Recent zbMATH articles in MSC 11Bhttps://zbmath.org/atom/cc/11B2022-09-13T20:28:31.338867ZUnknown authorWerkzeugNumber theory and combinatorics. A collection in honor of the mathematics of Ronald Grahamhttps://zbmath.org/1491.050022022-09-13T20:28:31.338867ZPublisher's description: This volume is dedicated to the work and memory of Professor Ronald L. Graham known as the architect of discrete mathematics and combinatorics and will consist of up to 20 contributions from top mathematicians reflecting on his work in combinatorics and number theory.
\begin{itemize}
\item Original contributions by leading experts in combinatorics and number theory
\item Includes essays and memories of Professor Graham from those that knew him
\item Of interest to researchers and graduate students working in combinatorics and number theory.
\end{itemize}
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Adeniran, Ayomikun; Snider, Lauren; Yan, Catherine}, Multivariate difference Gončarov polynomials, 1-20 [Zbl 07571262]
\textit{Butler, Steve (ed.); Hurlbert, Glenn (ed.)}, Foreword, V-VI [Zbl 07571285]
\textit{Allouche, J.-P.}, On an inequality in a 1970 paper of R. L. Graham, 21-26 [Zbl 07571263]
\textit{Alon, Noga; Alweiss, Ryan; Liu, Yang P.; Martinsson, Anders; Narayanan, Shyam}, Arithmetic progressions in sumsets of sparse sets, 27-33 [Zbl 07571264]
\textit{Bennett, Michael A.; Martin, Greg; O'bryant, Kevin}, Multidimensional Padé approximation of binomial functions: equalities, 35-64 [Zbl 07571265]
\textit{Blomberg, Lars; Shannon, S. R.; Sloane, N. J. A.}, Graphical enumeration and stained glass windows. I: Rectangular grids, 65-97 [Zbl 07571266]
\textit{Brown, Tom C.; Mohsenipour, Shahram}, Two extensions of Hilbert's cube lemma, 99-107 [Zbl 07571267]
\textit{Budden, Mark}, The Gallai-Ramsey number for a tree versus complete graphs, 109-113 [Zbl 07571268]
\textit{Buhler, Joe; Freiling, Chris; Graham, Ron; Kariv, Jonathan; Roche, James R.; Tiefenbruck, Mark; van Alten, Clint; Yeroshkin, Dmytro}, On Levine's notorious hat puzzle, 115-165 [Zbl 07571269]
\textit{Cooper, Joshua; Fickes, Grant}, Recurrence ranks and moment sequences, 167-186 [Zbl 07571270]
\textit{Dudek, Andrzej; Grytczuk, Jarosław; Ruciński, Andrzej}, On weak twins and up-and-down subpermutations, 187-202 [Zbl 07571271]
\textit{Farhangi, Sohail; Grytczuk, Jarosław}, Distance graphs and arithmetic progressions, 203-208 [Zbl 07571272]
\textit{Filaseta, Michael; Juillerat, Jacob}, Consecutive primes which are widely digitally delicate, 209-247 [Zbl 07571273]
\textit{Griggs, Jerrold R.}, Spanning trees and domination in hypercubes, 249-258 [Zbl 07571274]
\textit{Harborth, Heiko; Nienborg, Hauke}, Rook domination on hexagonal hexagon boards, 259-265 [Zbl 07571275]
\textit{Hindman, Neil; Strauss, Dona}, Strongly image partition regular matrices, 267-284 [Zbl 07571276]
\textit{Hopkins, Brian}, Introducing shift-constrained Rado numbers, 285-296 [Zbl 07571277]
\textit{Lichtman, Jared Duker}, Mertens' prime product formula, dissected, 297-310 [Zbl 07571278]
\textit{Nathanson, Melvyn B.}, Curious convergent series of integers with missing digits, 311-327 [Zbl 07571279]
\textit{Pomerance, Carl}, A note on Carmichael numbers in residue classes, 321-327 [Zbl 07571280]
\textit{Shkredov, I. D.; Solymosi, J.}, Tilted corners in integer grids, 329-338 [Zbl 07571281]
\textit{Alon, Noga (ed.); Brown, Tom C (ed.); Butler, Steve (ed.); Griggs, Jerrold R. (ed.); Hindman, Neil (ed.); Jungic, Veselin (ed.); Landman, Bruce M. (ed.); Nešetřil, Jaroslav (ed.)}, Remembrances [of Ron Graham], 339-360 [Zbl 07571282]
\textit{Butler, Steve}, A selected bibliography of Ron Graham, 355-360 [Zbl 07571283]On the maximum of the weighted binomial sum \(2^{-r}\sum_{i=0}^r\binom{m}{i}\)https://zbmath.org/1491.050102022-09-13T20:28:31.338867Z"Glasby, S. P."https://zbmath.org/authors/?q=ai:glasby.stephen-peter"Paseman, G. R."https://zbmath.org/authors/?q=ai:paseman.g-rFor a non-negative integer \(m \geq 0\), study of the function \(f_m(r) = \frac{1}{2^r} \sum_{i=0}^r \binom{m}{i}\) is of importance in coding and information theory. The authors mention that a back-of-the-envelope calculations by B. McKay in [\textit{S. P. Glasby}, ``On the maximum of the weighted binomial sum \(2^{-r}\sum_{i=0}^r \binom{m}{i}\)'', MathOverflow, Question 389857, \url{https://mathoverflow.net/questions/389857/maximum-of-the-weighted-binomial-sum-2-r-sum-i-0r-binommi}] indicate that the function has maximum value when \(r\) is close to \(m/3\). In this paper, the authors prove such a precise result and also give bounds for the maximal value. They deduce that the maximum value is asymptotic to \(\frac{3}{\sqrt{\pi m}} \bigg(\frac{3}{2} \bigg)^m\) as \(m \rightarrow \infty\). The methods are elementary.
