One-parameter generalization of dual-hyperbolic Pell numbers
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Dorota Bród
and
Iwona Włoch
Abstract
In this paper, we introduce one-parameter generalization of dual-hyperbolic Pell numbers – dual-hyperbolic r-Pell numbers. We give some identities for them related to Catalan identity, Cassini identity, d’Ocagne identity and convolution identity. Moreover, we define their matrix generators.
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(Communicated by István Gaál)
References
[1] Akar, M.—Yüce, S.—Şahin, S.: On the dual hyperbolic numbers and the complex hyperbolic numbers, Journal of Computer Science & Computational Mathematics 8(1) (2018); 10.20967/jcscm.2018.01.001Search in Google Scholar
[2] Aydin, F.T.: Dual-hyperbolic Pell quaternions, J. Discrete Math. Sci. Cryptogr. (2020); 10.1080/09720529.2020.1758367.Search in Google Scholar
[3] Bród, D.: On a new one parameter generalization of Pell numbers, Ann. Math. Sil. 33 (2019), 66–76.10.2478/amsil-2019-0011Search in Google Scholar
[4] Catarino, P.: On some identities and generating functions for k-Pell numbers, International Journal of Mathematical Analysis 7(38) (2013), 1877–1884.10.12988/ijma.2013.35131Search in Google Scholar
[5] Cihan, A.—Azak, A. Z.—Güngör, M. A.—Tosun, M.: A study on dual hyperbolic Fibonacci and Lucas numbers, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 27(1) (2019), 35–48.10.2478/auom-2019-0002Search in Google Scholar
[6] Clifford, W. K.: Preliminary sketch of biquaternions, Proc. Lond. Math. Soc. s1-4 (1871), 381–395.10.1112/plms/s1-4.1.381Search in Google Scholar
[7] Cockle, J.: On a new imaginary in algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 34 (1849), 37–47.10.1080/14786444908646169Search in Google Scholar
[8] Cockle, J.: On certain functions resembling quaternions, and on a new imaginary in algebra, Lond. Edinb. Dublin Philos. Mag. J. Sci. 33 (1848), 435–439.10.1080/14786444808646139Search in Google Scholar
[9] Cockle, J.: On impossible equations, on impossible quantities, and on tessarines, Lond. Edinb. Dublin Philos. Mag. J. Sci. 37 (1850), 281–283.10.1080/14786445008646598Search in Google Scholar
[10] Cockle, J.: On the symbols of algebra, and on the theory of tesarines, Lond. Edinb. Dublin Philos. Mag. J. Sci. 34 (1849), 406-410.10.1080/14786444908646257Search in Google Scholar
[11] Horadam, A. F.: Pell identities, Fibonacci Quart. 9(3) (1971), 245–263.Search in Google Scholar
[12] Kocer, E. G.—Tuglu, N.: The Binet formulas for the Pell and Pell-Lucas p-numbers, Ars Combin. 85 (2007), 3–17.Search in Google Scholar
[13] Piejko, K.—Włoch, I.: On k-distance Pell numbers in $3$-edge coloured graphs, J. Appl. Math. 2014 (2014), Art. ID 428020.10.1155/2014/428020Search in Google Scholar
[14] Szynal-Liana, A.—Włoch, I.: Hypercomplex Numbers of the Fibonacci type, Oficyna Wydawnicza Politechniki Rzeszowskiej, Rzeszów, 2019.Search in Google Scholar
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Articles in the same Issue
- Regular Papers
- Logical and algebraic properties of generalized orthomodular posets
- Integral prefilters and integral Eq-algebras
- Modules with fusion and implication based over distributive lattices: Representation and duality
- Localization of k × j-rough Heyting algebras
- Meet infinite distributivity for congruence lattices of direct sums of algebras
- Some exponential diophantine equations II: The equation x2 – Dy2 = kz for even k
- On extensions of the Open Door Lemma
- Nevanlinna theory for holomophic curves from annuli into semi-Abelian varieties
- Stability and feedback stabilizability of delay periodic differential equations with pairwise permutable matrix functions
- On the behaviour solutions of fractional and partial integro differential heat equations and its numerical solutions
- Oscillatory and asymptotic behavior of solutions of third-order quasi-linear neutral difference equations
- Some properties of Kantorovich variant of Chlodowsky-Szász operators induced by Boas-Buck type polynomials
- A Study on statistical versions of convergence of sequences of functions
- Fuzzy fixed point theorems and Ulam-Hyers stability of fuzzy set-valued maps
- Notes on affine Killing and two-Killing vector fields
- Prediction of the exponential fractional upper record-values
- On concomitants of generalized order statistics from generalized FGM family under a general setting
- One-parameter generalization of dual-hyperbolic Pell numbers
