Recent zbMATH articles in MSC 11Bhttps://www.zbmath.org/atom/cc/11B2021-02-27T13:50:00+00:00WerkzeugHalf Riordan array sequences.https://www.zbmath.org/1453.050082021-02-27T13:50:00+00:00"He, Tian-Xiao"https://www.zbmath.org/authors/?q=ai:he.tian-xiao|he.tianxiaoSummary: We construct sequences of vertical and horizontal halves of various Riordan arrays by using the vertical and horizontal halve Riordan array transform operators. The sequence characterizations of halves of Riordan arrays are presented and applied to the halve Riordan array transformation. The conditions of transforming a Riordan array to a pseudo-involution Riordan array by using the halve Riordan array transform are given. The condition of preserving the elements of a certain subgroup of the Riordan group under the halve Riordan array transformation are shown. Other properties of the halves of Riordan arrays and their entries such as related recurrence relations, double variable generating functions, combinatorial interpretations are also studied.Higher-order Fourier analysis and applications.https://www.zbmath.org/1453.680022021-02-27T13:50:00+00:00"Hatami, Hamed"https://www.zbmath.org/authors/?q=ai:hatami.hamed"Hatami, Pooya"https://www.zbmath.org/authors/?q=ai:hatami.pooya"Lovett, Shachar"https://www.zbmath.org/authors/?q=ai:lovett.shacharSummary: Fourier analysis has been extremely useful in many areas of mathematics. In the last several decades, it has been used extensively in theoretical computer science. Higher-order Fourier analysis is an extension of the classical Fourier analysis, where one allows to generalize the ``linear phases'' to higher degree polynomials. It has emerged from the seminal proof of Gowers of Szemerédi's theorem with improved quantitative bounds, and has been developed since, chiefly by the number theory community. In parallel, it has found applications also in theoretical computer science, mostly in algebraic property testing, coding theory and complexity theory.
The purpose of this book is to lay the foundations of higher-order Fourier analysis, aimed towards applications in theoretical computer science with a focus on algebraic property testing.An index-preserving bijection between marked tableaux and \(P_{n,2}\)-tableaux.https://www.zbmath.org/1453.051362021-02-27T13:50:00+00:00"Fulton, Samuel"https://www.zbmath.org/authors/?q=ai:fulton.samuel"O'Connor-Seville, Stephen"https://www.zbmath.org/authors/?q=ai:oconnor-seville.stephen"Welz, Matthew"https://www.zbmath.org/authors/?q=ai:welz.matthewSummary: \textit{J. R. Stembridge} [Discrete Math. 99, No. 1--3, 307--320 (1992; Zbl 0761.05097)] introduced marked tableaux and showed that the number of admissible marked tableaux of shape \(\lambda\vdash n\) is equal to the multiplicity of the irreducible Specht module \(S^\lambda\) in a certain representation of \(S_n\). Through their seemingly unrelated work with chromatic quasisymmetric functions, \textit{J. Shareshian} and \textit{M. L. Wachs} [in: Configuration spaces. Geometry, combinatorics and topology. Pisa: Edizioni della Normale. 433--460 (2012; Zbl 1328.05194)] established that this multiplicity of \(S^\lambda\) is also equal to the number of \(P_{n,2}\)-tableaux of shape \(\lambda\). Shareshian and Wachs went on to observe indirectly that the number of marked tableaux of shape \(\lambda\) and index \(j\) equals the number of \(P_{n,2}\)-tableaux of shape \(\lambda\) and index \(j\), while suggesting it might be interesting to find a bijective proof of this fact. In this paper, we present such a bijection. In particular, we develop an index-preserving bijection from the set of all marked tableaux of shape \(\lambda\) to the set of all \(P_{n,2}\)-tableaux of shape \(\lambda\).On products of shifts in arbitrary fields.https://www.zbmath.org/1453.110212021-02-27T13:50:00+00:00"Warren, Audie"https://www.zbmath.org/authors/?q=ai:warren.audieSummary: We adapt the approach of \textit{M. Rudnev}, \textit{G. Shakan} and \textit{I. D. Shkredov} [Proc. Am. Math. Soc. 148, No. 4, 1467--1479 (2020; Zbl 1442.11029)] to prove that in an arbitrary field \(\mathbb{F}\), for all \(A \subset \mathbb{F}\) finite with \(|A| < p^{1/4}\) if \(p:= \operatorname{char}(\mathbb{F})\) is positive, we have \[ |A(A+1)| \gg \frac{|A|^{11/9}}{(\log|A|)^{7/6}}, \quad |AA| + |(A+1)(A+1)| \gg \frac{|A|^{11/9}}{(\log|A|)^{7/6}}. \] This improves upon the exponent of \(\frac65\) given by an incidence theorem of \textit{S. Stevens} and \textit{F. de Zeeuw} [Bull. Lond. Math. Soc. 49, No. 5, 842--858 (2017; Zbl 1388.51002)].Cubic sums of \(q\)-binomial coefficients and the Fibonomial coefficients.https://www.zbmath.org/1453.110262021-02-27T13:50:00+00:00"Chu, Wenchang"https://www.zbmath.org/authors/?q=ai:chu.wenchang"Kılıç, Emrah"https://www.zbmath.org/authors/?q=ai:kilic.emrahSummary: Triple product sums on the generalized Fibonomial and Lucanomial coefficients are evaluated in closed forms by means of Bailey's summation formulae for two terminating well-poised \(_3\phi_2\)-series.A truncated identity of Euler and related \(q\)-congruences.https://www.zbmath.org/1453.050132021-02-27T13:50:00+00:00"Liu, Ji-Cai"https://www.zbmath.org/authors/?q=ai:liu.jicai"Huang, Zhong-Yu"https://www.zbmath.org/authors/?q=ai:huang.zhong-yuSummary: We discuss a truncated identity of Euler and present a combinatorial proof of it. We also derive two finite identities as corollaries. As an application, we establish two related \(q\)-congruences for sums of \(q\)-Catalan numbers, one of which has been proved by \textit{R. Tauraso} [Adv. Appl. Math. 48, No. 5, 603--614 (2012; Zbl 1270.11016)] by a different method.Associated binomial inversion for unified Stirling numbers and counting subspaces generated by subsets of a root system.https://www.zbmath.org/1453.110372021-02-27T13:50:00+00:00"Kamiyoshi, Tomohiro"https://www.zbmath.org/authors/?q=ai:kamiyoshi.tomohiro"Nagura, Makoto"https://www.zbmath.org/authors/?q=ai:nagura.makoto"Otani, Shin-Ichi"https://www.zbmath.org/authors/?q=ai:otani.shinichiSummary: We introduce an associated version of the binomial inversion for unified Stirling numbers defined by \textit{L. C. Hsu} and \textit{P. J. S. Shiue} [Adv. Appl. Math. 20, No. 3, 366--384 (1998; Zbl 0913.05006)]. This naturally appears when we count the number of subspaces generated by subsets of a root system. We count such subspaces of any dimension by using associated unified Stirling numbers, and then we will also give a combinatorial interpretation of our inversion formula. In particular, the well-known explicit formula for classical Stirling numbers of the second kind can be understood as a special case of our formula.On the number of rainbow solutions of linear equations in \(\mathbb{Z}/p\mathbb{Z}\).https://www.zbmath.org/1453.051312021-02-27T13:50:00+00:00"Huicochea, Mario"https://www.zbmath.org/authors/?q=ai:huicochea.marioSummary: Let \(n_1\), \(n_2\), \(n_3\), \(m\in\mathbb{Z}\) and \(p\) be a prime, and write \(b:=m+p\mathbb{Z}\) and \(a_i:=n_i+p\mathbb{Z}\) for each \(i\in\{1,2,3\}\). Given a partition of \(\mathbb{Z}/p\mathbb{Z}\) into nonempty subsets \(\mathbb{Z}/p\mathbb{Z}=A_1\cup A_2\cup A_3\), we say that \((s_1,s_2,s_3)\) is a rainbow solution of \(a_1x_1+a_2x_2+a_3x_3-b=0\) if it is a solution of this equation and \(A_i\cap\{s_1,s_2,s_3\}=\emptyset\) for each \(i\in\{1,2,3\}\); we denote by \(R\) the family of rainbow solutions of \(a_1x_1+a_2x_2+a_3x_3-b=0\). The first result of this paper is that if \(a_1a_2a_3=0+p\mathbb{Z}\) and the coefficients \(a_1\), \(a_2\), \(a_3\) are not equal, then \(|R|=\Omega(\min\{|A_1|,|A_2|,|A_3|\})\) where the constants are absolute. The second result of this paper is that if \(|A_1|\), \(|A_2|\), \(|A_3|\) are almost equal, \(n_1n_2n_3\neq 0=m\) and \(p\gg 0\), then \(|R|=\Omega(\min\{|A_1|,|A_2|,|A_3|\}^2)\) where the constants depend only on \(n_1\), \(n_2\), \(n_3\).On popular sums and differences for sets with small multiplicative doubling.https://www.zbmath.org/1453.110202021-02-27T13:50:00+00:00"Ol'mezov, K. I."https://www.zbmath.org/authors/?q=ai:olmezov.k-i"Semchenkov, A. S."https://www.zbmath.org/authors/?q=ai:semchenkov.a-s"Shkredov, I. D."https://www.zbmath.org/authors/?q=ai:shkredov.ilya-dThe authors prove quantitative estimates for the additive energy of finite subsets \(A\) of the field of real numbers with small multiplicative doubling \(|AA|\leq M|A|\). For instance (Theorem 5) if \(A\subset\mathbb{R}\) is finite with (*) \(|AA|\leq M|A|\) then \(E^+(A):=|\{(a,b,c,d)\in A^4 : a+b=c+d\}|\lesssim M^{7/3}|A|^{22/9}\). Using combinatorial and operator theory techniques they also reprove results on the sum sets and difference sets of \(A\) subject the condition (*) on which proofs the last author participated [J. Théor. Nombres Bordx. 31, No. 3, 573--602 (2019; Zbl 07246528); Sb. Math. 208, No. 12, 1854--1868 (2017; Zbl 1430.11019)]. The paper concludes with some comments and results on the estimates for fractional-rational expressions for sets \(A\subset\mathbb{F}\) in terms of its sums and products.
Reviewer: Štefan Porubský (Praha)Polynomial extensions of two Gibonacci delights and their graph-theoretic confirmations.https://www.zbmath.org/1453.110272021-02-27T13:50:00+00:00"Koshy, Thomas"https://www.zbmath.org/authors/?q=ai:koshy.thomasSummary: We extend two charming Gibonacci identities to their corresponding polynomial versions, investigate their geometric and Jacobsthal implications, and confirm the two charming Gibonacci and Jacobsthal polynomial identities using graph-theoretic tools.Shrinkage points of golden rectangle, Fibonacci spirals, and golden spirals.https://www.zbmath.org/1453.510092021-02-27T13:50:00+00:00"Duan, Jun-Sheng"https://www.zbmath.org/authors/?q=ai:duan.junshengSummary: We investigated the golden rectangle and the related Fibonacci spiral and golden spiral. The coordinates of the shrinkage points of a golden rectangle were derived. Properties of shrinkage points were discussed. Based on these properties, we conduct a comparison study for the Fibonacci spiral and golden spiral. Their similarities and differences were looked into by examining their polar coordinate equations, polar radii, arm-radius angles, and curvatures. The golden spiral has constant arm-radius angle and continuous curvature, while the Fibonacci spiral has cyclic varying arm-radius angle and discontinuous curvature.Permutations of \(\mathbb{N}\) generated by left-right filling algorithms.https://www.zbmath.org/1453.110392021-02-27T13:50:00+00:00"Dekking, F. M."https://www.zbmath.org/authors/?q=ai:dekking.michelKimberling introduced a permutation of \(\mathbb{N}\) obtained by a left-right filling procedure defined by \(\Pi(1)=1\) and for \(n\ge 2\), \(\Pi(n-L(n)):=n\) if \(\Pi(n-L(n))\) is not yet defined, and, otherwise \(\Pi(n+R(n)):=n\); where \(L,R: \mathbb{N}\to\mathbb{N}\) are two functions. For \(L(n)=R(n)=\lfloor n/2\rfloor\), the author provides a one-to-one connection of the corresponding permutation with a 3-automatic sequence which permits to provide many properties for this permutation. He also gives a framework to analyze other left-right filling algorithms establishing many relations between the corresponding permutations.
Reviewer: Michel Rigo (Liège)Solutions of polynomial equations in subgroups of \(\mathbb{F}_p^*\).https://www.zbmath.org/1453.111612021-02-27T13:50:00+00:00"Makarychev, Sergei"https://www.zbmath.org/authors/?q=ai:makarychev.sergei"Vyugin, Ilya"https://www.zbmath.org/authors/?q=ai:vyugin.ilya-vSummary: We present an upper bound on the number of solutions of an algebraic equation \(P(x,y)=0\) where \(x\) and \(y\) belong to the union of cosets of some subgroup of the multiplicative group \(\kappa^*\) of some field of positive characteristic. This bound generalizes the bound of \textit{P. Corvaja} and \textit{U. Zannier} [J. Eur. Math. Soc. (JEMS) 15, No. 5, 1927--1942 (2013; Zbl 1325.11060)] to the case of union of cosets. We also obtain the upper bounds on the generalization of additive energy.Statistically characterized subgroups of the circle.https://www.zbmath.org/1453.220032021-02-27T13:50:00+00:00"Dikranjan, Dikran"https://www.zbmath.org/authors/?q=ai:dikranjan.dikran-n"Das, Pratulananda"https://www.zbmath.org/authors/?q=ai:das.pratulananda"Bose, Kumardipta"https://www.zbmath.org/authors/?q=ai:bose.kumardiptaLet \(\mathbb T\) denote the circle group in additive notation. If \((a_n)\) is a sequence of integers, then the subgroup \(t_{(a_n)}(\mathbb T) = \{x\in \mathbb T: a_nx \to 0 \text{ in }\mathbb T\}\) of \(\mathbb T\) is called a characterized subgroup of \(\mathbb T\). The authors replace in this definition usual convergence by statistical convergence and obtain the notion of statistically characterized subgroup \(t^s_{(a_n)}(\mathbb T)\) of \(\mathbb T\). They mainly consider subgroups characterized by arithmetic sequences \((a_n)\), that is strictly increasing sequences \((a_n)\) in which \(a_0 =1\) and \(a_n|a_{n+1}\) for each \(n\in \mathbb N\). Several interesting results are obtained. The main results are:
(1) for any sequence of integers \((a_n)\), \(t^s_{(a_n)}(\mathbb T)\) is a Borel subgroup of \(\mathbb T\) and \(t_{(a_n)}(\mathbb T) \subset t^s_{(a_n)}(\mathbb T)\),
(2) for any arithmetic sequence \((a_n)\), \(t^s_{(a_n)}(\mathbb T) \neq t_{(a_n)}(\mathbb T)\) and the cardinality of \(t^s_{(a_n)}(\mathbb T)\) is continuum. Sufficient conditions for \(x\notin t^s_{(a_n)}(\mathbb T)\) are found. Eight open problems are posed.