Reviewer: Balasubramanian Sury (Bangalore)A new class of generating functions of binary products of Gaussian numbers and polynomialshttps://zbmath.org/1491.050142022-09-13T20:28:31.338867Z"Boughaba, Souhila"https://zbmath.org/authors/?q=ai:boughaba.souhila"Boussayoud, Ali"https://zbmath.org/authors/?q=ai:boussayoud.ali"Kerada, Mohamed"https://zbmath.org/authors/?q=ai:kerada.mohamedSummary: In this paper, we introduce an operator in order to derive some new symmetric properties of Gaussian Fibonacci numbers and Gaussian Lucas numbers. By making use of the operator defined in this paper, we give some new generating functions for Gaussian Fibonacci numbers, Gaussian Lucas numbers, Gaussian Pell numbers, Gaussian Pell Lucas numbers, Gaussian Jacobsthal numbers, Gaussian Jacobsthal polynomials, Gaussian Jacobsthal Lucas polynomials, and Gaussian
Pell polynomials.On the combined Jacobsthal-Padovan generalized quaternionshttps://zbmath.org/1491.050172022-09-13T20:28:31.338867Z"Gürses, Nurten"https://zbmath.org/authors/?q=ai:gurses.nurten"İşbilir, Zehra"https://zbmath.org/authors/?q=ai:isbilir.zehraSummary: In this article, we examine the combined Jacobsthal-Padovan (CJP) generalized quaternions with four special cases: Jacobsthal-Padovan, Jacobsthal-Perrin, adjusted Jacobsthal-Padovan and modified Jacobsthal-Padovan generalized quaternions. Then, recurrence relation, generating function, Binet-like formula and exponential generating function of these quaternions are examined. In addition to this, some new properties, special determinant equations, matrix formulas and summation formulas are discussed.Combinatorial identities involving reciprocals of the binomial and central binomial coefficients and harmonic numbershttps://zbmath.org/1491.050282022-09-13T20:28:31.338867Z"Batır, Necdet"https://zbmath.org/authors/?q=ai:batir.necdet"Chen, Kwang-Wu"https://zbmath.org/authors/?q=ai:chen.kwang-wuThe authors prove a general combinatorial formula involving the reciprocals of the binomial coefficients and the partial sums of an arbitrary sequence. The proof uses induction and a recurrence formula. Applications of this formula include several combinatorial identities involving reciprocals of the binomial and central binomial coefficients and harmonic numbers. The oldest examples to which the main result is applicable were already published in the textbook of \textit{R. L. Graham} et al. [Concrete mathematics: a foundation for computer science. 2nd ed. Amsterdam: Addison-Wesley Publishing Group (1994; Zbl 0836.00001)], most existing examples appeared in the work of \textit{R. Wituła} and \textit{D. Słota} [Asian-Eur. J. Math. 1, No. 3, 439--448 (2008; Zbl 1168.05005)]. The authors also provide some new applications.
Reviewer: Gabor Hetyei (Charlotte)Inverse relations and reciprocity laws involving partial Bell polynomials and related extensionshttps://zbmath.org/1491.050302022-09-13T20:28:31.338867Z"Schreiber, Alfred"https://zbmath.org/authors/?q=ai:schreiber.alfredSummary: The objective of this paper is, mainly, twofold: Firstly, to develop an algebraic setting for dealing
with Bell polynomials and related extensions. Secondly, based on the author's previous work on multivariate
Stirling polynomials [Discrete Math. 338, No. 12, 2462--2484 (2015; Zbl 1321.11030)], to present a number of new results related to different types of inverse relationships, among these (1) the use of multivariable Lah polynomials for characterizing self-orthogonal families of
polynomials that can be represented by Bell polynomials, (2) the introduction of `generalized Lagrange inversion polynomials' that invert functions characterized in a specific way by sequences of constants, (3) a general
reciprocity theorem according to which, in particular, the partial Bell polynomials \(B_{n,k}\) and their orthogonal
companions \(A_{n,k}\) belong to one single class of Stirling polynomials: \(A_{n,k} = (-1)^{n-k}B_{-k,-n}\). Moreover, of some
numerical statements (such as Stirling inversion, Schlömilch-Schläfli formulas) generalized polynomial versions
