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Journal articles on the topic 'Recurrence sequences with nonconstant coefficients'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

van der Poorten, A. J., and I. E. Shparlinski. "On linear recurrence sequences with polynomial coefficients." Glasgow Mathematical Journal 38, no. 2 (1996): 147–55. http://dx.doi.org/10.1017/s0017089500031372.

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We consider sequences (Ah)defined over the field ℚ of rational numbers and satisfying a linear homogeneous recurrence relationwith polynomial coefficients sj;. We shall assume without loss of generality, as we may, that the sj, are defined over ℤ and the initial values A0A]…, An−1 are integer numbers. Also, without loss of generality we may assume that S0 and Sn have no non-negative integer zero. Indeed, any other case can be reduced to this one by making a shift h → h – l – 1 where l is an upper bound for zeros of the corresponding polynomials (and which can be effectively estimated in terms of their heights)

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2

Rebenda, Josef, and Zuzana Pátíková. "Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays." Complexity 2020 (February 28, 2020): 1–12. http://dx.doi.org/10.1155/2020/2854574.

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An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD.

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3

Methfessel, C. "On the zeros of recurrence sequences with non-constant coefficients." Archiv der Mathematik 74, no. 3 (2000): 201–6. http://dx.doi.org/10.1007/s000130050431.

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4

Bertók, Csanád, Lajos Hajdu, István Pink, and Zsolt Rábai. "Linear combinations of prime powers in binary recurrence sequences." International Journal of Number Theory 13, no. 02 (2017): 261–71. http://dx.doi.org/10.1142/s1793042117500166.

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We give finiteness results concerning terms of linear recurrence sequences having a representation as a linear combination, with fixed coefficients, of powers of fixed primes. On one hand, under certain conditions, we give effective bounds for the terms of binary recurrence sequences with such a representation. On the other hand, in the case of some special binary recurrence sequences, all terms having a representation as sums of powers of [Formula: see text] and [Formula: see text] are explicitly determined.

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5

Natalini, Pierpaolo, and Paolo Ricci. "Higher order bell polynomials and the relevant integer sequences." Applicable Analysis and Discrete Mathematics 11, no. 2 (2017): 327–39. http://dx.doi.org/10.2298/aadm1702327n.

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The recurrence relation for the coefficients of higher order Bell polynomials, i.e. of the Bell polynomials relevant to nth derivative of a multiple composite function, is proved. Therefore, starting from this recurrence relation and by using the computer algebra program Mathematica?, some tables for complete higher order Bell polynomials and the relevant numbers are derived.

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6

Hubálovská, Marie, Štěpán Hubálovský, and Eva Trojovská. "On Homogeneous Combinations of Linear Recurrence Sequences." Mathematics 8, no. 12 (2020): 2152. http://dx.doi.org/10.3390/math8122152.

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Let (Fn)n≥0 be the Fibonacci sequence given by Fn+2=Fn+1+Fn, for n≥0, where F0=0 and F1=1. There are several interesting identities involving this sequence such as Fn2+Fn+12=F2n+1, for all n≥0. In 2012, Chaves, Marques and Togbé proved that if (Gm)m is a linear recurrence sequence (under weak assumptions) and Gn+1s+⋯+Gn+ℓs∈(Gm)m, for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ł and the parameters of (Gm)m. In this paper, we shall prove that if P(x1,…,xℓ) is an integer homogeneous s-degree polynomial (under weak hypotheses) and if P(Gn+1,…,Gn+ℓ)∈(Gm)m for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ℓ, the parameters of (Gm)m and the coefficients of P.

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7

Branquinho, Amílcar, Juan Garca-Ardila, and Francisco Marcellán. "Ratio asymptotics for biorthogonal matrix polynomials with unbounded recurrence coefficients." Applicable Analysis and Discrete Mathematics, no. 00 (2020): 51. http://dx.doi.org/10.2298/aadm190225051b.

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In this paper we study matrix biorthogonal polynomials sequences that satisfy a nonsymmetric three term recurrence relation with unbounded matrix coefficients. The outer ratio asymptotics for this family of matrix biorthogonal polynomials is derived under quite general assumptions. Some illustrative examples are considered.

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8

Dominici, Diego. "Recurrence coefficients of Toda-type orthogonal polynomials I. Asymptotic analysis." Bulletin of Mathematical Sciences 10, no. 02 (2020): 2050003. http://dx.doi.org/10.1142/s1664360720500034.

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We study the three-term recurrence coefficients [Formula: see text] of polynomial sequences orthogonal with respect to a perturbed linear functional depending on a variable [Formula: see text] We obtain power series expansions in [Formula: see text] and asymptotic expansions as [Formula: see text] We use our results to settle some conjectures proposed by Walter Van Assche and collaborators.

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9

ANDRICA, DORIN, and OVIDIU BAGDASAR. "On some results concerning the polygonal polynomials." Carpathian Journal of Mathematics 35, no. 1 (2019): 01–12. http://dx.doi.org/10.37193/cjm.2019.01.01.

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In this paper we define the nth polygonal polynomial and we investigate recurrence relations and exact integral formulae for the coefficients of Pn and for those of the Mahonianpolynomials. We also explore numerical properties of these coefficients, unraveling new meanings for old sequences and generating novel entries to the Online Encyclopedia of Integer Sequences (OEIS). Some open questions are also formulated.

