A New Type of Generalized Ernst Numbers

Yıl 2024, Cilt: 12 Sayı: 2, 90 - 98, 28.10.2024

Ece Gülşah Çolak , Nazmiye Gönül Bilgin , Yüksel Soykan

Öz

This study presents, the Gaussian generalized Ernst numbers as a new complex recursive number sequence. We also give Binet's formulas, Simson's formulas, generating functions for this sequence and we touch on Gaussian Ernst and Gaussian Ernst-Lucas numbers. Besides,we establish some identities and matrices associated with these sequences. This study's contribution to the literature is the constructed of an important generalization of generalized Ernst numbers that can be applied to different fields and the establishment of important equations regarding these numbers.

Anahtar Kelimeler

Ernst numbers, Gaussian Ernst numbers, Gaussian Ernst-Lucas numbers;, Gaussian generalized Ernst numbers.

Kaynakça

• [1] M. Asçı, E. Gürel, Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Numbers, Ars Combinatoria, 111 (2013), 53-63.
• [2] M. Asçı , E. Gürel, Gaussian Jacobsthal and Gaussian Jacobsthal Polynomials, Notes on Number Theory and Discrete Mathematics, 19 (2013), 25-36.
• [3] I. Bruce, A modified Tribonacci sequence, The Fibonacci Quarterly, 22(3) (1984), 244-246.
• [4] M. Catalani, Identities for Tribonacci-related sequences, (2002), arXiv:math/0209179v1 [math.CO].
• [5] P. Catarino, H. Campos, A note on Gaussian Modified Pell numbers, Journal of Information & Optimization Sciences, 39(6) (2018), 1363-1371.
• [6] G. Cerda-Morales, Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal-Lucas polynomials and their properties, Asian-European Journal of Mathematics, 14(5) (2021), 2150076.
• [7] E. Choi, Modular Tribonacci Numbers by Matrix Method, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 20(3) (2013), 207-221.
• [8] E.G. Çolak, N. GÖnül Bilgin, Y. Soykan, Gaussian Generalized John Numbers, Conference Proceedings of Science and Technology (CPOST), 6(1) (2023), 60-69.
• [9] M. Elia, Derived Sequences, The Tribonacci Recurrence and Cubic Forms, The Fibonacci Quarterly, 39(2) (2001), 107-115.
• [10] J. B. Fraleigh, A First Course In Abstract Algebra, (2nd ed.), Addison-Wesley, Reading, ISBN 0-201-01984-1, 1976.
• [11] E. Gürel, k-Order Gaussian Fibonacci and k-Order Gaussian Lucas Recurrence Relations, Ph.D. Thesis, Pamukkale University, 2015.
• [12] S. Halıcı , G. Cerda-Morales, On Quaternion-Gaussian Fibonacci Numbers and Their Properties, An. St. Univ. Ovidius Constanta, 29(1) (2021), 71-82.
• [13] S. Halıcı ,S. Öz, On Some Gaussian Pell and Pell-Lucas Numbers, Ordu U¨ niversitesi Bilim ve Teknoloji Dergisi, 6(1) (2016), 8-18.
• [14] S. Halıcı , S. Öz, On Gaussian Pell Polynomials and Their Some Properties, Palastine Journal of Mathematics, 7(1) (2018), 251-256.
• [15] C.J. Harman, Complex Fibonacci Numbers, The Fibonacci Quarterly 19 (1981), 82-86.
• [16] A. F. Horadam, Complex Fibonacci Numbers and Fibonacci Quaternions, American Mathematics Monthly, 70 (1963), 289-291.
• [17] J. H. Jordan, Gaussian Fibonacci and Lucas numbers, The Fibonacci Quarterly, 3 (1965), 315-318.
• [18] N. Karaaslan, Gaussian Bronze Lucas Numbers, BSEU Journal of Science, 9(1) (2022), 357-363.
• [19] M. Kumari, J. Tanti, K. Prasad, On Some New Families of k-Mersenne and Generalized k-Gaussian Mersenne Numbers and Their Polynomials.
• Contributions to Discrete Mathematics, 18(2) (2022), 244-260.
• [20] P. Y. Lin, De Moivre-Type Identities For The Tribonacci Numbers, The Fibonacci Quarterly, 26 (1988), 131-134.
• [21] E. Özkan, M. Taştan, On a New Family of Gauss k-Lucas Numbers and Their Polynomials, Asian-European Journal of Mathematics, 14(6) (2021), 2150101.
• [22] E. Özkan, M. Uysal, d-Gaussian Pell-Lucas Polynomials and Their Matrix Representations, Turkish Journal of Mathematics and Computer Science, 14(2) (2022), 262-270.
• [23] S. Pethe and A. F. Horadam, Generalized Gaussian Fibonacci Numbers, Bull. Austral. Math. Soc., 33 (1986), 37-48.
• [24] S. Pethe, Some Identities for Tribonacci Sequences, The Fibonacci Quarterly, 26 (1988), 144-151.
• [25] S. Pethe, A. F. Horadam, Generalised Gaussian Lucas Primordial numbers, The Fibonacci Quarterly, 26(1) (1988), 20-30.
• [26] A. Scott, T. Delaney, W. Hoggatt Jr., The Tribonacci sequence, The Fibonacci Quarterly, 15(3) (1977), 193–200.
• [27] A. Shannon, Tribonacci numbers and Pascal’s pyramid, The Fibonacci Quarterly, 15(3) (1977), 268-275.
• [28] Y. Soykan, E. Taşdemir, İ. Okumuş, M. Göcen, Gaussian Generalized Tribonacci Numbers, Journal of Progressive Research in Mathematics, 14(2) (2018).
• [29] Y. Soykan, Generalized Ernst Numbers, Asian Journal of Pure and Applied Mathematics, 4(1) (2022), 136-150.
• [30] Y. Soykan, Generalized Tribonacci Polynomials, Earthline Journal of Mathematical Science, 13(1) (2023), 1-120.
• [31] W. Spickerman, Binet’s formula for the Tribonacci sequence, The Fibonacci Quarterly, 20 (1981), 118-120.
• [32] D. Taşcı, H. Acar, Gaussian Tetranacci Numbers, Communications in Mathematics ans Applications, 8(3) (2017), 379-386.
• [33] D. Taşcı, H. Acar, Gaussian Padovan and Gaussian Pell-Padovan Numbers, Commun. Fac. Sci. Ank. Ser. A1 Math. Stat., 67(2) (2018), 82-88.
• [34] M. Taştan, N. Yılmaz, E. Özkan, A New Family of Gauss (k, t)-Horadam Numbers, Asian-European Journal of Mathematics, 15(12) (2022), 2250225.
• [35] T. Yağmur, N. Karaaslan, Gaussian Modified Pell Sequence and Gaussian Modified Pell Polynomial Sequence, Aksaray University Journal of Science and Engineering, 2(1) (2018), 63-72.
• [36] C. C. Yalavigi, Properties of Tribonacci numbers, The Fibonacci Quarterly, 10(3) (1972), 231-246.
• [37] M. Yaşar Kartal, Gaussian Bronze Fibonacci numbers, International Journal on Mathematics, Engineering and Natural Sciences, 4(13) (2020), 19-25.
• [38] N. Yılmaz, N. Taskara, Tribonacci and Tribonacci-Lucas Numbers via the Determinants of Special Matrices, Applied Mathematical Sciences, 8(39) (2014), 1947-1955.