are established. A number of well-known theorems (Jabotinsky, Mullin-Rota, Melzak, Comtet) are given new
proofs.Heine's transformation formula through \(q\)-difference equationshttps://zbmath.org/1491.050322022-09-13T20:28:31.338867Z"Arjika, Sama"https://zbmath.org/authors/?q=ai:arjika.sama"Chaudhary, M. P."https://zbmath.org/authors/?q=ai:chaudhary.mahendra-pal"Hounkonnou, M. N."https://zbmath.org/authors/?q=ai:hounkonnou.mahouton-norbertSummary: In this paper, we give an extension of the first Heine's transformation formula using \(q\)-difference equations. Further, we discussed a Ramanujan's theta function \(\psi(q)\) and deduced it as a particular case.Representations of degenerate Hermite polynomialshttps://zbmath.org/1491.050362022-09-13T20:28:31.338867Z"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-san"Jang, Lee-Chae"https://zbmath.org/authors/?q=ai:jang.lee-chae|jang.leechae"Lee, Hyunseok"https://zbmath.org/authors/?q=ai:lee.hyunseok"Kim, Hanyoung"https://zbmath.org/authors/?q=ai:kim.hanyoungSummary: The study on degenerate versions of some special numbers and polynomials, which began with Carlitz's pioneering work, has regained recent interests of some mathematicians. Motivated by this, we introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Recently, introduced was \(\lambda\)-umbral calculus where the usual exponential function appearing in the generating function of Sheffer sequence is replaced by the degenerate exponential function. Then, among other things, by using the formula of the \(\lambda\)-umbral calculus about expressing one \(\lambda\)-Sheffer polynomial in terms of another \(\lambda\)-Sheffer polynomials we represent the degenerate Hermite polynomials in terms of the higher-order degenerate Bernoulli, Euler, and Frobenius-Euler polynomials and vice versa.Exponential lower bounds on the generalized Erdős-Ginzburg-Ziv constanthttps://zbmath.org/1491.110122022-09-13T20:28:31.338867Z"Bitz, Jared"https://zbmath.org/authors/?q=ai:bitz.jared"Griffith, Sarah"https://zbmath.org/authors/?q=ai:griffith.sarah"He, Xiaoyu"https://zbmath.org/authors/?q=ai:he.xiaoyuSummary: For a finite abelian group \(G\), the generalized Erdős-Ginzburg-Ziv constant \(\mathsf{s}_k(G)\) is the smallest \(m\) such that a sequence of \(m\) elements in \(G\) always contains a \(k\)-element subsequence which sums to zero. If \(n=\exp(G)\) is the exponent of \(G\), the previously best known bounds for \(\mathsf{s}_{kn}(C_n^r)\) were linear in \(n\) and \(r\) when \(k\geq 2\). Via a probabilistic argument, we produce the exponential lower bound
\[
\mathsf{s}_{2n}(C_n^r) > \frac{n}{2}[1.25+o(1)]^r
\]
for \(n > 0\). For the general case, we show
\[
\mathsf{s}_{kn}(C_n^r) > \frac{kn}{4}(1+\frac{1}{ek+1}+o(1))^r.
\]Green's problem on additive complements of the squareshttps://zbmath.org/1491.110132022-09-13T20:28:31.338867Z"Ding, Yuchen"https://zbmath.org/authors/?q=ai:ding.yuchenSummary: Let \(A\) and \(B\) be two subsets of the nonnegative integers. We call \(A\) and \(B\) additive complements if all sufficiently large integers \(n\) can be written as \(a+b\), where \(a\in A\) and \(b\in B\). Let \(S=\{1^2,2^2,3^2,\cdots\}\) be the set of all square numbers. Ben Green was interested in the additive complement of \(S\). He asked whether there is an additive complement \(B=\{b_n\}_{n=1}^{\infty}\subseteq\mathbb{N}\) which satisfies \(b_n=\frac{\pi^2}{16}n^2+o(n^2)\). Recently, \textit{Y.-G. Chen} and \textit{J.-H. Fang} [J. Number Theory 180, 410--422 (201; Zbl 1421.11014)] proved that if \(B\) is such an additive complement, then
\[
\limsup_{n\rightarrow\infty}\,\frac{\frac{\pi^2}{16}n^2-b_n}{n^{1/2}\log n}\geq\sqrt{\frac{2}{\pi}}\frac{1}{\log 4}.
\]
They further conjectured that
\[
\limsup_{n\rightarrow\infty}\,\frac{\frac{\pi^2}{16}n^2-b_n}{n^{1/2}\log n}=+\infty.
\]
In this paper, we confirm this conjecture by giving a much more stronger result, i.e.,
\[
\limsup_{n\rightarrow\infty}\,\frac{\frac{\pi^2}{16}n^2-b_n}{n}\geq\frac{\pi }{4}.