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10

Dubickas, Artūras. "Intervals without primes near elements of linear recurrence sequences." International Journal of Number Theory 14, no. 02 (2018): 567–79. http://dx.doi.org/10.1142/s1793042118500355.

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Let [Formula: see text] be an unbounded sequence of integers satisfying a linear recurrence relation with integer coefficients. We show that for any [Formula: see text] there exist infinitely many [Formula: see text] for which [Formula: see text] consecutive integers [Formula: see text] are all divisible by certain primes. Moreover, if the sequence of integers [Formula: see text] satisfying a linear recurrence relation is unbounded and non-degenerate then for some constant [Formula: see text] the intervals [Formula: see text] do not contain prime numbers for infinitely many [Formula: see text]. Applying this argument to sequences of integer parts of powers of Pisot and Salem numbers [Formula: see text] we derive a similar result for those sequences as well which implies, for instance, that the shifted integer parts [Formula: see text], where [Formula: see text] and [Formula: see text] runs through some infinite arithmetic progression of positive integers, are all composite.

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11

Baake, Michael. "Small Oscillations, Sturm Sequences, and Orthogonal Polynomials." Zeitschrift für Naturforschung A 49, no. 3 (1994): 445–57. http://dx.doi.org/10.1515/zna-1994-0301.

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Abstract The relation between small oscillations of one-dimensional mechanical n-particle systems and the theory of orthogonal polynomials is investigated. It is shown how the polynomials provide a natural tool to determine the eigenfrequencies and eigencoordinates completely, where the existence of a specific two-termed recurrence formula is essential. Physical and mathematical statements are formulated in terms of the recurrence coefficients which can directly be obtained from the corresponding secular equation. Several results on Sturm sequences and orthogonal polynomials are presented with respect to the treatment of small oscillations. The relation to the numerical treatment of the generalized eigenvalue problem is discussed and further applications to physical problems from quantum mechanics, statistical mechanics, and spin systems are briefly outlined.

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12

Munarini, Emanuele. "A generalization of André-Jeannin’s symmetric identity." Pure Mathematics and Applications 27, no. 1 (2018): 98–118. http://dx.doi.org/10.1515/puma-2015-0028.

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Abstract In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers Wn, defined by a three-term recurrence Wn+2 = P Wn+1 − QWn with constant coefficients. In this paper, we extend this identity to sequences {an}n∈ℕ satisfying a three-term recurrence an+2 = pn+1an+1 + qn+1an with arbitrary coefficients. Then, we specialize such an identity to several q-polynomials of combinatorial interest, such as the q-Fibonacci, q-Lucas, q-Pell, q-Jacobsthal, q-Chebyshev and q-Morgan-Voyce polynomials.

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13

Dil, Ayhan, and Erkan Muniroğlu. "Applications of derivative and difference operators on some sequences." Applicable Analysis and Discrete Mathematics, no. 00 (2020): 30. http://dx.doi.org/10.2298/aadm190908030d.

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In this study, depending on the upper and the lower indices of the hyperharmonic number h(r), nonlinear recurrence relations are obtained. It is shown that generalized harmonic numbers and hyperharmonic numbers can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hyperharmonic numbers are given. Negative-ordered hyperharmonic numbers are defined and their alternative representations are given.

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14

Rebocho, Maria Das Neves. "Laguerre–Hahn orthogonal polynomials on the real line." Random Matrices: Theory and Applications 09, no. 01 (2019): 2040001. http://dx.doi.org/10.1142/s2010326320400018.

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A survey is given on sequences of orthogonal polynomials related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coefficients — the so-called Laguerre–Hahn class. The main goal is to describe analytical aspects, focusing on differential equations for those orthogonal polynomials, difference and differential equations for the recurrence coefficients, and distributional equations for the corresponding linear functionals.

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15

Essouabri, Driss, Kohji Matsumoto, and Hirofumi Tsumura. "Multiple Zeta-Functions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula." Canadian Journal of Mathematics 63, no. 2 (2011): 241–76. http://dx.doi.org/10.4153/cjm-2010-085-1.

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Abstract We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions fromwhich some generalizations of the classical sum formula can be deduced.

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16

Verde-Star, Luis. "Recurrence coefficients and difference equations of classical discrete orthogonal and q -orthogonal polynomial sequences." Linear Algebra and its Applications 440 (January 2014): 293–306. http://dx.doi.org/10.1016/j.laa.2013.10.051.

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17

Dou, Liping, Chengmin Hou, and Sui Sun Cheng. "Bifurcation Analysis for Nonlinear Recurrence Relations with Threshold Control and2k-Periodic Coefficients." Discrete Dynamics in Nature and Society 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/610345.

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A nonlinear recurrence involving a piecewise constant McCulloch-Pitts function and2k-periodic coefficient sequences is investigated. By allowing the threshold parameter to vary from 0 to∞, we work out a complete bifurcation analysis for the asymptotic behaviors of the corresponding solutions. Among other things, we show that each solution tends towards one of four different limits. Furthermore, the accompanying initial regions for each type of solutions can be determined. It is hoped that our analysis will provide motivation for further results for recurrent McCulloch-Pitts type neural networks.

