Horadam Numbers Research Articles

The determinant and a factorization of a Toeplitz matrix with some type of Horadam numbers entries

Abstract In this note, we define a symmetric Toeplitz matrix whose entries are from a particular Horadam sequence. We provide the determinant, the inverse and a factorization for such matrices and their special cases. These matrices are always nonsingular and their inverses are pentadiagonal matrices.

On unrestricted dual-generalized complex Horadam numbers

This research introduces a novel category of dual-generalized complex numbers, with components represented by unrestricted Horadam numbers. We present various recurrence relations, summation formulas, the Binet formula, and the generating function associated with these numbers. Additionally, a comprehensive bilinear index-reduction formula is derived, which encompasses Vajda’s, Catalan’s, Cassini’s, D’Ocagne’s, and Halton’s identities as specific cases.

On Horadam finite operator hybrid numbers

In this paper, we introduce and study a new hybrid number sequence with Horadam numbers and a finite operator, called Horadam finite operator hybrid numbers. We derive recurrence relation, Binet-like formula, ordinary generating function, exponential generating function, Poisson generating function, and summation formula for Horadam finite operator hybrid numbers. Moreover, we give matrix representation and Cassini's identities for these numbers.

On a generalization of dual-generalized complex Fibonacci quaternions

In this study, we introduce a new class of generalized quaternions whose components are dual-generalized complex Horadam numbers. We investigate some algebraic properties of them.

Reciprocal Series Involving Horadam Numbers

Reciprocal Series Involving Horadam Numbers

Reciprocal series involving Horadam numbers

UDC 511.176 We evaluate some new three-parameter families of finite reciprocal sums involving Horadam numbers. We are also able to state the results for the associated infinite sums. Some Fibonacci and Lucas sums are presented as examples.

Bi-periodic incomplete Horadam numbers

In this paper, we introduce bi-periodic~incomplete~Horadam numbers as a natural generalization of incomplete Horadam numbers. We study their basic properties and provide recurrence relations. In particular, we derive the generating function of these numbers.

Recurrence Relations for the Squares of the Horadam Numbers and Some Associated Consequences

Abstract We derive recurrence relations for the squares of the Horadam numbers w n 2 w_n^2 , where the Horadam sequence w n is such that the numbers w n , for n ∈ ℤ, are defined recursively by w 0 = a, w 1 = b, w n = pw n− 1 − qw n− 2 (n ≥ 2), where a, b, p and q are arbitrary complex numbers with p ≠ 0 and q ≠ 0. Some related results emanating from the recurrence relations such as reciprocal sums, partial sums, and sums with double binomial coefficients are also presented.

A new family of Gauss (k,t)-Horadam numbers

We present a new family of Gauss [Formula: see text]-Horadam numbers and obtain Binet formula of this family. We give the relationship between this family and the known Gauss [Formula: see text]-Horadam number. Then we prove the Cassini and Catalan identities for this family. Furthermore, we investigate the sums, the recurrence relations and generating functions of this family.

New families of Horadam numbers associated with finite operators and their applications

In this paper, by applying finite operator to the Horadam sequence, we define the Horadam finite operator sequences. We not only give some special cases of this new sequences but also investigate some properties of our new sequences such as recurrent relation, Binet‐like formula, summation formula, and generating function, respectively. We also present closed formula for the Horadam finite operator numbers and their special cases in terms of tridiagonal determinants. Moreover, by using the tridiagonal determinant, we once again verify the recurrence relation of the Horadam finite operator numbers.

Special formulas involving polygonal numbers and Horadam numbers

Some convolution-type identities involving polygonal numbers and Horadam numbers are derived. The method of proof is to properly relate the generating functions to each other. Additionally, we prove a general non-convolutional result involving these number families and discuss some of the consequences.

On a Generalization of Incomplete Fibonacci Quaternions

The aim of this article is to introduce a new class of quater- nions, namely, incomplete Horadam quaternions that are based on in- complete Horadam numbers which generalize the previously introduced incomplete Fibonacci and Lucas quaternions. Further, some identities including summation formulas and generating functions concerning these quaternions are also established.

Partial sum of the products of the Horadam numbers with subscripts in arithmetic progression

We evaluate the partial sum of the products of the terms of any two Horadam sequences with subscripts in arithmetic progression. Illustrative examples are drawn from six well-known Horadam sequences.

Two Generalizations of Dual-Hyperbolic Balancing Numbers

In this paper, we study two generalizations of dual-hyperbolic balancing numbers: dual-hyperbolic Horadam numbers and dual-hyperbolic k-balancing numbers. We give Catalan’s identity, Cassini’s identity, and d’Ocagne’s identity for them.

On the Third-Order Horadam and Geometric Mean Sequences

This paper, in considering aspects of the geometric mean sequence, offers new results connecting generalized Tribonacci and third-order Horadam numbers which are established and then proved independently.

On r-circulant matrices with Horadam numbers having arithmetic indices

On r-circulant matrices with Horadam numbers having arithmetic indices

A generalization of André-Jeannin’s symmetric identity

Abstract In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers Wn , defined by a three-term recurrence Wn +2 = P Wn +1 − QWn with constant coefficients. In this paper, we extend this identity to sequences {an}n ∈ℕ satisfying a three-term recurrence an +2 = pn +1 an +1 + qn +1 an with arbitrary coefficients. Then, we specialize such an identity to several q-polynomials of combinatorial interest, such as the q-Fibonacci, q-Lucas, q-Pell, q-Jacobsthal, q-Chebyshev and q-Morgan-Voyce polynomials.

Generalized Horadam Polynomials and Numbers

Abstract We consider the polynomials hn,m(x) (m ≥ 2) and the numbers hn,m (x = 1), which are the generalized Horadam polynomials and the generalized Horadam numbers, respectively. We also consider the polynomials h(s) n,m(x)- convolutions of the polynomials hn,m(x), and the sequence of numbers h(s) n,m- convolutions of the numbers hn,m, where s ≥ 0.

On some generalizations of Horadam’s numbers

In this paper, we introduce the incomplete Horadam numbers Wn(k), and hyper-Horadam numbers W(k)n, which generalize the Horadam?s numbers defined by the recurrence Wn = pWn-1 + qWn-2, with W0 = a and W1 = b. We give some combinatorial properties. As an application, we evaluate a lower and upper bounds for the spectral norms of r-circulant matrices associated with these two generalizations. Moreover, we establish a new bounds for the spectral norms of r-circulant matrices associated with Horadam?s numbers in terms of incomplete Horadam and hyper-Horadam numbers.

Inverse and Moore-Penrose inverse of Toeplitz matrices with classical Horadam numbers

Inverse and Moore-Penrose inverse of Toeplitz matrices with classical Horadam numbers