\]On generalized perfect difference sumsetshttps://zbmath.org/1491.110142022-09-13T20:28:31.338867Z"Fang, Jin-Hui"https://zbmath.org/authors/?q=ai:fang.jinhuiA variant of the proof of van der Waerden's theorem by Furstenberghttps://zbmath.org/1491.110152022-09-13T20:28:31.338867Z"Eyidoğan, Sadık"https://zbmath.org/authors/?q=ai:eyidogan.sadik"Özkurt, Ali Arslan"https://zbmath.org/authors/?q=ai:ozkurt.ali-arslanSummary: Let \(R\) be a commutative ring with identity. In this paper, for a given monotone decreasing positive sequence and an increasing sequence of subsets of \(R\), we will define a metric on \(R\) using them. Then, we will use this kind of metric to obtain a variant of the proof of van der Waerden's theorem by \textit{H. Furstenberg} [Recurrence in ergodic theory and combinatorial number theory. Princeton, New Jersey: Princeton University Press (1981; Zbl 0459.28023)].On arithmetic sums of Ahlfors-regular setshttps://zbmath.org/1491.110162022-09-13T20:28:31.338867Z"Orponen, Tuomas"https://zbmath.org/authors/?q=ai:orponen.tuomasLet \(A,B\subseteq\mathbb{R}\) be Ahlfors-regular sets with Hausdorff dimension \(\alpha\) resp. \(\beta\). The authors proves that the arithmetic sum \(A+\lambda B\) has Hausdorff dimension greater or eqaul to \(\alpha + \beta (1-\alpha)/(2-\alpha)\) outside a set of parameter values \(\lambda\) of Hausdorff dimension zero. This inserting result on Ahlfors-regular sets is stronger than the result on lower bounds for the dimension of arithmetic sums of arbitrary subsets of \(\mathbb{R}\) found in [\textit{J. Bourgain}, J. Anal. Math. 112, 193--236 (2010; Zbl 1234.11012)].
Reviewer: Jörg Neunhäuserer (Goslar)Non-commutative methods in additive combinatorics and number theoryhttps://zbmath.org/1491.110172022-09-13T20:28:31.338867Z"Shkredov, Ilya D."https://zbmath.org/authors/?q=ai:shkredov.ilya-dThis work is a survey on the area of arithmetic combinatorics, focussing on non-abelian results. Typically, additive combinatorics have dealt with problems both on the integers on abelian groups, but more recently there have been important breakthroughs on the study of questions in the non-abelian setting. For instance, the growth of sets on non-abelian groups has been a very recent trend of research with several breakthroughs on the last years.
This survey explore a wide variety of results on these area, and relates it with the abelian analogues. These includes problems on arithmetic combinatorics on itself (see for instance Section 4: The structure of sets with small doubling in an arbitrary group), as well as applications on other domains including incidence geometry, group theory and analytic number theory, among other.
Apart from a very detailed survey of the techniques (non-Fourier analysis, Balogh-Szeméredi-Gowers Theorem, etc), the author provides a very rich source of bibliography.
Reviewer: Juanjo Rué Perna (Barcelona)Linear recurrences of order at most two in small divisorshttps://zbmath.org/1491.110182022-09-13T20:28:31.338867Z"Chentouf, A. Anas"https://zbmath.org/authors/?q=ai:chentouf.a-anasIn this paper, small divisors of a positive integer \(n\) are divisors less than or equal to \(\sqrt{n}\). The author gives a complete characterization of numbers whose small divisors \(d_1 = 1 < d_2 < \ldots < d_{k}\) form a linear recurrence of order at most 2. In other words, for such numbers, there exist two integers \(a,b\) such that \(d_{i} = a d_{i-1} + b d_{i-2}\) for \(3 \leq i \leq k\). The method makes use of a tree representation of the sequences of small divisors. It turns out that all solutions with \(k \geq 5\) fall into one of the following cases:
\begin{itemize}
\item[1.] The small divisors are in geometric progression (\(b=0\));
\item[2.] The small divisors whose ranks have same parity are in geometric progression (\(a=0\));
\item[3.] The small divisors are in arithmetic progression (\(a=2\) and \(b=-1\)).
\end{itemize}
\textit{D. E. Iannucci} [Integers 18, Paper A77, 10 p. (2018; Zbl 1453.11016)] previously resolved the last case and found that \(n=60\) is the only solution of that kind with \(k \geq 5\).
Reviewer: Olivier Rozier (Paris)On a new generalization of Jacobsthal hybrid numbershttps://zbmath.org/1491.110192022-09-13T20:28:31.338867Z"Bród, D."https://zbmath.org/authors/?q=ai:brod.dorota"Szynal-Liana, A."https://zbmath.org/authors/?q=ai:szynal-liana.anettaSummary: We define a two-parameter generalization of Jacobsthal hybrid numbers. We give Binet formula, the generating functions and some identities for these numbers.More identities for Fibonacci and Lucas quaternionshttps://zbmath.org/1491.110202022-09-13T20:28:31.338867Z"Irmak, Nurettin"https://zbmath.org/authors/?q=ai:irmak.nurettinSummary: In this paper, we define the associate matrix as
\[
F= \left( \begin{matrix} 1+i+2j+3k & i+j+2k \\ i+j+2k & 1+j+k \end{matrix} \right).
