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1.
In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their
proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from
a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as
enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of
a number of the identities.
Received July 20, 2006 相似文献
2.
W.M. Abd-Elhameed N.A. Zeyada 《International Journal of Mathematical Education in Science & Technology》2017,48(1):102-107
This paper is concerned with developing a new class of generalized numbers. The main advantage of this class is that it generalizes the two classes of generalized Fibonacci numbers and generalized Pell numbers. Some new identities involving these generalized numbers are obtained. In addition, the two well-known identities of Sury and Marques which are recently developed are deduced as special cases. Moreover, some other interesting identities involving the celebrated Fibonacci, Lucas, Pell and Pell–Lucas numbers are also deduced. 相似文献
3.
Recently, Benjamin, Plott, and Sellers proved a variety of identities involving sums of Pell numbers combinatorially by interpreting
both sides of a given identity as enumerators of certain sets of tilings using white squares, black squares, and gray dominoes.
In this article, we state and prove q-analogues of several Pell identities via weighted tilings. 相似文献
4.
This paper presents two new identities involving generalized Fibonacci and generalized Lucas numbers. One of these identities generalize the two well-known identities of Sury and Marques which are recently developed. Some other interesting identities involving the famous numbers of Fibonacci, Lucas, Pell and Pell-Lucas numbers are also deduced as special cases of the two derived identities. Performing some mathematical operations on the introduced identities yield some other new identities involving generalized Fibonacci and generalized Lucas numbers. 相似文献
5.
By means of generating function and partial derivative methods, we investigate and establish several general summation formulas involving two classes of polynomials. The general results would apply to yield some identities for the Pell polynomials and Pell-Lucas polynomials, and other general polynomials can also be recovered in this paper. 相似文献
6.
《Discrete Mathematics》2002,257(1):125-142
We examine a pair of Rogers-Ramanujan type identities of Lebesgue, and give polynomial identities for which the original identities are limiting cases. The polynomial identities turn out to be q-analogs of the Pell sequence. Finally, we provide combinatorial interpretations for the identities. 相似文献
7.
Riordan矩阵的垂直一半和水平一半已经被许多学者分别研究过.本文给出了Riordan矩阵的$(m,r,s)$-halves的定义.利用此定义能够统一的讨论Riordan矩阵的垂直一半和水平一半.作为应用,通过对Pascal和Delannoy矩阵的$(m,r,s)$-halves的研究,可以得到了一些与Fibonacci, Pell和Jacobsthal序列相关的等式. 相似文献
8.
Meral Ya?ar 《Applied mathematics and computation》2012,218(10):6067-6071
In this paper, another proof of Pell identities is presented by using the determinant of tridiagonal matrix. It is calculated via the Laplace expansion. 相似文献
9.
In this paper, we consider infinite sums derived from the reciprocals of the Pell numbers. Then applying the floor function to the reciprocals of this sums, we obtain a new and interesting identity involving the Pell numbers. 相似文献
10.
In this paper, we consider the usual and generalized order-k Fibonacci and Pell recurrences, then we define a new recurrence, which we call generalized order-k F–P sequence. Also we present a systematic investigation of the generalized order-k F–P sequence. We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order-k F–P sequence by matrix methods. Further, we give the generating function and combinatorial representations of these numbers. Also we present an algorithm for computing the sums of the generalized order-k Pell numbers, as well as the Pell numbers themselves. 相似文献
11.
Zhizheng Zhang 《Discrete Mathematics》2006,306(21):2740-2754
As a generalization of Calkin's identity and its alternating form, we compute a kind of binomial identity involving some real number sequences and a partial sum of the binomial coefficients, from which many interesting identities follow. 相似文献
12.
Feng-Zhen Zhao 《Discrete Mathematics》2009,309(12):3830-3842
In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers. 相似文献
13.
《Discrete Mathematics》2022,345(9):112891
We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, also concerning Lucas, Fibonacci and Fine numbers. 相似文献
14.
Feng-Zhen Zhao Tianming Wang 《International Journal of Mathematical Education in Science & Technology》2013,44(6):913-919
In this note, using the theory of Pell equation, the authors discuss the integrity of certain series involving generalized Fibonacci and Lucas numbers. 相似文献
15.
《数学季刊》2020,(1)
The purpose of this paper is to give the extensions of some identities involving generalized Fibonacci and Lucas numbers with binomial coefficients.These results generalize the identities by Gulec,Taskara and Uslu in Appl.Math.Lett.23(2010) 68-72 and Appl.Math.Comput.220(2013) 482-486. 相似文献
16.
In this paper,we give several identities of finite sums and some infinite series involving powers and inverse of binomial coefficients,which extends the results of T.Trif. 相似文献
17.
Emanuele Munarini 《Discrete Mathematics》2019,342(8):2415-2428
In this paper, we introduce the Pell graphs, a new family of graphs similar to the Fibonacci cubes. They are defined on certain ternary strings (Pell strings) and turn out to be subgraphs of Fibonacci cubes of odd index. Moreover, as well as ordinary hypercubes and Fibonacci cubes, Pell graphs have several interesting structural and enumerative properties. Here, we determine some of them. Specifically, we obtain a canonical decomposition giving a recursive structure, some basic properties (bipartiteness and existence of maximal matchings), some metric properties (radius, diameter, center, periphery, medianicity), some properties on subhypercubes (cube coefficients and polynomials, cube indices, decomposition in subhypercubes), and, finally, the distribution of the degrees. 相似文献
18.
G. V. Voskresenskaya 《Journal of Mathematical Sciences》2012,182(4):444-455
The paper is concerned with Shimura sums related to modular forms with multiplicative coefficients which are products of Dedekind
η-functions of various arguments. Several identities involving Shimura sums are established. The type of identity obtained
depends on the splitting of primes in certain imaginary quadratic number fields. 相似文献
19.
We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials. 相似文献
20.
Tünde Kovács 《Periodica Mathematica Hungarica》2009,58(1):83-98
We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First
we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive
integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient
algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally,
partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values
of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.
相似文献