Reviewer: Ljubiša D. Kočinac (Niš)Repdigits as sums of four Fibonacci or Lucas numbers.https://www.zbmath.org/1453.110062021-02-27T13:50:00+00:00"Normenyo, Benedict Vasco"https://www.zbmath.org/authors/?q=ai:normenyo.benedict-vasco"Luca, Florian"https://www.zbmath.org/authors/?q=ai:luca.florian"Togb, Alain"https://www.zbmath.org/authors/?q=ai:togb.alainThe Fibonacci sequence \((F_n)_{n\ge 0}\), is defined by the linear recurrence \(F_0=0\), \(F_1=1\), and \(F_{n+2}=F_{n+1}+ F_n\) for all \(n\ge 0\). The Lucas sequence \((L_n)_{n\ge 0}\), is defined by the same recurrence but with different initial terms, \(L_0=2\) and \(L_1=1\).
In the paper under review, the authors find all numbers with repeated digits (also known as \textit{repdigits}) that can be written as sums of four Fibonacci or Lucas numbers. That is, they prove the following results.
Theorem 1. All nonnegative integer solutions \((m_1,m_2,m_3,m_4, n)\) of the Diophantine equation
\[
N=F_{m_1}+F_{m_2}+F_{m_3}+F_{m_4}=d\left(\frac{10^n -1}{9}\right) \quad \text{with} \quad d\in \{0, 1, \ldots, 9\}
\]
have
\begin{align*}
N\in &\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 99, 111, \\& \quad 222, 333, 555, 666, 777, 999, 1111, 2222, 11111, 66666\}.
\end{align*}
Theorem 2. All nonnegative integer solutions \((m_1,m_2,m_3,m_4, n)\) of the Diophantine equation
\[
N=L_{m_1}+L_{m_2}+L_{m_3}+L_{m_4}=d\left(\frac{10^n -1}{9}\right) \quad \text{with} \quad d\in \{0, 1, \ldots, 9\}
\]
have
\begin{align*}
N\in &\{4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, \\& \quad 222, 333, 555, 666, 999, 2222, 4444, 11111, 88888\}.
\end{align*}
The proof of Theorems 1 and 2 follow from a clever combination of techniques in Diophantine number theory, the elementary properties of Fibonacci and Lucas sequences, the theory of linear forms in nonzero logarithms of algebraic numbers á la Baker, and the Baker-Davenport reduction procedure. Computations are done in \textit{Maple}.
Reviewer: Mahadi Ddamulira (Kampala)Peters type polynomials and numbers and their generating functions: approach with \(p\)-adic integral method.https://www.zbmath.org/1453.110332021-02-27T13:50:00+00:00"Simsek, Yilmaz"https://www.zbmath.org/authors/?q=ai:simsek.yilmazSummary: The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and \(p\)-adic integrals method. (Peters polynomials \(s_n(x;\lambda,\mu)\) are defined by
\[
\frac{(1+t)^x}{(1+(1+t)^\lambda)^\mu}= \sum_{n=0}^\infty s_n(x;\lambda,\mu)\frac{t^n}{n!}.
\]
Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well-known special numbers and polynomials are presented. By using \(p\)-adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol-type Peters numbers and polynomials). By using these functions with their partial derivative equations and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well-known formulas. Finally, two open problems for interpolation functions for Apostol-type Peters numbers and polynomials are revealed.\(p\)-adic \(L\)-functions and classical congruences.https://www.zbmath.org/1453.110012021-02-27T13:50:00+00:00"Lin, Xianzu"https://www.zbmath.org/authors/?q=ai:lin.xianzuThis paper is structured as follows: after an introduction, Section 2 gives preliminaries that will be used throughout the article. In Section 3, the author reviews Barsky and Washington's \(p\)-adic expansion of power sums and its applications. In Section 4, the paper gives a similar \(p\)-adic expansion for sums of Lehmer's type and derives many corollaries. Sections 5 and 6 are devoted to extensions of Gauss's and Jacobi's congruences, and of Wilson's theorem, respectively.
Reviewer: Mouad Moutaoukil (Fès)New congruences and finite difference equations for generalized factorial functions.https://www.zbmath.org/1453.110122021-02-27T13:50:00+00:00"Schmidt, Maxie D."https://www.zbmath.org/authors/?q=ai:schmidt.maxie-dSummary: The generalized factorial product sequences we study in this article are defined symbolically by \(p_n(\alpha, R) = R(R + \alpha) \cdots (R + (n-1)\alpha)\) for any indeterminate \(R\). This definition of the generalized factorial functions \(p_n(\alpha, R)\) includes variants of the classical single and double factorial functions, multiple \(\alpha\)-factorial functions, \(n!_{(\alpha)}\), rising and falling factorial polynomials such as the Pochhammer symbol, and other well-known sequences as special cases. We use the rationality of the generalized \(h^{th}\) convergent functions, \(\text{Conv}_h(\alpha, R; z)\), to the infinite J-fraction expansions enumerating these generalized factorial product sequences first proved by the author in [``Mathematica summary notebook, supplementary reference, and computational documentation'', Preprint] to construct new congruences and h-order finite difference equations for generalized factorial functions. Applications of the new results we prove in this article include new finite sums and congruences for the \(\alpha\)-factorial functions, restatements of classical necessary and sufficient conditions of the primality of special integer subsequences and tuples, and new finite sums for the single and double factorial functions modulo integers \(h \ge 2\)Raabe's identity and covering systems.https://www.zbmath.org/1453.110172021-02-27T13:50:00+00:00"Movshovich, Yevgenya"https://www.zbmath.org/authors/?q=ai:movshovich.yevgenyaA system of arithmetic sequences (1) \(A_i=a_in+b_i\), \(n\in\mathbb{Z}\), \(a_i,b_i\) integers with \(0\leq b_i < a_i\), \(i=1,\dots,q\), \(q>1\), is said to be an exact (or disjoint) cover if every integer occurs in exactly one of the sequences. \textit{A. S. Fraenkel} [Discrete Math. 4, 359--366 (1973; Zbl 0257.10003)] found an equivalent characterization of (1) to be an exact cover in terms of values of Bernoulli polynomials. This generalization presents an extension of classical Raabe's multiplication identity. Fraenkel's result was later generalized by several authors, including the reviewer. In [Discrete Math. 38, 73--77 (1982; Zbl 0474.05001)] motivated by the role of Raabe's identity in the theory of \(p\)-adic distributions he indicated a possibility to interpret the above characterizations of unions of arithmetic progressions using a measure theoretic approach, when sequence \(A_i=a_in+b_i\), \(n\in\mathbb{Z}\) possesses the measure \(n_i^{m-1}B_m(a_i/n_i)\), for any fixed \(m\), \(B_M(X)\) denotes the \(m\)th Bernoulli polynomial. Without appealing to such a motivation the author of the present paper shows ``that the generalized Raabe identity is a sum of Raabe's identities for \(A_i\), and this sum is the direct source'' of the Fraenkel's type formulas. The author also (re)proves some applications and consequences related to covering systems.