\]
By the means of the matrix \(F\), we give several identities about Fibonacci and Lucas quaternions by matrix methods. Since there are two different determinant definitions of a quaternion square matrix (whose entries are quaternions), we obtain different Cassini identities for Fibonacci and Lucas quaternions apart from Cassini identities given in the papers [\textit{S. Halici}, Adv. Appl. Clifford Algebr. 22, No. 2, 321--327 (2012; Zbl 1329.11016)] and [\textit{M. Akyiğit} et al., 24, No. 3, 631--641 (2014; Zbl 1321.11020)].Products involving reciprocals of gibonacci polynomialshttps://zbmath.org/1491.110212022-09-13T20:28:31.338867Z"Koshy, Thomas"https://zbmath.org/authors/?q=ai:koshy.thomasSummary: We explore finite and infinite products involving reciprocals of gibonacci polynomials, and their Pell counterparts.On Gauss Fibonacci polynomials, on Gauss Lucas polynomials and their applicationshttps://zbmath.org/1491.110222022-09-13T20:28:31.338867Z"Özkan, Engin"https://zbmath.org/authors/?q=ai:ozkan.engin"Taştan, Merve"https://zbmath.org/authors/?q=ai:tastan.merveSummary: We define the Gauss Fibonacci polynomials. Then we give a formula for the Gauss Fibonacci polynomials by using the Fibonacci polynomials. The Gauss Lucas polynomials are described and the relation with Lucas polynomials are explained. We show that there is a relation between the Gauss Fibonacci polynomials and the Fibonacci polynomials. The Gauss Lucas polynomials are also given by using the Gauss Fibonacci polynomials. Some theorems like Cassini's theorem are proved for the polynomials. Their Binet's formulas are obtained. We also define the matrices of the Gauss Fibonacci polynomials and the Gauss Lucas polynomials. We examine properties of the matrices.Catalan transform of the incomplete Jacobsthal numbers and incomplete generalized Jacobsthal polynomialshttps://zbmath.org/1491.110232022-09-13T20:28:31.338867Z"Özkan, Engin"https://zbmath.org/authors/?q=ai:ozkan.engin"Uysal, Mine"https://zbmath.org/authors/?q=ai:uysal.mine"Kuloğlu, Bahar"https://zbmath.org/authors/?q=ai:kuloglu.baharSums of finite products of Pell polynomials in terms of hypergeometric functionshttps://zbmath.org/1491.110242022-09-13T20:28:31.338867Z"Patra, Asim"https://zbmath.org/authors/?q=ai:patra.asim"Panda, Gopal Krishna"https://zbmath.org/authors/?q=ai:panda.gopal-krishnaThe purpose of this article is to express sums of finite products of Pell polynomials \(P_n(x)\) in terms of hypergeometric functions. First, a fundamental connection between the Chebyshev polynomials of the second kind and the Pell polynomials is established. The new Chebyshev polynomials of the third and fourth kind as well as other special polynomials expressed in terms of hypergeometric functions are used in the proofs. Several special integrals and generating functions are used in the proofs. As special case, a formula for the \(r\)-th derivative of the Pell polynomial is derived. Obviously, many of these polynomials can be expressed in terms of each other.
Reviewer: Thomas Ernst (Uppsala)The generalized Lucas hybrinomials with two variableshttps://zbmath.org/1491.110252022-09-13T20:28:31.338867Z"Sevgi, Emre"https://zbmath.org/authors/?q=ai:sevgi.emreSummary: Özdemir defined the hybrid numbers as a generalization of complex, hyperbolic and dual numbers. In this research, we define the generalized Lucas hybrinomials with two variables. Also, we get the Binet formula, generating function and some properties for the generalized Lucas hybrinomials. Moreover, Catalan's, Cassini's and d'Ocagne's identities for these hybrinomials are obtained. Lastly, by the help of the matrix theory we derive the matrix representation of the generalized Lucas hybrinomials.Split complex bi-periodic Fibonacci and Lucas numbershttps://zbmath.org/1491.110262022-09-13T20:28:31.338867Z"Yilmaz, Nazmiye"https://zbmath.org/authors/?q=ai:yilmaz.nazmiyeSummary: The initial idea of this paper is to investigate the split complex bi-periodic Fibonacci and Lucas numbers by using SCFLN now on. We try to show some properties of SCFLN by taking into account the properties of the split complex numbers. Then, we present interesting relationships between SCFLN.Partial factorizations of products of binomial coefficientshttps://zbmath.org/1491.110272022-09-13T20:28:31.338867Z"Du, Lara"https://zbmath.org/authors/?q=ai:du.lara"Lagarias, Jeffrey C."https://zbmath.org/authors/?q=ai:lagarias.jeffrey-cLet \[\overline{G}_n =\prod_{k=0}^n \binom{n}{k},\] the product of the elements of the \(n\)th row of Pascal's triangle.
The authors study the partial factorizations of \(\overline{G}_n\) given by the product \(G(n, x)\) of all prime factors \(p\) of \(\overline{G}_n\) having \(p\leq x\), counted with multiplicity.
They show \[\log G(n, \alpha n)f_G(\alpha)n^2\quad\text{as } n \to \infty\] for a limit function \(f_G(\alpha)\) defined for \(0\leq\alpha\leq 1\). The the main results are deduced from study of functions \(A(n, x)\), \(B(n, x)\), that encode statistics of the base \(p\) radix expansions of the integer \(n\) (and smaller integers), where the base \(p\) ranges over primes \(p\leq x\). Asymptotics of \(A(n, x)\) and \(B(n, x)\) are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.
