Recent zbMATH articles in MSC 11B39https://zbmath.org/atom/cc/11B392021-06-15T18:09:00+00:00WerkzeugA unified treatment of certain classes of combinatorial identities.https://zbmath.org/1460.050212021-06-15T18:09:00+00:00"Batır, Necdet"https://zbmath.org/authors/?q=ai:batir.necdet"Sofo, Anthony"https://zbmath.org/authors/?q=ai:sofo.anthonySummary: We propose and prove some general combinatorics formulas. Applying these formulas, we obtain some new binomial harmonic and harmonic Fibonacci and Lucas number identities. We also recover some known identities included in the works of \textit{R. Frontczak} [J. Integer Seq. 23, No. 3, Article 20.3.2, 9 p. (2020; Zbl 1446.11026); ``Binomial sums with skew-harmonic numbers'', Preprint] and \textit{K. N. Boyadzhiev} [J. Integer Seq. 12, No. 6, Article ID 09.6.1, 8 p. (2009; Zbl 1213.11054); Adv. Appl. Discrete Math. 13, No. 1, 43--63 (2014; Zbl 1298.05015)].Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set.https://zbmath.org/1460.810262021-06-15T18:09:00+00:00"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Fillman, Jake"https://zbmath.org/authors/?q=ai:fillman.jake"Gorodetski, Anton"https://zbmath.org/authors/?q=ai:gorodetski.antonSummary: We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe-Sommerfeld criterion for sums of Cantor sets which may be of independent interest.On \(k\)-step Fibonacci functions and \(k\)-step Fibonacci numbers.https://zbmath.org/1460.110242021-06-15T18:09:00+00:00"Sriponpaew, Boonyong"https://zbmath.org/authors/?q=ai:sriponpaew.boonyong"Sassanapitax, Lee"https://zbmath.org/authors/?q=ai:sassanapitax.leeSummary: In this manuscript, we study \(k\)-step Fibonacci functions defined as \(f:\mathbb{R}\to\mathbb{R}\) such that \(f(x+k)= f(x+k-1)+ f(x+k-2)+\cdots+ f(x)\), for all \(x\in\mathbb{R}\). Moreover, we develop some properties of these functions including notions of \(f\)-even and \(f\)-odd functions. Furthermore, we show that \(\lim_{x\to\infty}\frac{f(x+i)}{f(x)}=\alpha^i\) for all \(i\in\mathbb{N}\), where \(\alpha\) is the \(k\)-nacci constant.On the period mod \(m\) of polynomially-recursive sequences: a case study.https://zbmath.org/1460.110252021-06-15T18:09:00+00:00"Banderier, Cyril"https://zbmath.org/authors/?q=ai:banderier.cyril"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florianIn the paper under review, the authors analyze the period mod \(m\) of a second order polynomially-recursive sequence. The problem originally comes from an enumeration of avoiding pattern permutations and appears to be linked with nice number theory notions (the Carmichael function, Wieferich primes, algebraic integers). They provide the mod \(a^k\) supercongruences, and generalized these results to a class of recurrences.
Reviewer: Uğur Duran (Iskenderun)Distribution of Wythoff sequences modulo one.https://zbmath.org/1460.110232021-06-15T18:09:00+00:00"Kawsumarng, Sutasinee"https://zbmath.org/authors/?q=ai:kawsumarng.sutasinee"Khemaratchatakumthorn, Tammatada"https://zbmath.org/authors/?q=ai:khemaratchatakumthorn.tammatada"Noppakaew, Passawan"https://zbmath.org/authors/?q=ai:noppakaew.passawan"Pongsriiam, Prapanpong"https://zbmath.org/authors/?q=ai:pongsriiam.prapanpongSummary: Let \(\alpha\) be the golden ratio and \(\beta\alpha= -1\). In the study of sumsets associated with Wythoff sequences, it is important to prove the inequality \(0< \{b\alpha\}+\beta^n<1\) for integers \(b\) and \(n\) in a certain range. In this article, we continue the investigation by replacing \(\{b\alpha\}+ \beta^n\) by \(\sqrt{5}\beta^{n-1}- \{b\alpha\}\).A new condition equivalent to the Ankeny-Artin-Chowla conjecture.https://zbmath.org/1460.111302021-06-15T18:09:00+00:00"Harrington, Joshua"https://zbmath.org/authors/?q=ai:harrington.joshua"Jones, Lenny"https://zbmath.org/authors/?q=ai:jones.lenny-kLet \(p\equiv 1\pmod{4}\) be a prime number and let \(\varepsilon=(t+u\sqrt{p})/2\) be the fundamental unit of \(\mathbb Q(\sqrt{p})\). The Ankeny-Artin-Chowla conjecture states that \(u\not\equiv 0\pmod{p}\). An analogous conjecture for primes \(p\equiv 3\pmod{4}\) is due to Mordell.