Reviewer: Štefan Porubský (Praha)The \(k\)-dimensional champagne pyramid.https://www.zbmath.org/1453.050122021-02-27T13:50:00+00:00"Baumann, Michael Heinrich"https://www.zbmath.org/authors/?q=ai:baumann.michael-heinrichSummary: Summenformeln, allen voran die Gauß'sche Summenformel \(\sum_{i=1}^{n}i=\frac{n}{2}(n+1)\), gehören zum Grundwissen eines jeden Mathematikers. Wir wollen in dieser Arbeit Verallgemeinerungen der Gauß'schen Summenformel der Form \(\sum_{i=1}^{n}\sum_{j=1}^{i}j\) betrachten, wie sie die Anzahl der Gläser in einer Champagnerpyramide der Höhe \(n\) beschreibt. Im Hauptteil der Arbeit werden wir die allgemeine Formel \(\sum_{n_{k-1}=1}^{n_{k}}\sum_{n_{k-2}=1}^{n_{k-1}}\dots\sum_{n_{2}=1}^{n_{3}}\sum_{n_{1}=1}^{n_{2}}n_{1}=\binom{n+k-1}{k}\) für die Anzahl der Gläser in einer \(k\)-dimensionalen Pyramide der Höhe \(n=n_{k}\) herleiten und beweisen.On some direct and inverse results concerning sums of dilates.https://www.zbmath.org/1453.110152021-02-27T13:50:00+00:00"Bhanja, Jagannath"https://www.zbmath.org/authors/?q=ai:bhanja.jagannath"Chaudhary, Shubham"https://www.zbmath.org/authors/?q=ai:chaudhary.shubham"Pandey, Ram Krishna"https://www.zbmath.org/authors/?q=ai:pandey.ram-krishna|pandey.ram-krishna.1Summary: Let $A$ and $B$ be two nonempty finite sets of integers and let $r$ be a positive integer. Define $A+r\cdot B:=\{a+rb: a\in A,\, b\in B \}$. In case $A=B$, \textit{G. A. Freiman} et al. [Eur. J. Comb. 40, 42--54 (2014; Zbl 1315.11089)] proved that $|A+r\cdot A|\geq 4|A|-4$ for $r\geq 3$. For $r=2$, they obtained an extended inverse result which states that if $|A|\geq 3$ and $|A+2\cdot A| < 4|A|-4$, then $A$ is a subset of an arithmetic progression of length at most $2|A|-3$. We present a new, self-contained proof of the direct result, $|A+r\cdot A|\geq 4|A|-4$ for $r\geq 3$. We also generalize the above extended inverse result to sums $A+2\cdot B$ for two sets $A$ and $B$.Three notes on Ser's and Hasse's representations for the zeta-functions.https://www.zbmath.org/1453.111062021-02-27T13:50:00+00:00"Blagouchine, Iaroslav V."https://www.zbmath.org/authors/?q=ai:blagouchine.iaroslav-v\textit{J. Ser} [C. R. Acad. Sci., Paris 182, 1075--1077 (1926; JFM 52.0338.02)] and
\textit{H. Hasse} [Math. Z. 32, 458--464 (1930; JFM 56.0894.03)] obtained series representations for the zeta-function and the Hurwitz zeta-function. The purpose of this article is to find several convergent series for similar Dirichlet series. In the process, connections with several other special numbers are obtained. The indirect connection to finite differences in formulas (26)--(28), (43), (44) is obtained via Gregory's coefficients and Cauchy numbers of the second kind. Relations to the digamma function and Euler's constant are established.
Finally, formulas with harmonic numbers
\[
H_n=\sum_{k=1}^n\frac 1k
\]
and Stirling numbers of the first kind are established. The supposed generating function (153) for these Stirling numbers
\[
\sum_{n=k}^{\infty}\frac {s(n,k)}{n!}z^n=\frac{(\ln(1+z))^k}{k!},\ k\in\mathbb{N}_0,
\]
which is also stated without proof in [\textit{I. V. Blagouchine}, J. Math. Anal. Appl. 442, No. 2, 404--434 (2016; Zbl 1339.33003)], is neither proved, nor easily found in the literature.
The proofs use Gregory-Newton interpolation series, integral representation of the Hurwitz zeta-function, integral formulas for the Gamma function. The paper has great mathematical and historical value and many of the formulas could certainly be \(q\)-deformed. The unusual name globally convergent series is used in the paper.
Reviewer: Thomas Ernst (Uppsala)On symmetric identities of Carlitz's type \(q\)-Daehee polynomials.https://www.zbmath.org/1453.110362021-02-27T13:50:00+00:00"Kim, Won Joo"https://www.zbmath.org/authors/?q=ai:kim.wonjoo"Jang, Lee-Chae"https://www.zbmath.org/authors/?q=ai:jang.lee-chae"Kim, Byung Moon"https://www.zbmath.org/authors/?q=ai:kim.byungmoonSummary: In this paper, we study Carlitz's type \(q\)-Daehee polynomials and investigate the symmetric identities for them by using the \(p\)-adic \(q\)-integral on \(\mathbb{Z}_p\) under the symmetry group of degree \(n\).A short proof of the canonical polynomial van der Waerden theorem.https://www.zbmath.org/1453.051302021-02-27T13:50:00+00:00"Fox, Jacob"https://www.zbmath.org/authors/?q=ai:fox.jacob"Wigderson, Yuval"https://www.zbmath.org/authors/?q=ai:wigderson.yuval"Zhao, Yufei"https://www.zbmath.org/authors/?q=ai:zhao.yufeiSummary: We present a short new proof of the canonical polynomial van der Waerden theorem, recently established by \textit{A. Girão} [``A canonical polynomial van der Waerden's theorem'', Preprint, \url{arXiv:2004.07766}].On the ratios and geometric boundaries of complex Horadam sequences.https://www.zbmath.org/1453.110982021-02-27T13:50:00+00:00"Bagdasar, Ovidiu"https://www.zbmath.org/authors/?q=ai:bagdasar.ovidiu-dumitru"Hedderwick, Eve"https://www.zbmath.org/authors/?q=ai:hedderwick.eve"Popa, Ioan-Lucian"https://www.zbmath.org/authors/?q=ai:popa.ioan-lucianSummary: Horadam sequences are second-order linear recurrences in the complex plane which depend on two initial conditions and two recurrence coefficients which are complex numbers. Recently, numerous papers have been devoted to the periodicity of these sequences, as well as to generalizations and applications. In this paper we investigate aspects related to the sequence of rations of consecutive terms and geometric bounds of Horadam sequences. We also propose some directions for further study.
