Search Results (21)

The purpose of this paper is to introduce the fractional Pell numbers, together with several properties, via a Grünwald–Letnikov fractional operator of orders $q\in (0,1)$ and $q\in (1,2)$. This paper also explores the fractional Pell–Lucas numbers and their properties.Due to the long-term memory property, fractional Pell sequences and fractional Pell–Lucas sequences present potential applications in modeling and computation. The closed form is deduced, and the numerical schemes are determined. The fractional characteristic equation is introduced, and it is shown that its solutions include a fractional silver ratio depending on the fractional order. In addition, the tiling problem and the concept of the fractional silver spiral are considered.A MATLAB program for applying the use of the fractional silver ratio is presented. Full article

Six well-known sequences (Fibonacci and Lucas numbers, Pell and Pell–Lucas polynomials, and Chebyshev polynomials) are characterized by quadratic linear recurrence relations. They are unified and reviewed under a common framework. Several useful properties (such as Binet-form expressions, Cassini identities, and Catalan formulae) and remarkable results concerning power sums, ordinary convolutions, and binomial convolutions are presented by employing the symmetric feature, series rearrangements, and the generating function approach. Most of the classical results concerning these six number/polynomial sequences are recorded as consequences. Full article

In this study, we introduce generalized k-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of k-order Pell polynomials, k-order Jacobsthal polynomials and k-order Jacobsthal F&L numbers. Within the scope of this research, a generalization of hybrid polynomials is given by moving them to the k-order. Hybrid polynomials defined by this generalization are called k-order F&L hybrinomials. A key aspect of our research is the establishment of the recurrence relations for generalized k-order F&L hybrinomials. After we give the recurrence relations for these hybrinomials, we obtain the generating functions of hybrinomials, shedding light on some of their important properties. Finally, we introduce the matrix representations of the generalized k-order F&L hybrinomials and give some properties of the matrix representations. Full article

In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family of the Horadam cubes, in a similar way the Lucas cubes relate to Fibonacci cubes or the Lucas-run graphs relate to Fibonacci-run graphs. As special cases, they also give rise to new graph families, such as Pell–Lucas cubes and Jacobsthal–Lucas cubes. We derive the several metric and enumerative properties of these cubes, including their diameter, periphery, radius, fundamental decomposition, number of edges, cube polynomials, and generating function of the cube polynomials. Full article

In this study, the $k$-Oresme and $k$-Oresme-Lucas sequences are defined, and some terms of these sequence are given. Then, the relations between the terms of the $k$-Oresme and $k$-Oresme-Lucas sequences are presented. In addition, we give these sequences the Binet formulas, generating functions, Cassini identity, Catalan identity etc. Moreover, the $k$-Oresme and $k$-Oresme-Lucas sequences are associated with Fibonacci, Pell numbers and Lucas, and Pell- Lucas numbers, respectively. Finally, the Catalan transforms of these sequences are given and Hankel transforms are applied to these Catalan sequences and associated with the terms of the sequence. Full article

In this paper, using the symmetrizing operator ${\delta }_{e_1{e}_2}^{2-l}$, we derive new generating functions of the products of $\left(p,q\right)$-modified Pell numbers with various bivariate polynomials, including Mersenne and Mersenne Lucas polynomials, Fibonacci and Lucas polynomials, bivariate Pell and bivariate Pell Lucas polynomials, bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials, bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, and bivariate complex Fibonacci and bivariate complex Lucas polynomials. Full article

The classical problem of finding all integers a and M such that the sums of M consecutive squared integers ${\left(a+i\right)}^2$ equal the squared integer $s^2$, where M is the number of terms in the sum, $a^2$ the first term and $a\ge 1$, $0\le i\le M-1$, yields remarkable regular linear features when plotting the values of M as a function of a. These linear features correspond to groupings of pairs of a values for successive same values of M found on either side of straight lines of equation $\mu M=2a+c$, where c is an integer constant and $\mu$ a parameter taking some rational values, called allowed values. We find expressions of a and s as a function of M for the allowed values of $\mu$ and M and parametric expressions of a, M, and s. Further, Pell equations deduced from the conditions of M are solved to find the allowed values of $\mu$ and to provide all solutions in a and M. These results yield new insights into the overall properties of the classical problem of the sums of consecutive squared integers equal to squared integers and allow us to solve this problem completely by providing all solutions in infinite families. Full article