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18

HOU, CHENGMIN, CHUNLAI WANG, and SUI SUN CHENG. "BIFURCATION ANALYSIS FOR A NONLINEAR RECURRENCE RELATION WITH THRESHOLD CONTROL AND PERIODIC COEFFICIENTS." International Journal of Bifurcation and Chaos 22, no. 03 (2012): 1250055. http://dx.doi.org/10.1142/s0218127412500551.

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A nonlinear recurrence involving a piecewise constant McCulloch–Pitts function and two 2-periodic coefficient sequences is investigated. By allowing the threshold parameter to vary from 0+ to +∞, we work out a complete bifurcation analysis for the asymptotic behaviors of the corresponding solutions. We show that there are four steady state solutions and that all solutions will tend to one of them. We hope that our results will be useful in further investigating neural networks involving the McCulloch–Pitts function with threshold and more general periodic coefficients.

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19

HONE, ANDREW N. W., and CHRISTINE SWART. "Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences." Mathematical Proceedings of the Cambridge Philosophical Society 145, no. 1 (2008): 65–85. http://dx.doi.org/10.1017/s030500410800114x.

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AbstractSomos 4 sequences are a family of sequences defined by a fourth-order quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. This connection is Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a by-product, non-periodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.

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20

Guo, Ji. "The Quotient Problem for Entire Functions." Canadian Mathematical Bulletin 62, no. 3 (2018): 479–89. http://dx.doi.org/10.4153/s0008439518000097.

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AbstractLet $\{\mathbf{F}(n)\}_{n\in \mathbb{N}}$ and $\{\mathbf{G}(n)\}_{n\in \mathbb{N}}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set ${\mathcal{N}}$ of natural numbers such that their ratio $\mathbf{F}(n)/\mathbf{G}(n)$ is an integer. In this paper we study an analogue of such a divisibility problem in the complex situation. Namely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences $F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$ and $G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$, where the $f_{i}$ and $g_{j}$ are nonconstant entire functions and the $a_{i}$ and $b_{j}$ are non-zero constants except that $a_{0}$ can be zero. We will show that the set ${\mathcal{N}}$ of natural numbers such that $F(n)/G(n)$ is an entire function is finite under the assumption that $f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$ is not constant for any non-trivial index set $(i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$.

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21

Mirashe, N., A. R. Moghaddamfar, S. H. Mozafari, S. M. H. Pooya, S. Navid Salehy, and S. Nima Salehy. "CONSTRUCTING NEW MATRICES AND INVESTIGATING THEIR DETERMINANTS." Asian-European Journal of Mathematics 01, no. 04 (2008): 575–88. http://dx.doi.org/10.1142/s1793557108000461.

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Let λ = (λi)i ≥ 1, μ = (μi)i ≥ 1, ν = (νi)i ≥ 1, ω = (ωi)i ≥ 1 and ψ = (ψi)i ≥ 1 be given sequences, and let (ai,j)i,j ≥ 1 be the doubly indexed sequence given by the recurrence [Formula: see text](i ≥ 3, j ≥ 2), with various choices for the two first rows a1,j, a2,j and first column ai,1. Note that the coefficients depend on the row index only. In this article we study the principal minors of doubly indexed sequences (ai,j)i,j ≥ 1 for certain sequences and certain initial conditions. Moreover, let (bi,j)i,j ≥ 1 be the doubly indexed sequence given by the recurrence [Formula: see text] with various choices for the first row b1,j and first column bi,1. We also study the principal minors of doubly indexed sequence (bi,j)i,j ≥ 1.

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22

Buşe, Constantin, Donal O’Regan, and Olivia Saierli. "Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients." Symmetry 11, no. 4 (2019): 512. http://dx.doi.org/10.3390/sym11040512.

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Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) , and ( c j ) (with j a non-negative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is as is given below. Assuming that the “monodromy matrix” A ( q ) has at least one multiple eigenvalue, we prove that the linear scalar recurrence x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } . Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle.

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23

Alfuraidan, Monther Rashed, and Ibrahim Nabeel Joudah. "On a New Formula for Fibonacci’s Family m-step Numbers and Some Applications." Mathematics 7, no. 9 (2019): 805. http://dx.doi.org/10.3390/math7090805.

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In this work, we obtain a new formula for Fibonacci’s family m-step sequences. We use our formula to find the nth term with less time complexity than the matrix multiplication method. Then, we extend our results for all linear homogeneous recurrence m-step relations with constant coefficients by using the last few terms of its corresponding Fibonacci’s family m-step sequence. As a computational number theory application, we develop a method to estimate the square roots.

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24

Huang, Hui, and Manuel Kauers. "D-finite numbers." International Journal of Number Theory 14, no. 07 (2018): 1827–48. http://dx.doi.org/10.1142/s1793042118501099.

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D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and P-recursive sequences. It consists of the limits of convergent P-recursive sequences. Typically, this class contains many well-known mathematical constants in addition to the algebraic numbers. Our definition of the class of D-finite numbers depends on two subrings of the field of complex numbers. We investigate how different choices of these two subrings affect the class. Moreover, we show that D-finite numbers are essentially limits of D-finite functions at the point one, and evaluating D-finite functions at non-singular algebraic points typically yields D-finite numbers. This result makes it easier to recognize certain numbers to be D-finite.

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25

Lyapin, A. P., and S. S. Akhtamova. "Recurrence relations for the sections of the generating series of the solution to the multidimensional difference equation." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 3 (2021): 414–23. http://dx.doi.org/10.35634/vm210305.