For the entire collection see [Zbl 1392.00002].Interpolation solutions of linear difference equations.https://www.zbmath.org/1453.390012021-02-27T13:50:00+00:00"Graça, Mário M."https://www.zbmath.org/authors/?q=ai:graca.mario-meirelesSummary: Given a linear and homogeneous difference equation, with constant coefficients, we begin by constructing a continuous function which is interpolatory of the difference equation solution. This function leads to a \(\mathbb{R}^2\) valued parametric function which we call, with some language abuse, `phase portrait' of the difference equation. The `phase portrait' proves to be an interesting tool in order to understand the dynamics of the solutions of a difference equation, similarly to the so called phase portrait in the context of systems of ordinary differential equations. As an illustration we present some examples where the referred interpolatory function is considered as well as phase portraits of certain second order difference equations connected to some Fibonacci type sequences.The \(r\)-Fubini-Lah numbers and polynomials.https://www.zbmath.org/1453.050112021-02-27T13:50:00+00:00"Rácz, Gabriella"https://www.zbmath.org/authors/?q=ai:racz.gabriellaSummary: In this paper we introduce and study the \(r\)-Fubini-Lah numbers and polynomials, in connection with the enumeration of those partitions of a finite set, where both the blocks and the partition itself are ordered, and \(r\) distinguished elements belong to distinct ordered blocks.The 2-adic complexity of a class of binary sequences with optimal autocorrelation magnitude.https://www.zbmath.org/1453.940532021-02-27T13:50:00+00:00"Sun, Yuhua"https://www.zbmath.org/authors/?q=ai:sun.yuhua"Yan, Tongjiang"https://www.zbmath.org/authors/?q=ai:yan.tongjiang"Chen, Zhixiong"https://www.zbmath.org/authors/?q=ai:chen.zhixiong"Wang, Lianhai"https://www.zbmath.org/authors/?q=ai:wang.lianhaiSummary: Recently, a class of binary sequences with optimal autocorrelation magnitude has been presented by \textit{W. Su} et al. [Des. Codes Cryptography 86, No. 6, 1329--1338 (2018; Zbl 1387.94060)], based on Ding-Helleseth-Lam sequences and interleaving technique. The linear complexity of this class of sequences has been proved to be large enough to resist the B-M Algorithm by \textit{C. Fan} [Des. Codes Cryptography 86, No. 10, 2441--2450 (2018; Zbl 1408.94923)]. In this paper, we study the 2-adic complexities of these sequences with period \(4p\) and show they are no less than \(2p\), i.e., its 2-adic complexity is large enough to resist the Rational Approximation Algorithm.A Roth-type theorem with mixed powers.https://www.zbmath.org/1453.111262021-02-27T13:50:00+00:00"Ching, Tak Wing"https://www.zbmath.org/authors/?q=ai:ching.tak-wingAuthor's abstract: Let \(c_1,c_2,c_3\) be nonzero integers such that \(c_1+c_2+c_3=0\). We consider the mixed power equation \(c_1(p_1^2+p_1'^3)+c_2(p_2^2+p_2'^3)+c_3(p_3^2+p_3'^3)=0\) where \(p_1,p_2,p_3\) belong to a certain set \(\mathcal{A}\) of primes and \(p_1',p_2',p_3'\) belong to another set \(\mathcal{A}'\) of primes. We prove a Roth-type result that whenever the densities of \(\mathcal{A}\) and \(\mathcal{A}'\) satisfy a certain lower bound, then the above equation has nontrivial solutions. The same method can be generalized to deduce analogous results for other equations involving mixed powers of higher degrees.
Reviewer: Giovanni Coppola (Napoli)Some further Hecke-type identities.https://www.zbmath.org/1453.110542021-02-27T13:50:00+00:00"Zhang, Zhizheng"https://www.zbmath.org/authors/?q=ai:zhang.zhizheng"Song, Hanfei"https://www.zbmath.org/authors/?q=ai:song.hanfeiTrees grown under young-age preferential attachment.https://www.zbmath.org/1453.050222021-02-27T13:50:00+00:00"Lyon, Merritt R."https://www.zbmath.org/authors/?q=ai:lyon.merritt-r"Mahmoud, Hosam M."https://www.zbmath.org/authors/?q=ai:mahmoud.hosam-mSummary: We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen-Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by `perturbed' Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.When the small divisors of a natural number are in arithmetic progression.https://www.zbmath.org/1453.110162021-02-27T13:50:00+00:00"Iannucci, Douglas E."https://www.zbmath.org/authors/?q=ai:iannucci.douglas-eSummary: We consider the positive divisors of a natural number \(n\) that do not exceed the square root of \(n\). We refer to these as the small divisors of \(n\). We find all natural numbers whose small divisors are in arithmetic progression.Arithmetic progressions in the trace of Brownian motion in space.https://www.zbmath.org/1453.601382021-02-27T13:50:00+00:00"Benjamini, Itai"https://www.zbmath.org/authors/?q=ai:benjamini.itai"Kozma, Gady"https://www.zbmath.org/authors/?q=ai:kozma.gadySummary: It is shown that the trace of three dimensional Brownian motion contains arithmetic progressions of length 5 and no arithmetic progressions of length 6 a.s.
For the entire collection see [Zbl 1446.00029].On the structure of distance sets over prime fields.https://www.zbmath.org/1453.111642021-02-27T13:50:00+00:00"Pham, Thang"https://www.zbmath.org/authors/?q=ai:pham.thang-van|pham-van-thang."Suk, Andrew"https://www.zbmath.org/authors/?q=ai:suk.andrewIn this paper the authors review the known Erdős-Falconer distance problem over finite fields [\textit{P. Erdős}, Am. Math. Mon. 53, 248--250 (1946; Zbl 0060.34805); \textit{A. Iosevich} and \textit{D. Koh}, SIAM J. Discrete Math. 23, No. 1, 123--135 (2008; Zbl 1190.52014); \textit{D. Koh} and \textit{C.-Y. Shen}, J. Number Theory 132, No. 11, 2455--2473 (2012; Zbl 1252.52013)]. Namely, Let $E$ be a set in $\mathbb F_{d}^{q}$, and let $\Delta(E)$ be the set of distinct distances determined by the pairs of points in $E$. How large does $E$ need to be to guarantee that $|\Delta(E)|\gg q$? The finite field variant of the Erdỏs distinct distances problem was first studied by \textit{J. Bourgain} et al. [Geom. Funct. Anal. 14, No. 1, 27--57 (2004; Zbl 1145.11306)]:
Theorem (Bourgain-Katz-Tao). Suppose $q\equiv 3\bmod 4$ is a prime. Let $E$ be a set in $F_{q}^2$. If $|E|=q^{\alpha}$ with $0<\alpha<2$, then we have $|\Delta(E)|\gg |E|^{(1/2)+\varepsilon}$, for some positive $\varepsilon=\varepsilon(\alpha)>0$. In a recent work, \textit{A. Iosevich} et al. [Mosc. J. Comb. Number Theory 8, No. 2, 103--115 (2019; Zbl 07063267)] proved that the exponent $d/2$ holds for the quotient set of the distance set, which is defined by
\[
((\Delta(E))/(\Delta(E))):=\{(a/b): a, b\in\Delta(E), b\neq 0\}
\]
Theorem (Iosevich-Koh-Parshall). Let $F_{q}$ be a finite field of order $q$, and let $E$ be a set in $F_{q}^{d}$.
(a) If $d\geq 2$ is even and $|E|\geq 9q^{d/2}$, then we have $((\Delta(E))/(\Delta(E)))=F_{q}$.
(b) If $d\geq 3$ is odd and $|E|\geq 6q^{d/2}$, then we have $\{0\}\cup F_{q}^+\subset((\Delta(E))/(\Delta(E)))$, where $F_{q}^+=\{x^2:x\in F_{q}, x\neq 0\}$. In this paper, the authors prove the following two theorems in two results for the case of quotient set, in even and old dimensions:
Theorem A. Let $F_{p}$ be a prime field, and let $A\subset F_{p}$. Then for $E=A^{d}\subset F_{p}^{d}$ with $d=2k$, $k\in N$, $k\geq 2$, we have $|((\Delta(E))/(\Delta(E)))|=|\{(a/b): a, b\in \Delta(E), b\neq 0\}|\geq (p/3)$, whenever $|E|\gg p^{(d/2)-\varepsilon}$ with $\varepsilon=(d/2).((2^{k}-2^{k-1}-1)/(2^{k}-1))$
Theorem B. Let $F_{p}$ be a prime field, and let $A\subset F_{p}$. Then for $E=A^{d}\subset F_{p}^{d}$ with $d=2k+1$, $k\in N$, $k\geq 2$, we have $|((\Delta(E))/(\Delta(E)))|=|\{(a/b): a, b\in\Delta(E), b\neq 0\}|\geq (p/3)$, whenever $|E|\gg p^{(d/2)-\varepsilon}$ with $\varepsilon=(d/2).((2^{k+2}-2^{k+1}-3)/(2^{k+3}-6))$.
