Arithmetic Triangle Research Articles

Some Identities on Generalized Arithmetic Triangles

Summary We generalize the arithmetic triangle of Blaise Pascal, Yang Hui, and others, by maintaining its recurrence relation, but replacing the traditional 1s on the boundary with arbitrary sequences. Within these structures, we examine some identities commonly studied in Pascal’s version, including those related to sums and alternating sums of entries in rows as well as the so-called hockey stick identities. We use recurrence relations and elementary generating functions to derive general results and see how these results can be used in some interesting special cases.

On hyper (r, q)-Fibonacci polynomials

Abstract Related to generalized arithmetic triangle, we introduce the hyper (r, q)-Fibonacci polynomials as the sum of these elements along a finite ray starting from a specific point, which generalize the hyper-Fibonacci polynomials. We give generating function, recurrence relations and we show some properties whose application allows us to extend the notion of Cassini determinant and to study some ratios. Moreover, we derive a connection between these polynomials and the incomplete (r, q)-Fibonacci polynomials defined in this paper.

On Recurrences in Generalized Arithmetic Triangle

AbstractIn the present paper, we consider the generalized arithmetic triangle called GAT which is structurally identical to Pascal’s triangle for which we keep the Pascal’s rule of addition and we replace both legs by two sequences (an)n≥1and (bn)n≥1witha0=b0= Ω. Our goal is to describe the recurrence relation associated to the sum of elements lying along a finite ray in this triangle. As consequences, we obtain some combinatorial properties and we establish that the sum of elements lying along a main rising diagonal is a convolution of generalized Fibonacci sequence and another sequence which one will determine. We also precise the corresponding generating function. Further, we establish some nice identities by using the Morgan-Voyce phenomenon. Finally, we generalize the Golden ratio.

An Arithmetic Triangle Arising From a One-Way Street Grid

Summary Pascal’s triangle arises by counting the number of shortest paths from (0, 0) to (n, k) on a square street grid. The length of the shortest path is the Manhattan distance from (0, 0) to (n, k). We consider the case of a square street grid of one-way streets, with successive parallel streets being oppositely directed. We investigate the associated distance function q (which is only a quasi-metric, since q(A, B) may differ from q(B, A)) and the arithmetic triangle obtained by counting shortest routes on the one-way grid from (0, 0) to (n, k).

Chudnovsky-Ramanujan type formulae for non-compact arithmetic triangle groups

We develop a uniform method to derive Chudnovsky-Ramanujan type formulae for triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we carry out the method systematically for non-compact arithmetic triangle groups and one non-Fuchsian covering. As a result, we derive all rational Ramanujan type series given by Chan-Cooper for levels 1-4, as well as two additional rational series of a similar form prescribed by Chan-Cooper for these levels, but not found in the paper of Chan-Cooper. These two additional series were first found by Z.-W. Sun in a slightly different form. We also derive additional rational series of a similar form, but not found in the papers of Chan-Cooper nor Z.-W. Sun.As an ingredient in the method, we give an algorithm to rigorously confirm the singular values of normalized Eisenstein series of weight 2 and give examples in higher class number, which may be of independent interest. We also prove some partial completeness results for Clausen type single summation series.

Linear recurrence sequence associated to rays of negatively extended Pascal triangle

We consider the extension of generalized arithmetic triangle to negative values of rows and we describe the recurrence relation associated to the sum of diagonal elements laying along finite rays. We also give the corresponding generating function. We conclude by an application to Fibonacci numbers and Morgan-Voyce polynomials with negative subscripts.

Використання властивості періодичності для генерування комбінаторних конфігурацій

Introduction. Sign combinatorial spaces that exist in two states: convolute (tranquility) and deployed (dynamics), are considered. Spaces, in particular biological, physical, informational and some others, for which the axioms of sign combinatorial spaces, are valid, have a combinatorial nature. When they are deployed, combinatorial numbers (Fibonacci numbers) are formed, through which logarithmic spirals appear in living nature. These spirals are formed due to the finite sequences that take place during the deployment of the agreed spaces and which are presented geometrically using polar coordinates. Formulation of the problem. The logarithmic spiral is geometrically represented through a “golden rectangle” in which one side is 1,618 times longer (“golden” number or golden section). The presence of the golden ratio in nature is manifested through Fibonacci numbers, which are formed from an arithmetic triangle from elements of finite sequences formed by the deployment of sign combinatorial spaces. But this spiral is transmitted through the “golden rectangle” indirectly. The problem is to trace its formation in nature through constructed sequences, the elements of which are represented by polar coordinates. The approach proposed. Using the finite sequences that are formed during the unfolding of sign combinatorial spaces and the representation of their elements in polar coordinates, we can trace the dynamics of the formation of logarithmic spirals in nature. Conclusion. Representation of biological space as a sign combinatorial space can explain various phenomena in nature. When unfolding these spaces from the convolute spaces finite sequences are formed, the sums of the members of which determine the number of combinatorial configurations in a subset of isomorphic combinatorial configurations and form an arithmetic triangle (Pascal’s triangle). Fibonacci numbers and, accordingly, a golden number are formed from an arithmetic triangle. The logarithmic spiral fits into a golden rectangle. The dynamics of the formation of the logarithmic spiral is traced due to the finite sequences formed as a result of the deployment of the sign combinatorial spaces, the elements of which are presented in polar coordinates.