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In this paper, we study the sections of the generating series for solutions to a linear multidimensional difference equation with constant coefficients and find recurrent relations for these sections. As a consequence, a multidimensional analogue of Moivre's theorem on the rationality of sections of the generating series depending on the form of the initial data of the Cauchy problem for a multidimensional difference equation is proved. For problems on the number of paths on an integer lattice, it is shown that the sections of their generating series represent the well-known sequences of polynomials (Fibonacci, Pell, etc.) with a suitable choice of steps.

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Hone, Andrew N. W., and Chloe Ward. "On the General Solution of the Heideman–Hogan Family of Recurrences." Proceedings of the Edinburgh Mathematical Society 61, no. 4 (2018): 1113–25. http://dx.doi.org/10.1017/s0013091518000196.

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AbstractWe consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan, and recently rediscovered in the theory of Laurent phenomenon algebras (a generalization of cluster algebras). All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences.

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27

Buşe, Constantin, Donal O’Regan, and Olivia Saierli. "Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues." Symmetry 11, no. 3 (2019): 339. http://dx.doi.org/10.3390/sym11030339.

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Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) and ( c j ) (with j nonnegative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is defined below. Assume that the eigenvalues x , y , z of the “monodromy matrix” A ( q ) verify the condition ( x − y ) ( y − z ) ( z − x ) ≠ 0 . We prove that the linear recurrence in C x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers–Ulam stable if and only if ( | x | − 1 ) ( | y | − 1 ) ( | z | − 1 ) ≠ 0 , i.e., the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } .

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28

WYSOCZAŃSKI, JANUSZ. "bm-INDEPENDENCE AND bm-CENTRAL LIMIT THEOREMS ASSOCIATED WITH SYMMETRIC CONES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 13, no. 03 (2010): 461–88. http://dx.doi.org/10.1142/s0219025710004115.

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We study the properties of the (noncommutative) bm-independence of algebras, indexed by partially ordered sets. The index sets are given by positive cones, in particular the symmetric cones, which include the positive-definite Hermitian matrices with complex or quaternionic entries. We formulate and prove the general versions of the bm-Central Limit Theorems for bm-independent random variables, indexed by lattices in such positive cones. The limit measures we obtain are symmetric and compactly supported on the real line. Their (even) moment sequences (gn)n≥0 satisfy the generalized recurrence for the Catalan numbers: [Formula: see text], where the coefficients γ(r) are computed by the Euler's beta-function of the first kind, related to the given positive cone. Example of a nonsymmetric cone, the Vinberg's cone, is also studied in this context.

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Buşe, Constantin, Vasile Lupulescu, and Donal O'Regan. "Hyers–Ulam stability for equations with differences and differential equations with time-dependent and periodic coefficients." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 5 (2019): 2175–88. http://dx.doi.org/10.1017/prm.2019.12.

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AbstractLetqbe a positive integer and let (an) and (bn) be two given ℂ-valued andq-periodic sequences. First we prove that the linear recurrence in ℂ0.1$$x_{n + 2} = a_nx_{n + 1} + b_nx_n,\quad n\in {\open Z}_+ $$is Hyers–Ulam stable if and only if the spectrum of the monodromy matrixTq: =Aq−1· · ·A0(i.e. the set of all its eigenvalues) does not intersect the unit circle Γ = {z∈ ℂ: |z| = 1}, i.e.Tqis hyperbolic. Here (and in as follows) we let0.2$$A_n = \left( {\matrix{ 0 & 1 \cr {b_n} & {a_n} \cr } } \right)\quad n\in {\open Z}_+ .$$Secondly we prove that the linear differential equation0.3$${x}^{\prime \prime}(t) = a(t){x}^{\prime}(t) + b(t)x(t),\quad t\in {\open R},$$(wherea(t) andb(t) are ℂ-valued continuous and 1-periodic functions defined on ℝ) is Hyers–Ulam stable if and only ifP(1) is hyperbolic; hereP(t) denotes the solution of the first-order matrix 2-dimensional differential system0.4$${X}^{\prime}(t) = A(t)X(t),\quad t\in {\open R},\quad X(0) = I_2,$$whereI2is the identity matrix of order 2 and0.5$$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$

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Mujahed, Hamzeh, and Benedek Nagy. "Exact Formula for Computing the Hyper-Wiener Index on Rows of Unit Cells of the Face-Centred Cubic Lattice." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 1 (2018): 169–87. http://dx.doi.org/10.2478/auom-2018-0011.

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Abstract Similarly to Wiener index, hyper-Wiener index of a connected graph is a widely applied topological index measuring the compactness of the structure described by the given graph. Hyper-Wiener index is the sum of the distances plus the squares of distances between all unordered pairs of vertices of a graph. These indices are used for predicting physicochemical properties of organic compounds. In this paper, the graphs of lines of unit cells of the face-centred cubic lattice are investigated. The graphs of face-centred cubic lattice contain cube points and face centres. Using mathematical induction, closed formulae are obtained to calculate the sum of distances between pairs of cube points, between face centres and between cube points and face centres. The sum of these formulae gives the hyper-Wiener index of graphs representing face-centred cubic grid with unit cells connected in a row. In connection to integer sequences, a recurrence relation is presented based on binomial coefficients.

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31

Loureiro, Ana F., and Walter Van Assche. "Three-fold symmetric Hahn-classical multiple orthogonal polynomials." Analysis and Applications 18, no. 02 (2019): 271–332. http://dx.doi.org/10.1142/s0219530519500106.