The main results of this paper are two equivalent conditions for the above conjectures in terms of Lucas polynomials \(L_n(x)\) and Vieta-Lucas polynomials \(L_n^-(x)\), which are defined by \(L_0(x)=L_0^-(x)=2\), \(L_1(x)=L_1^-(x)=x\) and \(L_n(x)=xL_{n-1}(x)+L_{n-2}(x)\), \(L_n^-(x)=xL_{n-1}^-(x)-L_{n-2}^-(x)\) for \(n\geq 2\). More precisely it is shown that the Ankeny-Artin-Chowla conjecture holds if and only if there exists an integer \(c>0\) such that \(a_1=L_p(c)\) is the smallest positive integer solution for \(a\) in the diophantine equation \(a^2+4=p^3b^2\). The result concerning the conjecture by Mordell is very similar, involving \(L_p^-(x)\) instead of \(L_p(x)\).
Reviewer: Alessandro Cobbe (Neubiberg)Relating certain weighted Fibonacci series to Bernoulli polynomials via the polylogarithm function.https://zbmath.org/1460.110212021-06-15T18:09:00+00:00"Adegoke, Kunle"https://zbmath.org/authors/?q=ai:adegoke.kunleSummary: We show that certain weighted Fibonacci and Lucas series can always be expressed as linear combinations of polylogarithms. In some special cases we evaluate the series in terms of Bernoulli polynomials, making use of the connection between these polynomials and the polylogarithm.New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile.https://zbmath.org/1460.110222021-06-15T18:09:00+00:00"Edwards, Kenneth"https://zbmath.org/authors/?q=ai:edwards.kenneth"Allen, Michael A."https://zbmath.org/authors/?q=ai:allen.michael-aSummary: We consider the tiling of an \(n\)-board (a board of size \(n \times 1)\) with squares of unit width and \((1,1)\)-fence tiles. A \((1,1)\)-fence tile is composed of two unit-width square sub-tiles separated by a gap of unit width. We show that the number of ways to tile an \(n\)-board using unit-width squares and \((1,1)\)-fence tiles is equal to a Fibonacci number squared when \(n\) is even and a golden rectangle number (the product of two consecutive Fibonacci numbers) when \(n\) is odd. We also show that the number of tilings of boards using \(n\) such square and fence tiles is a Jacobsthal number. Using combinatorial techniques we prove new identities involving golden rectangle and Jacobsthal numbers. Two of the identities involve entries in two Pascal-like triangles. One is a known triangle (with alternating ones and zeros along one side) whose \((n,k)\)th entry is the number of tilings using \(n\) tiles of which \(k\) are fence tiles. There is a simple relation between this triangle and the other which is the analogous triangle for tilings of an \(n\)-board. These triangles are related to Riordan arrays and we give a general procedure for finding which Riordan array(s) a triangle is related to. The resulting combinatorial interpretation of the Riordan arrays allows one to derive properties of them via combinatorial proof.Extraction of Cantor one-third set from Stern-Brocot sequence.https://zbmath.org/1460.110262021-06-15T18:09:00+00:00"Deepa, S."https://zbmath.org/authors/?q=ai:deepa.s-n"Gnanam, A."https://zbmath.org/authors/?q=ai:gnanam.aSummary: In this paper, we define and discuss the properties of Cantor one-third set and Stern-Brocot sequence with the principal aim of extracting the former from the latter.