Условия разрешимости задачи Неймана N2 для полигармонического уравнения в шаре

The N k Neumann-type class of problems for a polyharmonic equation in the unit ball is considered. The problems of this class generalize both the well-known Dirichlet problem and the Neumann problem. In a number of works, the set of the necessary conditions for the solvability of this problem has been found for the problems of such class, and it has been assumed that the most complete version of the found necessary conditions is also a set of the sufficient conditions for the solvability of the problem. This was a known fact with regard to the N 1 problem. In this study, an assumption that the found set of the necessary conditions coincides with the sufficient conditions of solvability of the N 2 problem for a homogeneous m -harmonic equation in a unit ball is proved. First, by changing the variables, the N 2 problem is reduced to a simpler N 0 . Dirichlet problem, the solution to which is considered to be known. Next, the conditions, under which the performed change of the variables is reversible, are found. The conditions found here are connected with the Dirichlet problem's solution having terms of the first order of smallness in the expansion in the neighborhood of zero. Finally, the previously obtained results are used, which concern the value of the m- harmonic function in the unit ball in the center of the ball with the values of the normal derivatives of this function at the boundary of the ball. These solvability conditions are transformed to the conditions associated with the values of the integrals over the sphere of polynomials in the normal derivatives of the desired solution on the unit sphere, the coefficients of which are the elements of the arithmetic Neumann triangle. The found conditions coincide with the previously obtained necessary conditions for the solvability of the N 2 problem.

Computing Sticks against Random Walk

A new deterministic model with the help of geometric constructions and computing sticks (not related to trajectories) is proposed for the new justification of consistency of the probabilistic approach to explain the random walk on a plane. A new, stepped form of the arithmetic triangle of Pascal based on the construction of horizontal and vertical lines (arrows) is suggested, a comparison is made with Pascal’s triangle of the usual form. A two-sided generalization of Pascal’s triangle is proposed. Geometric constructions and formulas for calculating the coefficients that fill in these new geometric (arithmetic) figures are given. Further types of generalization of the step-shaped Pascal triangle are proposed. Examples of generalized initial conditions and generalized recursive formulas for constructing various types of a generalized Pascal triangle are given.

On an arithmetic triangle of numbers arising from inverses of analytic functions

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).

Algebraic curves uniformized by congruence subgroups of triangle groups

We construct certain subgroups of hyperbolic triangle groups which we call \congruence subgroups. These groups include the classical congruence subgroups of SL2(Z), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the eld of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(Fq) regularly as Galois groups.

Proof of a conjecture of Kleinberg-Sawin-Speyer

Proof of a conjecture of Kleinberg-Sawin-Speyer, Discrete Analysis 2018:13, 7 pp. In the present paper the author constructs an $S_3$-symmetric probability distribution on $\{(a,b,c)\in\mathbb{Z}_{\geq 0}^3:a+b+c=p-1\}$ such that its marginal achieves the maximum entropy among all probability distributions on {0,1,…,p-1} with mean $(p-1)/3$. This proves a conjecture that appeared in the original preprint version of a [paper of Kleinberg, Sawin and Speyer](http://discreteanalysisjournal.com/article/3734-the-growth-rate-of-tri-colored-sum-free-sets), which is being published in Discrete Analysis alongside the present paper. Specifically, conditional on the existence of a probability distribution with the aforementioned properties, Kleinberg Sawin and Speyer established the existence of so-called tri-coloured sum-free sets in $(\mathbb{Z}/q\mathbb{Z})^n$ of size at least $\theta^{n-o(n)}$. A tri-coloured sum-free set in $(\mathbb{Z}/q\mathbb{Z})^n$ is a collection of triples $\{(a_i,b_i,c_i): a_i,b_i,c_i \in (\mathbb{Z}/q\mathbb{Z})^n\}$ such that $a_i+b_j+c_k=0$ if and only if $i=j=k$. A previous [paper by Blasiak, Church, Cohn, Grochow, Naslund, Sawin and Umans](http://discreteanalysisjournal.com/article/1245), also published in Discrete Analysis, had given an upper bound of $3\theta_q^n$ on the size of such a set, where $\theta_q < q$ is a constant only depending on $q$. The upper bound of Blasiak et al., obtained using the breakthrough polynomial method of Croot-Lev-Pach and Ellenberg-Gijswijt, is therefore shown to be approximately sharp by the result in the present paper. This matters for several reasons. First, it shows that this new polynomial method can lead to sharp bounds. In the original application to the cap set problem in $\mathbb{F}_3^n$, sharpness has not (yet) been established. Second, tri-coloured sum-free sets have connections to certain approaches to the fast matrix-multiplication problem, and to long-standing open problems in property testing. For example, Fox and Lovász used the Kleinberg-Sawin-Speyer result to prove that their polynomial bound for Green’s arithmetic triangle removal lemma, also obtained using a variant of the polynomial method, is essentially sharp. Moreover, Cohn, Kleinberg, Szegedy and Umans showed that it is impossible to achieve an exponent of $\omega=2$ for the matrix multiplication problem using sets satisfying the simultaneous triple-product property in the abelian groups $\mathbb{F}_p^n$. The present paper therefore contributes a fundamental ingredient to a result that cements the power of the Croot-Lev-Pach polynomial method.