Reviewer: Noureddine Daili (Sétif)Discretized sum-product estimates in matrix algebras.https://www.zbmath.org/1453.110192021-02-27T13:50:00+00:00"He, Weikun"https://www.zbmath.org/authors/?q=ai:he.weikunSummary: We generalize Bourgain's discretized sum-product theorem to matrix algebras
[\textit{J. Bourgain}, J. Anal. Math. 112, 193--236 (2010; Zbl 1234.11012)].Statistical distribution of the Stern sequence.https://www.zbmath.org/1453.110312021-02-27T13:50:00+00:00"Bettin, Sandro"https://www.zbmath.org/authors/?q=ai:bettin.sandro"Drappeau, Sary"https://www.zbmath.org/authors/?q=ai:drappeau.sary"Spiegelhofer, Lukas"https://www.zbmath.org/authors/?q=ai:spiegelhofer.lukasIn this long paper the authors study some statistical properties of the Stern sequence. The Stern sequence (or Stern's diatomic sequence) \((s_{n})_{n\in\mathbb{N}}\), is the recurrence sequence defined in the following way
\[
s_{0}=0, \;s_{1}=1,\;\text{and, for}\;n\geq 1, s_{2n}=s_{n},\;s_{2n+1}=s_{n}+s_{n+1}.
\]
One among many striking properties of the Stern sequence is the following: the sequence \((s_{n}/s_{n+1})_{n\in\mathbb{N}}\) enumerates the set of non-negative rational numbers.
The main result of the paper is the central limit theorem for the random variable \(\log S_{N}\), where \(S_{N}=s_{n}\) and \(n\) is taken uniformly from the set \(J_{N}=\mathbb{Z}\cap [2^{N},2^{N+1})\). More precisely, for some constants \(\alpha, \sigma>0\), as \(N\) tends to infinity, the values \(\log s_{n}\) are asymptotically distributed according to a Gaussian law with mean \(\alpha N\) and variance \(\sigma^{2}N\): for \(t\in\mathbb{R}\) satisfying \(t=O(N^{1/6})\) the following equality holds
\[
\mathbb{P}_{N}\left[\frac{\log S_{N}-\alpha N}{\sigma\sqrt{N}}\right]=\int_{-\infty}^{t}\frac{e^{-v^{2}/2}}{\sqrt{2\pi}}dv+O\left(\frac{(1+t^2)e^{-t^{2}/2}}{\sqrt{N}}\right).
\]
This result answer the question from the paper of \textit{J. Lansing} [J. Integer Seq. 17, No. 7, Article 14.7.5, 18 p. (2014; Zbl 1317.11024)].
Reviewer: Maciej Ulas (Kraków)The modular distribution of Stern's sequence.https://www.zbmath.org/1453.110242021-02-27T13:50:00+00:00"Deshouillers, Jean-Marc"https://www.zbmath.org/authors/?q=ai:deshouillers.jean-marc"Schinzel, Andrzej"https://www.zbmath.org/authors/?q=ai:schinzel.andrzejLet \((s_{n})\) be the Stern sequence, i.e., the recurrence sequences defined as:
\[
s_{0}=0, \;s_{1}=1,\quad\text{and, for}\quad n\geq 1, s_{2n}=s_{n},\;s_{2n+1}=s_{n}+s_{n+1}.
\]
The Stern sequence has an amusing property: the sequence \((s_{n}/s_{n+1})\) enumerates the set of non-negative rational numbers. Many other arithmetic properties of the Stern sequence is scattered through the literature.
In the paper under review the authors study the question concerning behaviour of the set, say \(S_{a,b,m}\), of those \(n\in\mathbb{N}\) such that \((s_{n},s_{n+1})\equiv (a,b)\pmod{m}\) for fixed non-negative integers \(a, b\) and an integer \(m\geq 2\). The main result of the paper states that the set \(S_{a,b,m}\) has a density which is non-zero in case of \(\gcd(a,b,m)=1\), and zero otherwise. The proof relies on properties of certain automaton related to the sequence \((s_{n}, s_{n+1})\pmod{m}\).
For the entire collection see [Zbl 1417.11001].
Reviewer: Maciej Ulas (Kraków)On third-order linear recurrent functions.https://www.zbmath.org/1453.110282021-02-27T13:50:00+00:00"Magnani, Kodjo Essonana"https://www.zbmath.org/authors/?q=ai:magnani.kodjo-essonanaSummary: A function \(\psi : \mathbb{R} \to \mathbb{R}\) is said to be a Tribonacci function with period \(p\) if \(\psi(x + 3 p) = \psi(x + 2 p) + \psi(x + p) + \psi(x)\), for all \(x \in \mathbb{R}\). In this paper, we present some properties on the Tribonacci functions with period \(p\). We show that if \(\psi\) is a Tribonacci function with period \(p\), then \(\lim_{x \to \infty} \left(\psi(x + p) / \psi(x)\right) = \beta \), where \(\beta\) is the root of the equation \(x^3 - x^2 - x - 1 = 0\) such that \(1 < \beta < 2\).On the Laurent property of the Somos-8 sequences.https://www.zbmath.org/1453.110232021-02-27T13:50:00+00:00"Bogoutdinova, Yu. G."https://www.zbmath.org/authors/?q=ai:bogoutdinova.yu-gSummary: \textit{S. Fomin} and \textit{A. Zelevinsky} [Adv. Appl. Math. 28, No. 2, 119--144 (2002; Zbl 1012.05012)] proved the Laurent property of the Somos-4, 5, 6, 7 sequences. In the paper the smallest number such that the corresponding element of the Somos-8 sequence is not a Laurent polynomial is calculated.Browkin's discriminator conjecture.https://www.zbmath.org/1453.110302021-02-27T13:50:00+00:00"Ciolan, Alexandru"https://www.zbmath.org/authors/?q=ai:ciolan.alexandru"Moree, Pieter"https://www.zbmath.org/authors/?q=ai:moree.pieterSummary: Let \(q\ge 5\) be a prime and put \(q^*=(-1)^{(q-1)/2}\cdot q\). We consider the integer sequence \(u_q(1),u_q(2),\ldots \) with \(u_q(j)=(3^j-q^*(-1)^j)/4\). No term in this sequence is repeated and thus for each \(n\) there is a smallest integer \(m\) such that \(u_q(1),\ldots, u_q(n)\) are pairwise incongruent modulo \(m\). We write \(D_q(n)=m\). The idea of considering the discriminator \(D_q(n)\) is due to Browkin who, in case 3 is a primitive root modulo \(q\), conjectured that the only values assumed by \(D_q(n)\) are powers of 2 and of \(q\). We show that this is true for \(n \ne 5\), but false for infinitely many \(q\) in case \(n=5\). We also determine \(D_q(n)\) in case 3 is not a primitive root modulo \(q\). Browkin's inspiration for his conjecture came from earlier work of Moree and \textit{A. Zumalacárregui} [J. Number Theory 160, 646--665 (2016; Zbl 1396.11140)], who determined \(D_5(n)\) for $n\ge 1$, thus proving a conjecture of Sălăjan. For a fixed prime \(q\) their approach is easily generalized, but requires some innovations in order to deal with all primes \(q\ge 7\) and all \(n\ge 1\). Interestingly enough, Fermat and Mirimanoff primes play a special role in this.On an arithmetic triangle of numbers arising from inverses of analytic functions.https://www.zbmath.org/1453.110382021-02-27T13:50:00+00:00"Bagdasaryan, Armen G."https://www.zbmath.org/authors/?q=ai:bagdasaryan.armen-g"Bagdasar, Ovidiu"https://www.zbmath.org/authors/?q=ai:bagdasar.ovidiu-dumitruSummary: The Lagrange inversion formula is a fundamental tool in combinatorics. In this work, we investigate an inversion formula for analytic functions, which does not require taking limits. By applying this formula to certain functions we have found an interesting arithmetic triangle for which we give a recurrence formula. We then explore the links between these numbers, Pascal's triangle, and Bernoulli's numbers, for which we obtain a new explicit formula. Furthermore, we present power series and asymptotic expansions of some elementary and special functions, and some links to the Online Encyclopedia of Integer Sequences (OEIS).