Many post-secondary institutions have implemented anti-poverty programs to address students’ basic needs insecurities. This study examined the provision of 17 types of basic needs programs at Texas Hispanic-serving institutions over the course of the COVID-19 pandemic with the aim to identify changes in the number and types of programs offered as well as factors that may influence the presence of specific types of basic needs programs on campus. While the average number of basic needs programs per institution varied little over time, the specific types of programs that were offered changed. Institution type as a 2-year or 4-year institution was associated with providing on-campus mental health services, on-campus physical health services, and after-school care for students’ children at pre-pandemic and anticipated post-pandemic time points and employing students and free food or meal vouchers at the pre-pandemic time point. The percentage of students receiving Pell Grants was associated with basic needs programs to assist students applying for public services and referrals to off-campus health services pre-pandemic and anticipated post-pandemic. The presence of an on-campus free food pantry was associated with the percentage of students receiving Pell Grants at the anticipated post-pandemic time point only. Over the course of the pandemic, there were changes to the types of basic needs programs offered. Some types of basic needs programs were associated with institutional and/or student characteristics. Given the continued presence of basic needs programs through the course of the pandemic and into the post-pandemic period, the use of these kinds of programs and services to support students, while influenced by external factors such as the pandemic, appears institutionally established as a way to facilitate going to college for students in need. Full article

For an integer $k\ge 2$, let ${(P_n^{(k)})}_n$ be the k-generalized Pell sequence which starts with $0,\dots ,0,1$ (k terms), and each term afterwards is given by $P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}$. In this paper, we determine all solutions of the Markov equation $x^2+y^2+z^2=3xyz$, with x, y, and z being k-generalized Pell numbers. This paper continues and extends a previous work of Kafle, Srinivasan and Togbé, who found all Markov triples with Pell components. Full article

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. Full article

Wheat (Triticum aestivum L.) cultivation has been globally restricted by many plant viruses such as the Wheat streak mosaic virus (WSMV), Barley stripe mosaic virus (BSMV), and Brome mosaic virus (BMV). Herein, the transcriptome of wheat was in silico analyzed under mono- (WSMV, BSMV, or BMV), bi- (BMV&BSMV, BMV&WSMV, and BSMV&WSMV), and tripartite (WSMV, BSMV, and BMV) infections using the RNA-seq technique. Total numbers of 1616/270, 5243/690 and 5589/2183 differentially expressed genes (DEGs) were up/down-regulated during the bipartite infection of BMV&BSMV, BMV&WSMV and BSMV&WSMV, respectively, while the tripartite infection resulted in the up/down-regulation of 6110/2424 DEGs. The NAC and bHLH were the most commonly presented transcription factor (TF) families in WSMV, BMV, and BSMV infection, while C2H2, bHLH, and NAC were the TF families involved in BMV&WSMV, BMV&BSMV, and BSMV&WSMV infections, respectively. The RLK-Pelle_DLSV was the most commonly expressed protein kinase (PK) family in all infection patterns. Promoter analysis showed that the motifs involved in gene expression, CUL4 RING ubiquitin ligase complex, stress response, brassinosteroid response, and energy-related pathways were significantly induced in wheat plants under bipartite infections. The gene expression network analysis showed that a defense-related gene, i.e., allene oxide synthase (AOS) gene, serves as a crucial hub in tripartite infections. Full article

The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind ${}_2{F}_1\left(z\right)$ are included in all connection coefficients for a specific z. Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed. Full article