Fibonacci words in hyperbolic Pascal triangles

Abstract The hyperbolic Pascal triangle HPT 4,q (q ≥ 5) is a new mathematical construction, which is a geometrical generalization of Pascal’s arithmetical triangle. In the present study we show that a natural pattern of rows of HPT 4,5 is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the hyperbolic Pascal triangles. The geometrical properties of a HPT 4,q imply a graph structure between the finite Fibonacci words.

A tight bound for Green's arithmetic triangle removal lemma in vector spaces

Let p be a fixed prime. A triangle in Fpn is an ordered triple (x,y,z) of points satisfying x+y+z=0. Let N=pn=|Fpn|. Green proved an arithmetic triangle removal lemma which says that for every ϵ>0 and prime p, there is a δ>0 such that if X,Y,Z⊂Fpn and the number of triangles in X×Y×Z is at most δN2, then we can delete ϵN elements from X, Y, and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1/δ can be taken to be an exponential tower of twos of height logarithmic in 1/ϵ.We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in Fpn. We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is an explicit number Cp such that we may take δ=(ϵ/3)Cp, and we must have δ≤ϵCp−o(1). In particular, C2=1+1/(5/3−log2⁡3)≈13.239, and C3=1+1/c3 with c3=1−log⁡blog⁡3, b=a−2/3+a1/3+a4/3, and a=33−18, which gives C3≈13.901. The proof uses the essentially sharp bound on multicolored sum-free sets due to work of Kleinberg–Sawin–Speyer, Norin, and Pebody, which builds on the recent breakthrough on the cap set problem by Croot–Lev–Pach, and the subsequent work by Ellenberg–Gijswijt, Blasiak–Church–Cohn–Grochow–Naslund–Sawin–Umans, and Alon.

Quantum unique ergodicity and the number of nodal domains of eigenfunctions

We prove that the Hecke-Maass eigenforms for a compact arithmetic triangle group have a growing number of nodal domains as the eigenvalue tends to + ∞ +\infty . More generally the same is proved for eigenfunctions on negatively curved surfaces that are even or odd with respect to a geodesic symmetry and for which quantum unique ergodicity holds.

Arithmetic Triangle

The product of the first $n$ terms of an arithmetic progression may be developed in a polynomial of $n$ terms. Each one of them presents a coefficient $C_{nk}$ that is independent from the initial term and the common difference of the progression. The most interesting point is that one may construct an "Arithmetic Triangle'', displaying these coefficients, in a similar way one does with Pascal's Triangle. Moreover, some remarkable properties, mainly concerning factorials, characterize the Triangle. Other related `triangles' -- eventually treated as matrices -- also display curious facts, in their linear \emph{modus operandi}, such as successive "descendances''.

On the arithmetic dimension of triangle groups

Let $\Delta=\Delta(a,b,c)$ be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles $\pi/a,\pi/b,\pi/c$ drawn on the hyperbolic plane. We define the arithmetic dimension of $\Delta$ to be the number of split real places of the quaternion algebra generated by $\Delta$ over its (totally real) invariant trace field. Takeuchi has determined explicitly all triples $(a,b,c)$ with arithmetic dimension $1$, corresponding to the arithmetic triangle groups. We show more generally that the number of triples with fixed arithmetic dimension is finite, and we present an efficient algorithm to completely enumerate the list of triples of bounded arithmetic dimension.

To the Hilbert class field from the hypergeometric modular function

In this article we make an explicit approach to the problem: “For a given CM field M, construct its maximal unramified abelian extension C(M) by the adjunction of special values of certain modular functions” in some restricted cases with [M:Q]≥4. We make our argument based on Shimura's main result on the complex multiplication theory of his article in 1967. His main result treats CM fields embedded in a quaternion algebra B over a totally real number field F. We determine the modular function which gives the canonical model for all B's coming from arithmetic triangle groups. That is our main theorem. As its application, we make an explicit case-study for B corresponding to the arithmetic triangle group Δ(3,3,5). By using the modular function of K. Koike obtained in 2003, we show several examples of the Hilbert class fields as an application of our theorem to this triangle group.

On the arithmetic triangle arising from the solvability conditions for the Neumann problem

We study the arithmetic triangle arising from the solvability conditions for the Neumann problem for the polyharmonic equation in the unit ball. Recurrence relations for the elements of this triangle are obtained.

On an Arithmetical Triangle

We derive and explicitly solve recurrent equations with boundary conditions for coefficients in the solvability condition for the Neumann problem for the polyharmonic equation in a unit ball. Bibliography: 9 titles.