Sum Formulas For Generalized Tetranacci Numbers: Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{k} and ∑_{k=1}ⁿx^{k}W_{-k}

In this paper, closed forms of the sum formulas ∑_{k=0}ⁿx^{k}W_{k} and ∑_{k=1}ⁿx^{k}W_{-k} for generalized Tetranacci numbers are presented. As special cases, we give summation formulas of Tetranacci, Tetranacci-Lucas, and other fourth-order recurrence sequences

A Study on Sum Formulas of Generalized Pentanacci Sequence: Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{k} and ∑_{k=1}ⁿx^{k}W_{-k}

In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}x^{k}W_{k}$ and $\sum_{k=1}^{n}x^{k}W_{-k}$ for generalized Pentanacci numbers are presented. As special cases, we give summation formulas of Pentanacci, Pentanacci-Lucas, and other fifth-order recurrence sequences

The m−Order Linear Recursive Quaternions

This study considers the m−order linear recursive sequences yielding some well-known sequences (such as the Fibonacci, Lucas, Pell, Jacobsthal, Padovan, and Perrin sequences). Also, the Binet-like formulas and generating functions of the m−order linear recursive sequences have been derived. Then, we define the m−order linear recursive quaternions, and give the Binet-like formulas and generating functions for them

A Study on Sum Formulas for Generalized Tribonacci Numbers: Closed Forms of the Sum Formulas ∑_{k=0}ⁿkx^{k}W_{k}, ∑_{k=1}ⁿkx^{k}W_{-k}

In this paper, closed forms of the sum formulas ∑_{k=0}ⁿkx^{k}W_{k}, ∑_{k=1}ⁿkx^{k}W_{-k} for generalized Tribonacci numbers are presented. As special cases, we give summation formulas of Tribonacci, Tribonacci-Lucas, Padovan, Perrin, Narayana and some other third order linear recurrance sequences

On the bicomplex Gaussian Fibonacci and Gaussian Lucas numbers

We give the bicomplex Gaussian Fibonacci and the bicomplex Gaussian Lucas numbers and establish the generating functions and Binet’s formulas related to these numbers. Also, we present the summation formula, matrix representation and Honsberger identity and their relationship between these numbers. Finally, we show the relationships among the bicomplex Gaussian Fibonacci, the bicomplex Gaussian Lucas, Gaussian Fibonacci, Gaussian Lucas and Fibonacci numbers

Studies On the Recurrence Properties of Generalized Pentanacci Sequence

In this paper, we investigate the recurrence properties of the generalized Pentanacci sequence and present how the generalized Pentanacci sequence at negative indices can be expressed by the sequence itself at positive indices

On Gaussian Leonardo Hybrid Polynomials

In the present paper, we first study the Gaussian Leonardo numbers and Gaussian Leonardo hybrid numbers. We give some new results for the Gaussian Leonardo numbers, including relations with the Gaussian Fibonacci and Gaussian Lucas numbers, and also give some new results for the Gaussian Leonardo hybrid numbers, including relations with the Gaussian Fibonacci and Gaussian Lucas hybrid numbers. For the proofs, we use the symmetric and antisymmetric properties of the Fibonacci and Lucas numbers. Then, we introduce the Gaussian Leonardo polynomials, which can be considered as a generalization of the Gaussian Leonardo numbers. After that, we introduce the Gaussian Leonardo hybrid polynomials, using the Gaussian Leonardo polynomials as coefficients instead of real numbers in hybrid numbers. Moreover, we obtain the recurrence relations, generating functions, Binet-like formulas, Vajda-like identities, Catalan-like identities, Cassini-like identities, and d’Ocagne-like identities for the Gaussian Leonardo polynomials and hybrid polynomials, respectively