scholarly journals 132-avoiding two-stack sortable permutations, Fibonacci numbers, and Pell numbers

2004 ◽  
Vol 143 (1-3) ◽  
pp. 72-83 ◽  
Author(s):  
Eric S. Egge ◽  
Toufik Mansour
Keyword(s):  
Fibonacci Numbers ◽  
Pell Numbers

scholarly journals Symmetric and generating functions of generalized (p,q)-numbers

2021 ◽  
Vol 48 (4) ◽  
Author(s):  
Nabiha Saba ◽  
◽  
Ali Boussayoud ◽  
Abdelhamid Abderrezzak ◽  
◽  
...  
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Lucas Polynomials ◽  
Binet’S Formula

In this paper, we will firstly define a new generalization of numbers (p, q) and then derive the appropriate Binet's formula and generating functions concerning (p,q)-Fibonacci numbers, (p,q)- Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Also, some useful generating functions are provided for the products of (p,q)-numbers with bivariate complex Fibonacci and Lucas polynomials.

scholarly journals Summing a Family of Generalized Pell Numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Linear Combination ◽  
Computer Algebra ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Partial Sums ◽  
Pell Numbers ◽  
New Family ◽  
Binet Formula

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.

scholarly journals In Praise of an Elementary Identity of Euler

10.37236/2009 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Gaurav Bhatnagar
Keyword(s):  
Hypergeometric Series ◽  
Fibonacci Numbers ◽  
Basic Hypergeometric Series ◽  
Binomial Theorem ◽  
Pell Numbers ◽  
Generalized Hypergeometric Series ◽  
Pentagonal Number Theorem ◽  
Wz Method ◽  
Elementary Identity

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu–Vandermonde sum, the Pfaff–Saalschütz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8\phi_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.

scholarly journals Construction of a new class of generating functions of binary products of some special numbers and polynomials

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 1001-1013
Author(s):  
Souhila Boughaba ◽  
Ali Boussayoud ◽  
Serkan Araci ◽  
Mohamed Kerada ◽  
Mehmet Acikgoz
Keyword(s):  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Pell Numbers ◽  
New Class ◽  
Special Numbers And Polynomials ◽  
Symmetric Properties

In this paper, we derive some new symmetric properties of k-Fibonacci numbers by making use of symmetrizing operator. We also give some new generating functions for the products of some special numbers such as k-Fibonacci numbers, k-Pell numbers, Jacobsthal numbers, Fibonacci polynomials and Chebyshev polynomials.

Divisibility properties for Fibonacci and related numbers

2013 ◽  
Vol 97 (540) ◽  
pp. 461-464
Author(s):  
Jawad Sadek ◽  
Russell Euler
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Unified Approach ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Image Position ◽  
Divisibility Properties

Although it is an old one, the fascinating world of Fibonnaci numbers and Lucas numbers continues to provide rich areas of investigation for professional and amateur mathematicians. We revisit divisibility properties for t0hose numbers along with the closely related Pell numbers and Pell-Lucas numbers by providing a unified approach for our investigation.For non-negative integers n, the recurrence relation defined bywith initial conditionscan be used to study the Pell (Pn), Fibonacci (Fn), Lucas (Ln), and Pell-Lucas (Qn) numbers in a unified way. In particular, if a = 0, b = 1 and c = 1, then (1) defines the Fibonacci numbers xn = Fn. If a = 2, b = 1 and c = 1, then xn = Ln. If a = 0, b = 1 and c = 2, then xn = Pn. If a =b = c = 2, then xn = Qn [1].

Fibonacci numbers in generalized Pell sequences

2020 ◽  
Vol 70 (5) ◽  
pp. 1057-1068
Author(s):  
Jhon J. Bravo ◽  
Jose L. Herrera
Keyword(s):  
Lower Bounds ◽  
Continued Fractions ◽  
Fibonacci Numbers ◽  
Independent Interest ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Pell Numbers

AbstractIn this paper, by using lower bounds for linear forms in logarithms of algebraic numbers and the theory of continued fractions, we find all Fibonacci numbers that appear in generalized Pell sequences. Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for Fibonacci numbers in the Pell sequence.

scholarly journals GENERATING FUNCTIONS OF EVEN AND ODD GAUSSIAN NUMBERS AND POLYNOMIALS

2021 ◽  
Vol 21 (1) ◽  
pp. 125-144
Author(s):  
NABIHA SABA ◽  
ALI BOUSSAYOUD ◽  
MOHAMED KERADA
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
New Class ◽  
Gaussian Polynomials

In this study, we introduce a new class of generating functions of odd and even Gaussian (p,q)-Fibonacci numbers, Gaussian (p,q)-Lucas numbers, Gaussian (p,q)-Pell numbers, Gaussian (p,q)-Pell Lucas numbers, Gaussian Jacobsthal numbers and Gaussian Jacobsthal Lucas numbers and we will recover the new generating functions of some Gaussian polynomials at odd and even terms. The technique used her is based on the theory of the so called symmetric functions.

scholarly journals Tiling a Strip with Triangles

10.37236/3478 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
John Bodeen ◽  
Steve Butler ◽  
Taekyoung Kim ◽  
Xiyuan Sun ◽  
Shenzhi Wang
Keyword(s):  
Fibonacci Numbers ◽  
Pell Numbers

In this paper, we examine the tilings of a $2\times n$ "triangular strip" with triangles. These tilings have connections with Fibonacci numbers, Pell numbers, and other known sequences. We derive several different recurrences, establish some properties of these numbers, and give a refined count for these tilings (i.e., by the number and type of triangles used) and establish several properties of these refined counts.

Palindromic Closures and Thue-Morse Substitution for Markoff Numbers

2017 ◽  
Vol 12 (2) ◽  
pp. 25-35 ◽  
Author(s):  
Christophe Reutenauer ◽  
Laurent Vuillon
Keyword(s):  
Fibonacci Numbers ◽  
Recursive Construction ◽  
Pell Numbers ◽  
New Formula ◽  
Markoff Numbers

Abstract We state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word v ∈ {a, b}∗ such that m − 2 is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word av. This construction gives a new recursive construction of the Markoff numbers by the lengths of the words involved in the palindromic closure. This construction interpolates between the Fibonacci numbers and the Pell numbers.

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