Incomplete Fibonacci and Lucas numbers

Filipponi, Piero

doi:10.1007/bf02845088

Cited by 26 publications

(17 citation statements)

References 3 publications

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“…The incomplete tribonacci number, denoted by t (s) n , is defined as the value of T (s) n (x) at x = 1. Incomplete Fibonacci numbers and polynomials have also been considered and are defined in a comparable way; see, e.g., [5,8]. Some combinatorial identities for the incomplete Fibonacci numbers were given in [3] and a bi-periodic generalization was studied in [9].…”

Section: Introductionmentioning

confidence: 99%

Some combinatorial identities for the r-Dowling polynomials

Shattuck¹

2019

NNTDM

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The incomplete tribonacci polynomials, denoted by T (s) n (x), generalize the usual tribonacci polynomials and were introduced in [10], where several algebraic identities were shown. In this paper, we provide a combinatorial interpretation for T (s) n (x) in terms of weighted linear tilings involving three types of tiles. This allows one not only to supply combinatorial proofs of the identities for T (s) n (x) appearing in [10] but also to derive additional identities. In the final section, we provide a formula for the ordinary generating function of the sequence T (s) n (x) for a fixed s, which was requested in [10]. Our derivation is combinatorial in nature and makes use of an identity relating T (s) n (x) to T n (x).

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Section: Introductionmentioning

confidence: 99%

Some combinatorial identities for the r-Dowling polynomials

Shattuck¹

2019

NNTDM

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“…where [n] denotes the integer part of n. The connection between ordinary and incomplete Fibonacci and Lucas numbers are also given in [7] as…”

Section: Fibonacci and Lucas Numbersmentioning

confidence: 99%

A symmetric algorithm for hyperharmonic and Fibonacci numbers

Dil

Mező

2008

Applied Mathematics and Computation

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In this work, we introduce a symmetric algorithm obtained by the recurrence relation a k n = a k n−1 +a k−1 n . We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacciand Lucas numbers. An explicit formulae for hyperharmonic numbers, general generating functions of the Fibonacci-and Lucas numbers are obtained.Besides we define "hyperfibonacci numbers", "hyperlucas numbers". Using these new concepts, some relations between ordinary and incomplete Fibonacci-and Lucas numbers are investigated.

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“…with initial conditions F 0 = 0, F 1 = 1 and L 0 = 2, L 1 = 1, respectively. In [9], Filipponi introduced a generalization of the Fibonacci numbers. Accordingly, the incomplete Fibonacci and Lucas numbers are determined by: where n ∈ N. Note that F n ( n−1 2 ) = F n and L n ( n 2 ) = L n .…”

Section: Introductionmentioning

confidence: 99%

Incomplete q-Chebyshev polynomials

Ercan¹,

Firengiz²,

Tuglu³

2018

Filomat

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In this paper, we get the generating functions of q-Chebyshev polynomials using ηz operator, which is ηz (f (z)) = f (qz) for any given function f (z). Also considering explicit formulas of q-Chebyshev polynomials, we give new generalizations of q-Chebyshev polynomials called incomplete q-Chebyshev polynomials of the first and second kind. We obtain recurrence relations and several properties of these polynomials. We show that there are connections between incomplete q-Chebyshev polynomials and the some well-known polynomials.

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Incomplete Fibonacci and Lucas numbers

Cited by 26 publications

References 3 publications

Some combinatorial identities for the r-Dowling polynomials

Some combinatorial identities for the r-Dowling polynomials

A symmetric algorithm for hyperharmonic and Fibonacci numbers

Incomplete q-Chebyshev polynomials

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