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We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as [Formula: see text]-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a [Formula: see text]-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them [Formula: see text]-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

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32

Irimiciuc, Stefan Andrei, Andrei Zala, Dan Dimitriu, et al. "Novel Approach for EEG Signal Analysis in a Multifractal Paradigm of Motions. Epileptic and Eclamptic Seizures as Scale Transitions." Symmetry 13, no. 6 (2021): 1024. http://dx.doi.org/10.3390/sym13061024.

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Two different operational procedures are proposed for evaluating and predicting the onset of epileptic and eclamptic seizures. The first procedure analyzes the electrical activity of the brain (EEG signals) using nonlinear dynamic methods (the time variations of the standard deviation, the variance, the skewness and the kurtosis; the evolution in time of the spatial–temporal entropy; the variations of the Lyapunov coefficients, etc.). The second operational procedure reconstructs any type of EEG signal through harmonic mappings from the usual space to the hyperbolic one using the time homographic invariance of a multifractal-type Schrödinger equation in the framework of the scale relativity theory (i.e., in a multifractal paradigm of motions). More precisely, the explicit differential descriptions of the brain activity in the form of 2 × 2 matrices with real elements disclose, through the in-phase coherences at various scale resolutions (i.e., as scale transitions), the multitude of brain neuronal dynamics, especially sequences of epileptic and eclamptic seizures. These two operational procedures are not mutually exclusive, but rather become complementary, offering valuable information concerning epileptic and eclamptic seizures. In such context, the prediction of epileptic and eclamptic seizures becomes fundamental for patients not responding to medical treatment and also presenting an increased rate of seizure recurrence.

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Kirsanov, Mikhail N. "Analytical calculation of deformations of a truss for a long span covering." Vestnik MGSU, no. 10 (October 2020): 1399–406. http://dx.doi.org/10.22227/1997-0935.2020.10.1399-1406.

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Introduction. The method of induction based on the number of panels is employed to derive formulas designated for deflection of a square in plan hinged rod structure, which has supports on its sides and which consists of individual pyramidal rod elements. The truss is statically determinable and symmetrical. Some features of the solution are identified on the curves constructed according to derived formulas. Materials and methods. The analysis of forces arising in the rods of the covering is performed symbolically using the method of joint isolation and operators of the Maple symbolic math engine. The deflection in the centre of the covering is found by the Maxwell–Mohr’s formula. The rigidity of truss rods is assumed to be the same. The analysis of a sequence of analytical calculations of trusses, having different numbers of panels, is employed to identify coefficients, designated for deflection and reaction at the supports, in the final design formula. The induction method is employed for this purpose. Common members of sequences of coefficients are derived from the solution of linear recurrence equations made using Maple operators. Results. Solutions, obtained for three types of loads, are polynomial in terms of the number of panels. To illustrate the solutions and their qualitative analysis, curves describing the dependence of deflection on the number of panels are made. The author identified the quadratic asymptotics of the solution with respect to the number of panels. The quadratic asymptotics is linear in height. Conclusions. Formulas are obtained for calculating deflection and reactions of covering supports having an arbitrary number of panels and dimensions if exposed to three types of loads. The presence of extremum points on solution curves is shown. The dependences, identified by the author, are designated both for evaluating the accuracy of numerical solutions and for solving problems of structural optimization in terms of rigidity.

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Kirsanov, Mikhail N. "Analytical assessment of the frequency of natural vibrations of a truss with an arbitrary number of panels." Structural Mechanics of Engineering Constructions and Buildings 16, no. 5 (2020): 351–60. http://dx.doi.org/10.22363/1815-5235-2020-16-5-351-360.

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The aim of the work is to derive a formula for the dependence of the first frequency of the natural oscillations of a planar statically determinate beam truss with parallel belts on the number of panels, sizes and masses concentrated in the nodes of the lower truss belt. Truss has a triangular lattice with vertical racks. The solution uses Maple computer math system operators. Methods. The basis for the upper estimate of the desired oscillation frequency of a regular truss is the energy method. As a form of deflection of the truss taken deflection from the action of a uniformly distributed load. Only vertical mass movements are assumed. The amplitude values of the deflection of the truss is calculated by the Maxwell - Mohrs formula. The forces in the rods are determined in symbolic form by the method of cutting nodes. The dependence of the solution on the number of panels is obtained by an inductive generalization of a series of solutions for trusses with a successively increasing number of panels. For sequences of coefficients of the desired formula, fourth-order homogeneous linear recurrence equations are compiled and solved. Results. The solution is compared with the numerical one, obtained from the analysis of the entire spectrum of natural frequencies of oscillations of the mass system located at the nodes of the truss. The frequency equation is compiled and solved using Eigenvalue search operators in the Maple system. It is shown that the obtained analytical estimate differs from the numerical solution by a fraction of a percent. Moreover, with an increase in the number of panels, the error of the energy method decreases monotonically. A simpler lower bound for the oscillation frequency according to the Dunkerley method is presented. The accuracy of the lower estimate is much lower than the upper estimate, depending on the size and number of panels.