For the entire collection see [Zbl 1409.68021].On some new arithmetic functions involving prime divisors and perfect powers.https://www.zbmath.org/1453.110022021-02-27T13:50:00+00:00"Bagdasar, Ovidiu"https://www.zbmath.org/authors/?q=ai:bagdasar.ovidiu-dumitru"Tatt, Ralph"https://www.zbmath.org/authors/?q=ai:tatt.ralphSummary: Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora's theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 443--551 (1995; Zbl 0823.11029)]) or Catalan (solved in 2002 by \textit{P. Mihăilescu} [J. Reine Angew. Math. 572, 167--195 (2004; Zbl 1067.11017)]). The purpose of this paper is two-fold. First, we present some new integer sequences \(a(n)\), counting the positive integers smaller than \(n\), having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers \(i^j\) obtained for \(1 \leq i\), \(j \leq n\). Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [The OnLine Encyclopedia of Integer Sequences, \url{http://oeis.org, OEIS Foundation Inc. 2011}]. Finally, we discuss some other novel integer sequences.
For the entire collection see [Zbl 1409.68021].On a paper of Erdős and Szekeres.https://www.zbmath.org/1453.110402021-02-27T13:50:00+00:00"Bourgain, Jean"https://www.zbmath.org/authors/?q=ai:bourgain.jean"Chang, Mei-Chu"https://www.zbmath.org/authors/?q=ai:chang.mei-chuLet \(M(a_1,\ldots,a_n)= \displaystyle\max_{|z|=1} \prod_{k=1}^n (1-z^{a_k})\) and \(f(n) = \displaystyle\min_{a_1,\ldots,a_n} M(a_1,\ldots,a_n)\), where \(a_1 \leq a_2 \leq \cdots \leq a_n\) are positive integers. \textit{P. Erdős} and \textit{G. Szekeres} [Acad. Serbe Sci., Publ. Inst. Math. 13, 29--34 (1959; Zbl 0097.03302)] (the paper is quoted incorrectly in the present paper) proved \(f(n)\geq \sqrt {2n}\), \(\lim f(n)^{1/n} = 1\) and expected \(f(n)<\exp(n^{1-c})\) for some positive \(c<1\). Their results were subject to several improvements. In the present paper the study is restricted to the condition \(a_1 < \dots < a_n\). The authors prove, among other results, that
\[ M(a_1,\ldots,a_n)<\exp(c\sqrt{n\log n}\log\log n)\quad\text{if }n\asymp N/2, \tag{1} \]
\[ M(a_1,\ldots,a_n)>\exp(\tau n)\quad\text{if }n>(1-\tau)N \tag{2} \]
for a suitable positive constant \(\tau\) and if \(\{a_1 < \dots < a_n\}\subset \{1,\dots,N\}\) in both cases,
\[ \log M(a_1,\ldots,a_n)\gg m^{1/2-\varepsilon}/\sqrt{\log n}\tag{3} \]
provided the numbers \(\{a_1 < \cdots < a_n\}\) do not admit non-trivial \(0\), \(1\), \(-1\) relations.
Reviewer: Štefan Porubský (Praha)Colored multipermutations and a combinatorial generalization of Worpitzky's identity.https://www.zbmath.org/1453.050022021-02-27T13:50:00+00:00"Engbers, John"https://www.zbmath.org/authors/?q=ai:engbers.john"Pantone, Jay"https://www.zbmath.org/authors/?q=ai:pantone.jay"Stocker, Christopher"https://www.zbmath.org/authors/?q=ai:stocker.christopher-jSummary: Worpitzky's identity, first presented in [``Studien über die Bernoullischen und Eulerischen Zahlen'', J. Reine Angew. Math. 94, 203--232 (1883)], expresses \(n^p\) in terms of the Eulerian numbers and binomial coefficients: \(n^p=\sum^{p-1}_{i=0}\left\langle \begin{matrix} p \\ i \end{matrix}\right\rangle\binom{n+i}{p}\). \textit{C. Pita-Ruiz} [Integers 18, Paper A17, 42 p. (2018; Zbl 1435.11046)] recently defined numbers \(A_{a,b,r}(p,i)\) implicitly to satisfy a generalized Worpitzky identity \(\binom{an+b}{r}^p=\sum^{rp}_{i=0}A_{a,b,r}(p,i)\binom{n+rp-i}{rp}\), and asked whether there is a combinatorial interpretation of the numbers \(A_{a,b,r}(p,i)\).
We provide such a combinatorial interpretation by defining a notion of
descents in colored multipermutations, and then proving that \(A_{a,b,r}(p,i)\) is equal to the number of colored multipermutations of \(\{1^r,2^r,\dots,p^r\}\) with \(a\) colors and \(i\) weak descents. We use this to give combinatorial proofs of several identities involving \(A_{a,b,r}(p,i)\), including the aforementioned generalized Worpitzky identity.Some properties and applications of non-trivial divisor functions.https://www.zbmath.org/1453.110052021-02-27T13:50:00+00:00"Hill, S. L."https://www.zbmath.org/authors/?q=ai:hill.sally-l"Huxley, M. N."https://www.zbmath.org/authors/?q=ai:huxley.martin-n"Lettington, M. C."https://www.zbmath.org/authors/?q=ai:lettington.matthew-c"Schmidt, K. M."https://www.zbmath.org/authors/?q=ai:schmidt.karl-michaelSummary: The \(j\)th divisor function \(d_j\), which counts the ordered factorisations of a positive integer into \(j\) positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative \(j\)th non-trivial divisor function \(c_j\), which counts the ordered factorisations of a positive integer into \(j\) factors each of which is greater than or equal to 2, is rather less well studied. Additionally, we consider the associated divisor function \(c_j^{(r)}\), for \(r\ge 0\), whose definition is motivated by the sum-over divisors recurrence for \(d_j\). We give an overview of properties of \(d_j, c_j\) and \(c_j^{(r)}\), specifically regarding their Dirichlet series and generating functions as well as representations in terms of binomial coefficient sums and hypergeometric series. Noting general inequalities between the three types of divisor function, we then observe how their ratios can be expressed as binomial coefficient sums and hypergeometric series, and find explicit Dirichlet series and Euler products for some of these. As an illustrative application of the non-trivial and associated divisor functions, we show how they can be used to count principal reversible square matrices of the type considered by Ollerenshaw and Brée and so sum-and-distance systems of integers.Lucas polynomials and the fixed points of the Gauss map.https://www.zbmath.org/1453.110252021-02-27T13:50:00+00:00"Al-Fadhel, Tariq A."https://www.zbmath.org/authors/?q=ai:al-fadhel.tariq-aSummary: This paper shows that if \([0;\overline{n}]\) is a fixed point of the Gauss map, then every odd power of it is also a fixed point, and can be calculated from the \(B_n\)'s of \([0;\overline{n}]\).