Reviewer: Guo-Shuai Mao (Nanjing)On Motzkin numbers and central trinomial coefficientshttps://zbmath.org/1491.110282022-09-13T20:28:31.338867Z"Sun, Zhi-Wei"https://zbmath.org/authors/?q=ai:sun.zhiwei|sun.zhi-wei.1|sun.zhi-weiSummary: The Motzkin numbers \(M_n = \sum^n_{k = 0} \binom{n}{2 k} \binom{2 k}{k} / (k + 1) (n = 0, 1, 2, \ldots )\) and the central trinomial coefficients \(T_n (n = 0, 1, 2, \ldots )\) given by the constant term of \(( 1 + x + x^{- 1} )^n\), have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with \(n\) any positive integer:
\[
\begin{aligned}
\frac{ 2}{ n} &\sum_{k = 1}^n(2 k + 1) M_k^2 \in \mathbb{Z},\\
\frac{ n^2 ( n^2 - 1 )}{ 6} &\Bigg| \sum_{k = 0}^{n - 1} k(k + 1)(8 k + 9) T_k T_{k + 1},
\end{aligned}
\]
and also
\[
\sum_{k = 0}^{n - 1}(k + 1)(k + 2)(2 k + 3) M_k^2 3^{n - 1 - k} = n(n + 1)(n + 2) M_n M_{n - 1}.
\]Asymptotic behavior of Bernoulli-Dunkl and Euler-Dunkl polynomials and their zeroshttps://zbmath.org/1491.110292022-09-13T20:28:31.338867Z"Mínguez Ceniceros, Judit"https://zbmath.org/authors/?q=ai:minguez-ceniceros.judit"Varona, Juan L."https://zbmath.org/authors/?q=ai:varona-malumbres.juan-luisThe purpose of this article is to study the asymptotic behavior of the recently introduced Bernoulli-Dunkl and Euler-Dunkl polynomials.
These polynomials are defined with so-called \(E_{\alpha}\) functions instead of exponential functions in the numerator. Dilcher showed asympotic behaviour of the Bernoulli and the Euler polynomials in terms of trigonometric functions and the authors show corresponding asymptotic behavior of the Bernoulli-Dunkl and the Euler-Dunkl polynomials. The coefficients of the asymptotic expansions as well as the right hand sides are Bessel functions. The first positive zero of \(J_{\alpha}(z)\) appears as coefficients in function arguments as well as in numerators and denominators. Several formulas for Bernoulli-Dunkl polynomials, which are similar to Bernoulli polynomials, where the Dunkl operator replaces the derivative, and new coefficients replace the binomial coefficients are used to prove the main theorems. Graphs are shown which display the differences between the new polynomials and Bernoulli polynomials. And similar for Euler-Dunkl polynomials. The zeros for Bernoulli-Dunkl and Euler-Dunkl polynomials are shown in graphs.
Reviewer: Thomas Ernst (Uppsala)Semiorthogonality of geometric polynomialshttps://zbmath.org/1491.110302022-09-13T20:28:31.338867Z"Kargın, Levent"https://zbmath.org/authors/?q=ai:kargin.levent"Çay, Emre"https://zbmath.org/authors/?q=ai:cay.emreGeometric polynomials are also termed as Fubini polynomials which is most commonly used in literature.
Special polynomials and numbers have significant roles in various branches of mathematics, theoretical physics, and engineering. The problems arising in mathematical physics and engineering are framed in terms of differential equations. Most of these equations can only be treated by using various families of special polynomials which provide new means of mathematical analysis. They are widely used in computational models of scientific and engineering problems. In addition, these special polynomials allow the derivation of different useful identities in a straightforward way and help in introducing new families of special polynomials.
The Fubini-type polynomials (or geometric-type polynomials) appear in combinatorial mathematics and play an important role in the theory and applications of mathematics, thus many number theory and combinatorics experts have extensively studied their properties and obtained series of interesting results. The Fubini-type numbers and polynomials are related Bernoulli numbers with diverse extensions and proven to be an effective tool in different topics in combinatorics and analysis.
In the related paper, the authors have examined the semiorthogonality of Fubini and higher order Fubini polynomials. They showed that the integrals of products of the higher order Fubini polynomials can be evaluated in terms of Bernoulli numbers, which means that the higher order geometric polynomials are also semiorthogonal. As applications, they have given some new explicit formulas for Bernoulli and \(p\)-Bernoulli numbers.
Reviewer: Uğur Duran (Iskenderun)2-variable Fubini-degenerate Apostol-type polynomialshttps://zbmath.org/1491.110312022-09-13T20:28:31.338867Z"Nahid, Tabinda"https://zbmath.org/authors/?q=ai:nahid.tabinda"Ryoo, Cheon Seoung"https://zbmath.org/authors/?q=ai:ryoo.cheon-seoungSpectral theory of regular sequenceshttps://zbmath.org/1491.110322022-09-13T20:28:31.338867Z"Coons, Michael"https://zbmath.org/authors/?q=ai:coons.michael"Evans, James"https://zbmath.org/authors/?q=ai:evans.james-a|evans.james-b|evans.james-e|evans.james-w|evans.james-r|evans.james-d"Mañibo, Neil"https://zbmath.org/authors/?q=ai:manibo.neilRegular sequences are a well-studied generalization to arbitrary (unbounded) alphabets of fixed points of constant-length substitutions (also called morphisms) defined over a finite alphabet, that is, of automatic sequences. They can be defined by the fact that the vector space generated by the \(k\)-kernel of the sequence is finite-dimensional.
From the abstract: ``Using the harmonic analysis of measures associated with substitutions as motivation, we study the limiting asymptotics of regular sequences by constructing a systematic measure-theoretic framework surrounding them. The constructed measures are generalisations of mass distributions supported on attractors of iterated function systems.''
