitions and multinomial coefficients can be generated using this generalization. The partitioned binomial theorem and a new formula for the partition function p(n) are obtained in this way.
27 citations
TL;DR: In this paper, generalized symmetric functions are used to find explicit formulas of the Fibonacci numbers, and of the Tchebychev polynomials of first and second kinds.
Abstract: In this paper, we calculate the generating functions by using the con- cepts of symmetric functions. Although the methods cited in previous works are in principle constructive, we are concerned here only with the question of manipulating combinatorial objects, known as symmetric op- erators. The proposed generalized symmetric functions can be used to find explicit formulas of the Fibonacci numbers, and of the Tchebychev polynomials of first and second kinds.
25 citations
TL;DR: In this article, two sequences called (s, t)-Jacobsthal, ( s, t) Jacobsthal Lucas are defined by considering the usual Jacob Sthal and Jacob Stalhal Lucas numbers and established some properties of these sequences and some important relationships between them.
Abstract: Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this study, two sequences called (s, t)-Jacobsthal, (s, t)-Jacobsthal Lucas are defined by considering the usual Jacobsthal and Jacobsthal Lucas numbers. After that, we establish some properties of these sequences and some important relationships between (s, t)-Jacobsthal sequence and (s, t)-Jacobsthal Lucas sequence.
24 citations
21 Dec 2016
TL;DR: From Binet’s formula of Mersenne sequence, some properties for this sequence are obtained such as the generating matrix, tridiagonal matrices and circulant type matrices.
Abstract: From Binet’s formula of Mersenne sequence we get some properties for this sequence. Mersenne, Jacobsthal and Jacobsthal-Lucas sequences are considered in order to achieve some relations between them, sums and certain products involving terms of these sequences. We also present some results with matrices involving Mersenne numbers such as the generating matrix, tridiagonal matrices and circulant type matrices.
21 citations