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Malnassy, Greg, Susan Geyer, Noreen Fulton, et al. "Comparison Of Deep Sequencing and Allele-Specific Oligonucleotide PCR Methods For MRD Quantitation In Acute Lymphoblastic Leukemia and Mantle Cell Lymphoma: CALGB 10403 and CALGB 59909 (Alliance)." Blood 122, no. 21 (2013): 2547. http://dx.doi.org/10.1182/blood.v122.21.2547.2547.

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Abstract Background The assessment of minimal residual disease (MRD) is a key component of prognosis and monitoring in acute lymphoblastic leukemia (ALL) and mantle cell lymphoma (MCL). Allele-specific oligonucleotide (ASO)-PCR can be used to assess MRD; however, this technique requires preparation of clonotype-specific primers for each patient, which is laborious and time-consuming. We demonstrated the utility of sequencing-based MRD assessment in ALL (Faham et al., Blood 2012). This quantitative approach relies on amplification and sequencing of immunoglobulin and T-cell receptor gene segments using consensus primers and can address some of the limitations associated with traditional MRD detection. Here we compared the ability of the sequencing and ASO-PCR methods to identify clonal cancer gene rearrangements at diagnosis and evaluated the concordance of MRD detection in bone marrow (BM) or blood samples from 37 patients (pts) with ALL and 22 pts with MCL entered onto prospective CALGB treatment trials, 10403 and 59909 (Alliance), respectively . Methods Using the quantitative ASO-PCR and sequencing assays, we analyzed diagnostic blood and BM samples from ALL pts for clonal rearrangements of immunoglobulin (IGH-VDJ, IGH-DJ, IGK) and T cell receptor (TRB, TRD, TRG) genes. We then assessed MRD at the IGH and/or TRG locus in 84 follow-up samples from ALL pts. Similarly, we analyzed samples from 22 MCL pts for immunoglobulin (IGH-VDJ, IGH-DJ, IGK) clonal gene rearrangements and measured MRD at the IGH and/or IGK locus in 114 follow-up samples. Sensitivity of the ASO-PCR vs sequencing was 1 X 10 4-5 vs 1 X 106, respectively. Concordance between sequencing and ASO-PCR MRD assessment on serial samples collected during and post-treatment was evaluated across all pts and within disease groups using concordance correlation coefficients for repeated measures (CCC-RM). Concordance in identification of detectable vs. undetectable MRD by both methods was also evaluated using Kappa statistics. Results Using the sequencing platform, high frequency clonal rearrangements were observed in at least two receptors in 97% and 95% of pts with ALL and MCL, respectively. Selected ALL samples were known to have IGH-VDJ or TRG clonal rearrangements by ASO-PCR; however, sequencing revealed additional clonal rearrangements in 36/37 (97%) pts with ALL. Good concordance was observed with identification of MRD positive vs. negative between the methods (K=0.62; p<0.0001), where better agreement was observed within the ALL (K=0.63) than within the MCL (K=0.59) cohorts. Across both diseases, the sequencing method proved a more sensitive measure and CCC-RM was 0.82, reflecting the fact that 28 pts with undetectable MRD by ASO-PCR methods had low detectable MRD measures by the sequencing approach. The sequencing platform also provided new insights into kinetics of relapse. In one ALL case (Fig 1), we observed three high frequency IGH-VDJ clones, two of which are related based on VDJ replacement model. Dramatic reduction in frequency of one clone (clone 3, Fig 1) was observed following initial induction chemotherapy in both BM and blood. Only this single clone was monitored using ASO-PCR, and the patient appeared to be in molecular remission. However, the other two high frequency clones were chemo-resistant, and sequencing MRD monitoring revealed no reduction in clone frequency in either the BM or blood. The patient experienced early relapse at day 56, which may have resulted from the expansion of a clone carrying one or both of the clonal sequences that was not monitored by ASO-PCR. Thus, the sequencing method can be used to monitor response to treatment at the individual clone level. Conclusions This study demonstrates concordance between identification of detectable MRD in ALL and MCL by sequencing and ASO-PCR Methods. The sequencing approach offers improvements over ASO-PCR including the ability to monitor multiple clonal sequences, faster turnaround time (results in 1 week), and greater sensitivity. The clinical significance of this greater sensitivity remains to be tested prospectively and must be correlated with clinical results. Nevertheless, the sequencing method represents an alternative approach for clinical MRD monitoring which could fundamentally improve the ability to monitor disease progression and recurrence in patients with lymphoid malignancies. Disclosures: Carlton: Sequenta, Inc.: Employment, Equity Ownership. Weng:Sequenta, Inc.: Employment, Equity Ownership. Faham:Sequenta, Inc.: Employment, Equity Ownership, Membership on an entity’s Board of Directors or advisory committees.

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Kauers, Manuel. "Shift Equivalence of P-finite Sequences." Electronic Journal of Combinatorics 13, no. 1 (2006). http://dx.doi.org/10.37236/1126.

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We present an algorithm which decides the shift equivalence problem for P-finite sequences. A sequence is called P-finite if it satisfies a homogeneous linear recurrence equation with polynomial coefficients. Two sequences are called shift equivalent if shifting one of the sequences $s$ times makes it identical to the other, for some integer $s$. Our algorithm computes, for any two P-finite sequences, given via recurrence equation and initial values, all integers $s$ such that shifting the first sequence $s$ times yields the second.

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Gerhold, Stefan. "On Some Non-Holonomic Sequences." Electronic Journal of Combinatorics 11, no. 1 (2004). http://dx.doi.org/10.37236/1840.