Reviewer: Michel Rigo (Liège)Three-parameter mock theta functionshttps://zbmath.org/1491.110482022-09-13T20:28:31.338867Z"Cui, Su-Ping"https://zbmath.org/authors/?q=ai:cui.su-ping"Gu, Nancy S. S."https://zbmath.org/authors/?q=ai:gu.nancy-shan-shan"Hou, Qing-Hu"https://zbmath.org/authors/?q=ai:hou.qinghu"Su, Chen-Yang"https://zbmath.org/authors/?q=ai:su.chenyangSummary: Mock theta functions were first introduced by Ramanujan. Historically, mock theta functions can be represented as Eulerian forms, Appell-Lerch sums, Hecke-type double sums, and Fourier coefficients of meromorphic Jacobi forms. In this paper, in view of the \(q\)-Zeilberger algorithm and the Watson-Whipple transformation formula, we establish five three-parameter mock theta functions in Eulerian forms, and express them by Appell-Lerch sums. Especially, the main results generalize some two-parameter mock theta functions. For example, setting \((m, q, x) \to(1, q^{1 / 2}, x q^{- 1 / 2})\) in
\[
\sum_{n = 0}^\infty \frac{ (- q^2; q^2)_n q^{n^2 + (2 m - 1) n}}{ (x q^m, x^{- 1} q^m; q^2)_{n + 1}},
\] we derive the universal mock theta function \(g_2(x, q)\).Density of sequences of the form \(x_n=f(n)^n\) in \([0,1]\)https://zbmath.org/1491.110702022-09-13T20:28:31.338867Z"Saunders, J. C."https://zbmath.org/authors/?q=ai:saunders.j-cIn this paper, the author shows that certain sequences $x_n$ satisfying $\log x_n=ng(n)$, where $g$ is a real analytic function on the interval $[-1,1]$ are dense subsets of the interval $[0,1]$. The proofs make use of Diophantine approximation type arguments, continued fractions and an adaptation of Zaharescu's breakthrough on the small values of square multiples modulo 1.
Reviewer: Emre Alkan (İstanbul)Multilinear exponential sums with a general class of weightshttps://zbmath.org/1491.110752022-09-13T20:28:31.338867Z"Kerr, Bryce"https://zbmath.org/authors/?q=ai:kerr.bryce"Macourt, Simon"https://zbmath.org/authors/?q=ai:macourt.simonIn the paper under review, the authors study weighted multilinear exponential sums \[ S(\mathcal{X}_1,\dots,\mathcal{X}_n;\omega_1,\dots,\omega_n):=\sum_{x_1\in \mathcal{X}_1}\cdots\sum_{x_n\in \mathcal{X}_n}\omega_1(\mathbf{x})\cdots\omega_n(\mathbf{x}) e_p(x_1x_2\cdots x_n),\] where \(\mathcal{X}_1,\dots,\mathcal{X}_n\) are subsets of \(\mathbb{F}_p^*\), \(e_p(\theta)=\exp(2\pi i\theta /p)\) with \(p\) a prime.
\textit{J. Bourgain} [Geom. Funct. Anal. 18, No. 5, 1477--1502 (2009; Zbl 1162.11043)] showed that if \(|\mathcal{X}_i|>p^{\varepsilon}\) and \(|\mathcal{X}_1|\cdots|\mathcal{X}_n|\geq p^{1+\varepsilon}\) then \[ S(\mathcal{X}_1,\dots,\mathcal{X}_n;1,\dots,1)\ll p^{-\delta}|\mathcal{X}_1|\cdots|\mathcal{X}_n|\] with \(\delta>0\) depending on \(\varepsilon\).
The main result of the authors is a bound on the general weighted sum \(S(\mathcal{X}_1,\dots,\mathcal{X}_n;\omega_1,\dots,\omega_n)\). The expression is too complicated to restate here -- the authors give various examples where the bound is nontrivial (mostly depending on the sizes of the sets \(\mathcal{X}_i\)). The method of proof uses the recent improvement due to \textit{I. D. Shkredov} [Trans. Mosc. Math. Soc. 2018, 231--281 (2018; Zbl 1473.11034); translation from Tr. Mosk. Mat. O.-va 79, No. 2, 271--334 (2018)] of Bourgain's result which is based on geometric incidence estimates of \textit{M. Rudnev} [Combinatorica 38, No. 1, 219--254 (2018; Zbl 1413.51001)].
