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Öğe Bazı özel kuaterniyon sayı dizilerinin ve polinomlarının cebirsel özellikleri(Düzce Üniversitesi, 2020) Kaplan, Faruk; Öztürk, Arzu ÖzkoçBu çalışmada ilk olarak bazı özel sayı dizileri, polinomları ve kuaterniyonlarının günümüze dek tarihsel süreci incelendi. Daha sonra üreteç matrislerin temelini oluşturan Q-matrisi hatırlatıldı. İkinci bölümde, yine özel sayı dizileri, polinomlar ve kuaterniyonlar ile ilgili temel bilgiler verildi. Üçüncü bölümde ise, iki değişkenli Fibonacci kuaterniyon polinomları ve iki değişkenli Lucas kuaterniyon polinomları tanıtıldı ve temel özellikleri incelendi. Dahası bu kuaterniyon polinomların Binet formülleri, üreteç fonksiyonları, Catalan, Cassini, d'Ocagne gibi özel özdeşlikler ile çeşitli toplam formülleri ve matris gösterimleri ispatlarıyla verildi. Sonraki bölümde, kuaterniyon dizilerinin bir genellemesi olan Horadam kuaterniyonlarından hareketle çe¸sitli toplam ve binom formülleri, özdeşlikler ve matris gösterimleri elde edildi. Son bölümde ise sonuç ve öneriler sunuldu.Öğe Binomial Transforms of k-Narayana Sequences and Some Properties(Fuat USTA, 2024) Kaplan, Faruk; Öztürk, Arzu ÖzkoçThe aim of the study is to obtain new binomial transforms for the $k-$ Narayana sequence. The first of these is the binomial transform, which is its normal form, and in the first step, after finding the recurrence relation of this new binomial transform, the generating function and Binet formula were obtained. Finally, Pascal's triangle was calculated. In the rest of the article, $k-$binomial transform was performed for the $k-$ Narayana sequence and the recurrence relation, generating function, Binet formula and Pascal's triangle were examined for the new sequence obtained. Then, by performing the falling binomial transform and the rising binomial transform, the features listed above were found again for these sequences.Öğe On Gaussian Quadra Fibona-Pell Sequence and A Quaternion Sequence Formed by the Terms of This Sequence(Duzce University, 2025) Öztürk, Arzu Özkoç; Kaplan, FarukIn this study, the Gaussian quadra Fibona-Pell sequence is proposed and examined. The quadra Fibona-Pell sequence is first extended to define the Gaussian quadra Fibona-Pell sequence. Then the generating function, Binet-like formula, and some identities are represented. In addition, some formulas related to the Gaussian quadra Fibona-Pell sequence and some matrices containing terms of the sequence are studied. Finally we define a quaternion sequence formed by the terms of Gaussian quadra Fibona-Pell sequence.Öğe On the binomial transforms of the Horadam quaternion sequences(Wiley, 2021) Kaplan, Faruk; Ozkoc Ozturk, ArzuThe main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a different type of binomial transform.Öğe SOME PROPERTIES OF BIVATIATE FIBONACCI AND LUCAS QUATERNION POLYNOMIALS(Univ Nis, 2020) Ozturk, Arzu Ozkoc; Kaplan, FarukIn this work, we introduce bivariate Fibonacci quaternion polynomials and bivariate Lucas quaternion polynomials. We present generating function, Binet formula, matrix representation, binomial formulas and some basic identities for the bivariate Fibonacci and Lucas quaternion polynomial sequences. Moreover we give various kinds of sums for these quaternion polynomials.