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A sequence of complex numbers is holonomic if it satisfies a linear recurrence with polynomial coefficients. A power series is holonomic if it satisfies a linear differential equation with polynomial coefficients, which is equivalent to its coefficient sequence being holonomic. It is well known that all algebraic power series are holonomic. We show that the analogous statement for sequences is false by proving that the sequence $\{\sqrt{n}\}_n$ is not holonomic. In addition, we show that $\{n^n\}_n$, the Lambert $W$ function and $\{\log{n}\}_n$ are not holonomic, where in the case of $\{\log{n}\}_n$ we have to rely on an open conjecture from transcendental number theory.

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Chen, William Y. C., Arthur L. B. Yang, and Elaine L. F. Zhou. "Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences." Electronic Journal of Combinatorics 17, no. 1 (2010). http://dx.doi.org/10.37236/486.

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The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let $P(x)$ be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of $P(x+1)$, which leads to the log-concavity of $P(x+c)$ for any $c\geq 1$ due to Llamas and Martínez-Bernal. As a consequence, we obtain the ratio monotonicity of the Boros-Moll polynomials obtained by Chen and Xia without resorting to the recurrence relations of the coefficients.

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Bousquet-Mélou, Mireille, James Propp, and Julian West. "Perfect Matchings for the Three-Term Gale-Robinson Sequences." Electronic Journal of Combinatorics 16, no. 1 (2009). http://dx.doi.org/10.37236/214.

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In 1991, David Gale and Raphael Robinson, building on explorations carried out by Michael Somos in the 1980s, introduced a three-parameter family of rational recurrence relations, each of which (with suitable initial conditions) appeared to give rise to a sequence of integers, even though a priori the recurrence might produce non-integral rational numbers. Throughout the '90s, proofs of integrality were known only for individual special cases. In the early '00s, Sergey Fomin and Andrei Zelevinsky proved Gale and Robinson's integrality conjecture. They actually proved much more, and in particular, that certain bivariate rational functions that generalize Gale-Robinson numbers are actually polynomials with integer coefficients. However, their proof did not offer any enumerative interpretation of the Gale-Robinson numbers/polynomials. Here we provide such an interpretation in the setting of perfect matchings of graphs, which makes integrality/polynomiality obvious. Moreover, this interpretation implies that the coefficients of the Gale-Robinson polynomials are positive, as Fomin and Zelevinsky conjectured.

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Visontai, Mirkó. "Some Remarks on the Joint Distribution of Descents and Inverse Descents." Electronic Journal of Combinatorics 20, no. 1 (2013). http://dx.doi.org/10.37236/2135.

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We study the joint distribution of descents and inverse descents over the set of permutations of $n$ letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that give the recurrence for these generating functions. As a result we devise a recurrence for the coefficients in question but are unable to settle the conjecture. We examine generalizations of the conjecture and obtain a type $B$ analog of the recurrence satisfied by the two-variable generating function. We also exhibit some connections to cyclic descents and cyclic inverse descents. Finally, we propose a combinatorial model for the joint distribution of descents and inverse descents in terms of statistics on inversion sequences.

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Ostafe, Alina, and Igor E. Shparlinski. "On the Skolem problem and some related questions for parametric families of linear recurrence sequences." Canadian Journal of Mathematics, February 8, 2021, 1–20. http://dx.doi.org/10.4153/s0008414x21000080.

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Abstract We show that in a parametric family of linear recurrence sequences $a_1(\alpha ) f_1(\alpha )^n + \cdots + a_k(\alpha ) f_k(\alpha )^n$ with the coefficients $a_i$ and characteristic roots $f_i$ , $i=1, \ldots ,k$ , given by rational functions over some number field, for all but a set of elements $\alpha $ of bounded height in the algebraic closure of ${\mathbb Q}$ , the Skolem problem is solvable, and the existence of a zero in such a sequence can be effectively decided. We also discuss several related questions.

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Schneider, Jonathan. "Polynomial Sequences of Binomial-Type Arising in Graph Theory." Electronic Journal of Combinatorics 21, no. 1 (2014). http://dx.doi.org/10.37236/3702.

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In this paper, we show that the solution to a large class of "tiling'' problems is given by a polynomial sequence of binomial type. More specifically, we show that the number of ways to place a fixed set of polyominos on an $n\times n$ toroidal chessboard such that no two polyominos overlap is eventually a polynomial in $n$, and that certain sets of these polynomials satisfy binomial-type recurrences. We exhibit generalizations of this theorem to higher dimensions and other lattices. Finally, we apply the techniques developed in this paper to resolve an open question about the structure of coefficients of chromatic polynomials of certain grid graphs (namely that they also satisfy a binomial-type recurrence).

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Liu, Lily Li, and Ya-Nan Li. "Recurrence Relations for the Linear Transformation Preserving the Strong $q$-Log-Convexity." Electronic Journal of Combinatorics 23, no. 3 (2016). http://dx.doi.org/10.37236/5913.

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Let $[T(n,k)]_{n,k\geqslant0}$ be a triangle of positive numbers satisfying the three-term recurrence relation\[T(n,k)=(a_1n+a_2k+a_3)T(n-1,k)+(b_1n+b_2k+b_3)T(n-1,k-1).\]In this paper, we give a new sufficient condition for linear transformations\[Z_n(q)=\sum_{k=0}^{n}T(n,k)X_k(q)\]that preserves the strong $q$-log-convexity of polynomials sequences. As applications, we show linear transformations, given by matrices of the binomial coefficients, the Stirling numbers of the first kind and second kind, the Whitney numbers of the first kind and second kind, preserving the strong $q$-log-convexity in a unified manner.

