In this paper, we introduce the split k-Fibonacci and k-Lucas quaternions. We obtain the Binet formulas, generating functions and exponential generating functions of these quaternions. Moreover, we give the Catalan, Cassini and d'Ocagne identities for the split k-Fibonacci and k-Lucas quaternions.

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In this paper, we give the exponential generating functions for the generalized Fibonacci and generalized Lucas quaternions, respectively. Moreover, we give some new formulas for binomial sums of these quaternions by using their Binet forms. MSC: 11B39

Irmak recently asked an open question related to divisibility properties of Fibonacci and Lucas quaternions [4, p. 374]. In this paper, we give an answer to Fibonacci and Lucas hybrid number version of this question.

In the paper, the authors find closed formulas and recurrent relations for bi-periodic Fibonacci polynomials and for bi-periodic Lucas polynomials in terms of the Hessenberg determinants. Consequently, the authors derive closed formulas and recurrent relations for the Fibonacci, Lucas, bi-periodic Fibonacci, and bi-periodic Lucas numbers in terms of the Hessenberg determinants.

In the paper, the authors find closed formulas and recurrent relations for bi-periodic Fibonacci polynomials and for bi-periodic Lucas polynomials in terms of the Hessenberg determinants. Consequently, the authors derive closed formulas and recurrent relations for the Fibonacci, Lucas, bi-periodic Fibonacci, and bi-periodic Lucas numbers in terms of the Hessenberg determinants.

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Bu makalede, quadrapell kuaterniyonları tanıtılmıştır. Bu kuaterniyonların bazı kombinatorik özellikleri verilmiştir. İlave olarak, bu kuaterniyonlara binomiyal dönüşüm uygulanmış ve bazı cebirsel özellikler türetilmiştir.