Reviewer: Thomas Stoll (Vandœuvre-lès Nancy)Max and min matrices with hyper-Fibonacci numbershttps://zbmath.org/1491.150352022-09-13T20:28:31.338867Z"Solmaz, Tuğçe"https://zbmath.org/authors/?q=ai:solmaz.tugce"Bahşi, Mustafa"https://zbmath.org/authors/?q=ai:bahsi.mustafaGroups containing locally maximal product-free sets of size 4https://zbmath.org/1491.200582022-09-13T20:28:31.338867Z"Anabanti, C. S."https://zbmath.org/authors/?q=ai:anabanti.chimere-stanleySummary: Every locally maximal product-free set \(S\) in a finite group \(G\) satisfies \(G=S\cup SS \cup S^{-1}S \cup SS^{-1}\cup \sqrt{S}\), where \(SS=\{xy\mid x, y\in S\}\), \(S^{-1}S=\{x^{-1}y\mid x, y\in S\}\), \(SS^{-1}=\{xy^{-1}\mid x, y\in S\}\) and \(\sqrt{S}=\{x\in G\mid x^2\in S\}\). To better understand locally maximal product-free sets, \textit{E. A. Bertram} [Discrete Math. 44, 31--43 (1983; Zbl 0506.05060)] asked whether every locally maximal product-free set \(S\) in a finite abelian group satisfy \(|\sqrt{S}|\leqslant 2|S|\). This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size \(4\), continuing the work of \textit{A. P. Street} and \textit{E. G. Whitehead jun.} [J. Comb. Theory, Ser. A 17, 219--226 (1974; Zbl 0288.05020); Lect. Notes Math. 403, 109--124 (1974; Zbl 0318.05010)] and \textit{M. Giudici} and \textit{S. Hart} [Electron. J. Comb. 16, No. 1, Research Paper R59, 17 p. (2009; Zbl 1168.20009)] on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size \(4\), and conclude with a conjecture on the size \(4\) problem as well as an open problem on the general case.The Fibonacci-circulant sequences in the binary polyhedral groupshttps://zbmath.org/1491.200772022-09-13T20:28:31.338867Z"Karaduman, Erdal"https://zbmath.org/authors/?q=ai:karaduman.erdal"Deveci, Omur"https://zbmath.org/authors/?q=ai:deveci.omurSummary: In 2017 \textit{Ö. Deveci} et al. [Iran. J. Sci. Technol., Trans. A, Sci. 41, No. 4, 1033--1038 (2017; Zbl 1391.11026)] defined the Fibonacci-circulant sequences of the first and second kinds as shown, respectively:
\[
x_n^1 = -x_{n-1}^1+x_{n-2}^1-x_{n-3}^1\text{ for }n \geq 4,\text{ where } x_1^1=x_2^1=0\text{ and } x_3^1=1
\]
and
\[
x_n^2 = -x_{n-3}^2-x_{n-4}^2+x_{n-5}^2\text{ for }n \geq 6, \text{ where }x_1^2 = x_2^2 = x_3^2=x_4^2=0\text{ and }x_5^2=1
\]
Also, they extended the Fibonacci-circulant sequences of the first and second kinds to groups. In this paper, we obtain the periods of the Fibonacci-circulant sequences of the first and second kinds in the binary polyhedral groups.\( \mu \)-statistical convergence and the space of functions \(\mu \)-stat continuous on the segmenthttps://zbmath.org/1491.400032022-09-13T20:28:31.338867Z"Sadigova, S. R."https://zbmath.org/authors/?q=ai:sadigova.sabina-rahibThe author introduces $\mu$-statistical density of a point and the concept of $\mu$-statistical fundamentality at a point and also states that this method is equivalent to the concept of $\mu$-stat convergence. On the other hand, the concept of $\mu$-stat continuity is defined. Some properties of the space of all $\mu$-stat continuous functions are examined.
Reviewer: Emre Taş (Kırşehir)A note on discrete degenerate random variableshttps://zbmath.org/1491.600162022-09-13T20:28:31.338867Z"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-san"Jang, Lee-Chae"https://zbmath.org/authors/?q=ai:jang.lee-chae|jang.leechae"Kim, H. Y."https://zbmath.org/authors/?q=ai:kim.hwan-y|kim.ho-yeun|kim.heon-young|kim.hae-young|kim.hyun-young|kim.ha-young|kim.hye-yong|kim.hyun-yul|kim.ho-youn|kim.hun-young|kim.ho-yon|kim.hae-yong|kim.hye-yeonSummary: In this paper, we introduce two discrete degenerate random variables, namely the degenerate binomial and degenerate Poisson random variables. We deduce the expectations of the degenerate binomial random variables. We compute the generating function of the moments of the degenerate Poisson random variables, which leads us to define the new type degenerate Bell polynomials, and hence obtain explicit expressions for the moments of those random variables in terms of such polynomials. We also get the variances of the degenerate Poisson random variables. Finally, we illustrate two examples of the degenerate Poisson random variables.Periods of iterations of functions with restricted preimage sizeshttps://zbmath.org/1491.600172022-09-13T20:28:31.338867Z"Martins, Rodrigo S. V."https://zbmath.org/authors/?q=ai:martins.rodrigo-s-v"Panario, Daniel"https://zbmath.org/authors/?q=ai:panario.daniel"Qureshi, Claudio"https://zbmath.org/authors/?q=ai:qureshi.claudio-m"Schmutz, Eric"https://zbmath.org/authors/?q=ai:schmutz.ericYet another way of calculating moments of the Kesten's distribution and its consequences for Catalan numbers and Catalan triangleshttps://zbmath.org/1491.600322022-09-13T20:28:31.338867Z"Szabłowski, Paweł J."https://zbmath.org/authors/?q=ai:szablowski.pawel-jerzySummary: We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, also concerning Lucas, Fibonacci and Fine numbers.Investigating Catalan numbers with Pascal's trianglehttps://zbmath.org/1491.970042022-09-13T20:28:31.338867Z"Hong, Dae S."https://zbmath.org/authors/?q=ai:hong.dae-s(no abstract)Sequences of ratios of a convex quadrilateralhttps://zbmath.org/1491.970072022-09-13T20:28:31.338867Z"Laudano, Francesco"https://zbmath.org/authors/?q=ai:laudano.francesco(no abstract)