Moreover, this paper shows the relation between odd Lucas polynomials and the odd powers of the fixed point of the Gauss map to conclude that the set of odd Lucas polynomials with composition forms a commutative monoid.Multi-poly-Bernoulli numbers and related zeta functions.https://www.zbmath.org/1453.110352021-02-27T13:50:00+00:00"Kaneko, Masanobu"https://www.zbmath.org/authors/?q=ai:kaneko.masanobu"Tsumura, Hirofumi"https://www.zbmath.org/authors/?q=ai:tsumura.hirofumiSummary: We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at nonpositive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the \(\xi\)-function defined by Arakawa and Kaneko [\textit{T. Arakawa} et al., Bernoulli numbers and zeta functions. With an appendix by Don Zagier. Tokyo: Springer (2014; Zbl 1312.11015)]. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.Generalized rascal triangles.https://www.zbmath.org/1453.110322021-02-27T13:50:00+00:00"Hotchkiss, Philip K."https://www.zbmath.org/authors/?q=ai:hotchkiss.philip-kSummary: The ``rascal triangle'' was introduced by three middle school students in 2010 [\textit{A. Anggoro}, \textit{E. Liu}, and \textit{A. Tulloch}, College Math. J. 41, 393--395 (2010; \url{doi:10.4169/074683410X521991})]. In this paper we describe number triangles that are generalizations of the rascal triangle, and show that these generalized rascal triangles are characterized by arithmetic sequences on all diagonals, as well as rascal-like multiplication and addition rules.A two-by-two matrix representation of a generalized Fibonacci sequence.https://www.zbmath.org/1453.110292021-02-27T13:50:00+00:00"Wani, Arfat Ahmad"https://www.zbmath.org/authors/?q=ai:wani.arfat-ahmad"Rathore, G. P. S."https://www.zbmath.org/authors/?q=ai:rathore.g-p-s"Badshah, V. H."https://www.zbmath.org/authors/?q=ai:badshah.v-h"Sisodiya, Kiran"https://www.zbmath.org/authors/?q=ai:sisodiya.kiran-singhSummary: The Fibonacci sequence is a well-known example of second order recurrence sequence, which belongs to a particular class of recursive sequences. In this article, other generalized Fibonacci sequence is introduced and defined by \(H_{k,n+1}=2H_{k,n}+kH_{k,n-1}\), \(n\geq 1\), \(H_{k,0}=2\), \(H_{k,1}=1\) and \(k\) is the positive real number. Also \(n\)-th power of the generating matrix for this generalized Fibonacci sequence is established and some basic properties of this sequence are obtained by matrix methods.Some notes on alternating power sums of arithmetic progressions.https://www.zbmath.org/1453.110342021-02-27T13:50:00+00:00"Bazsó, András"https://www.zbmath.org/authors/?q=ai:bazso.andras"Mező, István"https://www.zbmath.org/authors/?q=ai:mezo.istvanCenturies ago, many mathematicians have studied the sums of powers of positive integers or the alternating sums of powers of positive integers. Many new techniques and methods have been discovered to find these sums from years ago to the present. The well-known novel formulas for these sums of powers of positive integers is given by the following formula involving the Bernoulli
\[
\sum\limits_{j=1}^{n-1}j^{m}=\frac{B_{m+1}(n)-B_{m+1}}{m+1},
\]
and the alternating sum of powers of positive integers is given by the following formula involving the Euler polynomials and the Euler numbers:
\[
\sum\limits_{j=1}^{n-1}(-1)^{j}j^{m}=\frac{(-1)^{n+1}E_{m}(n)+E_{m}}{2},
\]
where $m$ and $n$ are positive integers and with $n>1$, $B_{m+1}(n)$, $B_{m+1}$, and $E_{m+1}(n)$, $E_{m+1}$ denote the Bernoulli polynomials and numbers, and the Euler polynomials and numbers. The sum of powers of positive integers formula is also known as the Faulhaber formulas which was firstly given by Johann Faulhaber (5 May 1580--10 September 1635) who was a German mathematician. In this paper the authors have studied on the alternating power sum:
\[
r^n-(m+r)^n+(2m+r)^n-\cdots +-1)^{\ell-1}((\ell-1)m + r) ^n,
\]
which is expressed in terms of the Stirling numbers of the first kind and the \(r\)-Whitney numbers of the second kind. The authors have also given a necessary and sufficient condition for the integrality of the coefficients of the polynomial extensions of the above alternating power sum.
Reviewer: Yilmaz Simsek (Antalya)A new sum-product estimate in prime fields.https://www.zbmath.org/1453.110182021-02-27T13:50:00+00:00"Chen, Changhao"https://www.zbmath.org/authors/?q=ai:chen.changhao"Kerr, Bryce"https://www.zbmath.org/authors/?q=ai:kerr.bryce"Mohammadi, Ali"https://www.zbmath.org/authors/?q=ai:mohammadi.aliSummary: We obtain a new sum-product estimate in prime fields for sets of large cardinality. In particular, we show that if \(A\subseteq \mathbb{F}_p\) satisfies \(|A|\leq p^{64/117}\) then
\[ \max \{|A\pm A|,|AA|\} \gtrsim |A|^{39/32}.\]
Our argument builds on and improves some recent results of \textit{G. Shakan} and \textit{I. D. Shkredov} ['Breaking the 6/5 threshold for sums and products modulo a prime', Preprint, 2018, \url{arXiv:1806.07091}] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy \(E^+(P)\) of some subset \(P\subseteq A+A\). Our main novelty comes from reducing the estimation of \(E^+(P)\) to a point-plane incidence bound of \textit{M. Rudnev} [Combinatorica 38, No. 1, 219--254 (2017; Zbl 1413.51001)] rather than a point-line incidence bound used by Shakan and Shkredov.On boundaries of attractors in dynamical systems.https://www.zbmath.org/1453.280132021-02-27T13:50:00+00:00"Niralda P. C., Nitha"https://www.zbmath.org/authors/?q=ai:niralda-p-c.nitha"Mathew, Sunil"https://www.zbmath.org/authors/?q=ai:mathew-karakkad.sunil"Secelean, Nicolae Adrian"https://www.zbmath.org/authors/?q=ai:secelean.nicolae-adrianSummary: \textit{Fractal geometry} is one of the beautiful and challenging branches of mathematics. \textit{Self similarity} is an important property, exhibited by most of the fractals. Several forms of self similarity have been discussed in the literature. \textit{Iterated Function System (IFS)} is a mathematical scheme to generate fractals. There are several variants of IFSs such as condensation IFS, countable IFS, etc. In this paper, certain properties of self similar sets, using the concept of boundary are discussed. The notion of boundaries like \textit{similarity boundary} and \textit{dynamical boundary} are extended to condensation IFSs. The relationships and measure theoretic properties of boundaries in dynamical systems are analyzed. Self similar sets are characterized using the Hausdorff measure of their boundaries towards the end.On subsequences of even elements of Somos-6 sequences.https://www.zbmath.org/1453.110222021-02-27T13:50:00+00:00"Avdeeva, M. O."https://www.zbmath.org/authors/?q=ai:avdeeva.m-o"Bykovskii, V. A."https://www.zbmath.org/authors/?q=ai:bykovskii.v-aSummary: It is known that the subsequences of Somos-4 and Somos-5 sequences are Somos-4 sequences. In the paper, it is proved that this property is broken for Somos-6 sequences.