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Hetyei, Gábor. "Orthogonal Polynomials Represented by $CW$-Spheres." Electronic Journal of Combinatorics 11, no. 2 (2004). http://dx.doi.org/10.37236/1861.

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Given a sequence $\{Q_n(x)\}_{n=0}^{\infty}$ of symmetric orthogonal polynomials, defined by a recurrence formula $Q_n(x)=\nu_n\cdot x\cdot Q_{n-1}(x)-(\nu_n-1)\cdot Q_{n-2}(x)$ with integer $\nu_i$'s satisfying $\nu_i\geq 2$, we construct a sequence of nested Eulerian posets whose $ce$-index is a non-commutative generalization of these polynomials. Using spherical shellings and direct calculations of the $cd$-coefficients of the associated Eulerian posets we obtain two new proofs for a bound on the true interval of orthogonality of $\{Q_n(x)\}_{n=0}^{\infty}$. Either argument can replace the use of the theory of chain sequences. Our $cd$-index calculations allow us to represent the orthogonal polynomials as an explicit positive combination of terms of the form $x^{n-2r}(x^2-1)^r$. Both proofs may be extended to the case when the $\nu_i$'s are not integers and the second proof is still valid when only $\nu_i>1$ is required. The construction provides a new "limited testing ground" for Stanley's non-negativity conjecture for Gorenstein$^*$ posets, and suggests the existence of strong links between the theory of orthogonal polynomials and flag-enumeration in Eulerian posets.

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Kirsanov, Mikhail N. "ANALYSIS OF THE DEFLECTION OF A TRUSS WITH A DECORATIVE LATTICE." Stroitel stvo nauka i obrazovanie [Construction Science and Education], no. 1 (March 31, 2019). http://dx.doi.org/10.22227/2305-5502.2019.1.1.

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Introduction. A scheme is proposed for a planar symmetric statically determinate beam truss with a rectilinear lower belt, struts, multidirectional braces and a polygonal outline of the upper belt. The belts of the truss are rectilinear, the hinges are ideal. The truss belongs to the class of regular trusses having periodic cells. The supporting rods are not deformable. The truss is evenly loaded around the nodes of the lower belt. Materials and methods. The task is to deduce the dependence of the deflection of the truss on the number of panels in the span. The deflection is obtained from the Maxwell-Mora formula under the assumption that all the rods have the same rigidity. Forces in the structural rods from the effective uniform load and from the unit vertical in the middle of the span are determined by the method of cutting the nodes. The matrix of the system of linear equations of node equilibrium is made up of the cosines of the forces with the coordinate axes. To compile a system of equations and solve it, the program of symbolic mathematics Maple is used. To obtain the general formula, a number of problems of trusses with a number of panels from 2 to 29 are solved. Sequences of the coefficients of the deflection formula have common terms for which homogeneous recurrence equations are also compiled using the methods of the Maple system using specialized operators. Results. The solutions of recurrence equations have the form of polynomials with coefficients that depend on the parity of the number of panels and contain trigonometric functions. The graphs of the solutions obtained are constructed and analysed. Sharp changes of deflection characteristic for such truss and their non-monotonic character are noted. It is shown that for a fixed, independent on the number of panels, length of the span and the total load, the relative deflection with increasing number of panels first decreases, then varies little. Conclusions. The asymptotic property of the solution is obtained by the methods of the Maple system: an inclined asymptote is found. The slope is calculated using the analytical capabilities of Maple. A simple formula is derived for the horizontal displacement of the mobile support from the action of the load. The dependence is monotonic. The height of the truss is included in the denominator of the formula.

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Hellou, Mustapha, and Franck Lominé. "Stokes Flow Within Networks of Flow Branches." Journal of Fluids Engineering 140, no. 12 (2018). http://dx.doi.org/10.1115/1.4040832.

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Stokes flow in the branches of structured looped networks with successive identical square loops and T-junction branches is studied. Analytical expressions of the flow rate in the branches are determined for network of one, two, three, or four loops with junction head loss neglected relative to regular one. Then, a general expression of the flow rate is deduced for networks with more loops. This expression contains particularly a sequence of coefficients obeying to a recurrence formula. This sequence is a part of the fusion of Fibonacci and Tribonacci sequences. Furthermore, a general formula that expresses the quotient of flow rate in successive junction flow branches is given. The limit of this quotient for an infinite number of junction branches is found to be equal to 2+3. When the inlet and outlet flow rates are equal, this quotient obeys to a sequence of invariant numbers whatever the ratio of flow rate in the outlet branches is. Thus, the flow rate distribution for any configuration of inlet and outlet flow rates can be calculated. This study is also performed using Hardy–Cross method and a commercial solver of Navier-Stokes equation. The analytical results are approached very well with Hardy–Cross method. The numerical resolution agrees also with analytical results. However, the difference with the numerical results becomes significant for low flow rate in the junction branches. The flow streamlines are then determined for some inlet and outlet flow rate configurations. They particularly illustrate that recirculation flow takes place in branches of low flow rate.

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