THE GENERALIZED BINET FORMULA, REPRESENTATION AND SUMS OF THE GENERALIZED ORDER-$k$ PELL NUMBERS

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

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On the computing of the generalized order-k Pell numbers in log time

Applied Mathematics and Computation ◽

10.1016/j.amc.2005.12.053 ◽

2006 ◽

Vol 181 (1) ◽

pp. 511-515 ◽

Cited By ~ 3

Author(s):

E. Kilic ◽

B. Altunkaynak ◽

D. Tasci

Keyword(s):

Pell Numbers ◽

Generalized Order

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On the connections between Pell numbers and Fibonacci p-numbers

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.1.148-160 ◽

2021 ◽

Vol 27 (1) ◽

pp. 148-160

Author(s):

Anthony G. Shannon ◽

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Özgür Erdağ ◽

Ömür Deveci ◽

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...

Keyword(s):

Pell Numbers ◽

Binet Formula ◽

Generating Matrix ◽

Combinatorial Representation

In this paper, we define the Fibonacci–Pell p-sequence and then we discuss the connection of the Fibonacci–Pell p-sequence with the Pell and Fibonacci p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Fibonacci–Pell p-numbers by the aid of the n-th power of the generating matrix of the Fibonacci–Pell p-sequence. Furthermore, we derive relationships between the Fibonacci–Pell p-numbers and their permanent, determinant and sums of certain matrices.

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Bi-Periodic Pell Sequence

Academic Journal of Applied Mathematical Sciences ◽

10.32861/ajams.67.136.144 ◽

2020 ◽

pp. 136-144

Author(s):

S. Uygun ◽

H. Karataş

Keyword(s):

Generating Function ◽

Convergence Properties ◽

Pell Numbers ◽

Binet Formula

In this study, we introduce a new generalization of the Pell numbers which is called bi-periodic Pell sequences. We then proceed to find the Binet formula as well as the generating function for this sequence. The well-known Cassini, Catalan and the D’ocagne’s identities as well as some related binomial summation and sum formulas are also given. The convergence properties of the consecutive terms of this sequence is also examined.

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Information Measures for Generalized Order Statistics and Their Concomitants under General Framework from Huang-Kotz FGM Bivariate Distribution

Entropy ◽

10.3390/e23030335 ◽

2021 ◽

Vol 23 (3) ◽

pp. 335

Author(s):

Mohamed A. Abd Elgawad ◽

Haroon M. Barakat ◽

Shengwu Xiong ◽

Salem A. Alyami

Keyword(s):

Order Statistics ◽

General Framework ◽

Bivariate Distribution ◽

Generalized Order Statistics ◽

Information Measures ◽

Progressive Type ◽

Information Number ◽

Fisher Information Number ◽

Dual Generalized Order Statistics ◽

Generalized Order

In this paper, we study the concomitants of dual generalized order statistics (and consequently generalized order statistics) when the parameters γ1,⋯,γn are assumed to be pairwise different from Huang–Kotz Farlie–Gumble–Morgenstern bivariate distribution. Some useful recurrence relations between single and product moments of concomitants are obtained. Moreover, Shannon’s entropy and the Fisher information number measures are derived. Finally, these measures are extensively studied for some well-known distributions such as exponential, Pareto and power distributions. The main motivation of the study of the concomitants of generalized order statistics (as an important practical kind to order the bivariate data) under this general framework is to enable researchers in different fields of statistics to use some of the important models contained in these generalized order statistics only under this general framework. These extended models are frequently used in the reliability theory, such as the progressive type-II censored order statistics.

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Load-Sharing Reliability Models with Different Component Sensitivities to Other Components’ Working States

Advances in Applied Probability ◽

10.1017/apr.2020.49 ◽

2021 ◽

Vol 53 (1) ◽

pp. 107-132

Author(s):

Tomasz Rychlik ◽

Fabio Spizzichino

Keyword(s):

Order Statistics ◽

Load Sharing ◽

Single Component ◽

Generalized Order Statistics ◽

Hazard Rates ◽

System Operation ◽

Reliability Models ◽

Mixtures Of Distributions ◽

Failure Times ◽

Generalized Order

AbstractWe study the distributions of component and system lifetimes under the time-homogeneous load-sharing model, where the multivariate conditional hazard rates of working components depend only on the set of failed components, and not on their failure moments or the time elapsed from the start of system operation. Then we analyze its time-heterogeneous extension, in which the distributions of consecutive failure times, single component lifetimes, and system lifetimes coincide with mixtures of distributions of generalized order statistics. Finally we focus on some specific forms of the time-nonhomogeneous load-sharing model.

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