Tangent Numbers Research Articles

Factorial transformation for some classical combinatorial sequences

Factorial transformation known from Euler’s time is a very powerful tool for summation of divergent power series. We use factorial series for summation of ordinary power generating functions for some classical combinatorial sequences. These sequences increase very rapidly, so OGFs for them diverge and mostly unknown in a closed form. We demonstrate that factorial series for them are summable and expressed in known functions. We consider among others Stirling, Bernoulli, Bell, Euler and Tangent numbers. We compare factorial transformation with other summation techniques such as Pade approximations, transformation to continued fractions, and Borel integral summation. This allowed us to derive some new identities for GFs and express their integral representations in a closed form.

Nonlinear differential equations associated with degenerate tangent numbers

Nonlinear differential equations associated with degenerate tangent numbers

The Monotonicity and Log-Behaviour of Some Functions Related to the Euler Gamma Function

AbstractThe aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functionsis considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequenceis proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functionsis also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

DIFFERENTIAL EQUATIONS ASSOCIATED WITH TANGENT NUMBERS

DIFFERENTIAL EQUATIONS ASSOCIATED WITH TANGENT NUMBERS

On the Generalized Degenerate Tangent Numbers and Polynomials

In [7], Ryoo introduced the generalized tangent numbers and polynomials. In this paper, our goal is to give generating functions of the degenerate generalized tangent numbers and polynomials. We also obtain some explicit formulas for degenerate generalized tangent numbers and polynomials.

СЕЧЕНИЯ ЧИСЛОВОЙ ПРИЗМЫ, СВЯЗАННЫЕ С ПОЛИНОМАМИ БЕССЕЛЯ

The integer set previously obtained by the author in the study of moments and cumulants of three-parameter probability distribution of the hyperbolic cosine type is considered. This distribution is a generalization of Meixner two-parameter distribution. Moments of distribution at specific parameters vary as a certain class of polynomials with the corresponding coefficients. On the basis of the differential ratio of polynomials, recurrence formulas for their coefficients are received. The set of polynomial coefficients { U ( n ; k , j )} that depends on three indices, and which is formed by these formulas, is the object of study. The set is structured in the form of a numeric prism. When fixing one or two indices or functional connection between the indices, different sections of numerical prisms are obtained: number triangles or number sequences. Among the sections of the numerical prism are both known (Stirling triangle, tangential numbers, secant numbers, etc.) and new integer sets. Classic Bessel triangle enters into the considered numerical prism as a section { U (2 n–j ; n , j )}, where n = 0, 1, 2, …, j = 0, 1, 2, … n . In this section the sequences classified as coefficients in the Bessel polynomials are determined. Based on the theoretical developments related to the Bessel polynomials, dependences and relations for a number of elements of numerical prism are found and justified. The obtained results also allow putting sequences through the values of hypergeometric functions and modified Bessel functions of the second kind. Considered set differs in the ease of construction, and its study has revealed previously unknown properties and relations of various mathematical objects (sequences, polynomials, functions, etc.), particularly related to the Bessel polynomials

Notes on degenerate twisted (h,q)-tangent numbers and polynomials associated with the p-adic integral on Zp

Notes on degenerate twisted (h,q)-tangent numbers and polynomials associated with the p-adic integral on Zp

Closed formulas for computing higher-order derivatives of functions involving exponential functions

For integers k ≥ 1 and n ≥ 0, the functions 1/(1−λeαt)k and the derivatives (1/(1−λeαt))(n) can be expressed each other by linear combinations. Based on this viewpoint, we find several new closed formulas for higher-order derivatives of trigonometric and hyperbolic functions, derive a higher-order convolution formula for the tangent numbers, and generalize a recurrence relation for the tangent numbers.

Derivatives of tangent function and tangent numbers

In the paper, by induction, the Faà di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the sine and cosine functions, obtains explicit formulas for two Bell polynomials of the second kind for successive derivatives of sine and cosine functions, presents curious identities for the sine function, discovers explicit formulas and recurrence relations for the tangent numbers, the Bernoulli numbers, the Genocchi numbers, special values of the Euler polynomials at zero, and special values of the Riemann zeta function at even numbers, and comments on five different forms of higher order derivatives for the tangent function and on derivative polynomials of the tangent, cotangent, secant, cosecant, hyperbolic tangent, and hyperbolic cotangent functions.

Regular systems of paths and families of convex sets in convex position

In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies have at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies have at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Toth, and generalizes a theorem of Bisztriczky and Fejes Toth. Our results on families of convex bodies are consequences of more general Ramsey-type results about the crossing patterns of systems of graphs of continuous functions $f:[0,1] \to \mathbb{R}$. On our way towards proving the Pach-Toth conjecture we obtain a combinatorial characterization of such systems of graphs in which all subsystems of equal size induce equivalent crossing patterns. These highly organized structures are what we call regular systems of paths and they are natural generalizations of the notions of cups and caps from the famous theorem of Erdos and Szekeres. The characterization of regular systems is combinatorial and introduces some auxiliary structures which may be of independent interest.

The Complete cd-Index of Boolean Lattices

Let $[u,v]$ be a Bruhat interval of a Coxeter group such that the Bruhat graph $BG(u,v)$ of $[u,v]$ is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of $[u,v]$. Since in this case the complete cd-index and the cd-index of $[u,v]$ coincide, we also obtain a new combinatorial interpretation for the coefficients of the cd-index of Boolean lattices. To this end, we label an edge in $BG(u,v)$ by a pair of nonnegative integers and show that there is a one-to-one correspondence between such sequences of nonnegative integer pairs and Bruhat paths in $BG(u,v)$. Based on this labeling, we construct a flip $\mathcal{F}$ on the set of Bruhat paths in $BG(u,v)$, which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip $\mathcal{F}$ is compatible with any given reflection order and also satisfies the flip condition for any cd-monomial $M$. Thus by results of Karu, the coefficient of $M$ enumerates certain Bruhat paths in $BG(u,v)$, and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip $\mathcal{F}$. We enumerate the number of cd-monomials in the complete cd-index of $[u,v]$ in terms of Entringer numbers, which are refined enumerations of Euler numbers. We also give a refined enumeration of the coefficient of d${}^n$ in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.

Computing Shortest Paths among Curved Obstacles in the Plane

A fundamental problem in computational geometry is to compute an obstacle-avoiding Euclidean shortest path between two points in the plane. The case of this problem on polygonal obstacles is well studied. In this article, we consider the problem version on curved obstacles, which are commonly modeled as splinegons . A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge (polygons are special splinegons), and the combinatorial complexity of each curved edge is assumed to be O (1). Given in the plane two points s and t and a set s of h pairwise disjoint splinegons with a total of n vertices, after a bounded degree decomposition of S is obtained, we compute a shortest s -to- t path avoiding the splinegons in O ( n + h log h + k ) time, where k is a parameter sensitive to the geometric structures of the input and is upper bounded by O ( h 2 ). The bounded degree decomposition of S , which is similar to the triangulation of the polygonal domains, can be computed in O ( n log n ) time or O ( n + h log 1 + ϵ h ) time for any ϵ > 0. In particular, when all splinegons are convex, the decomposition can be computed in O ( n + h log h ) time and k is linear to the number of common tangents in the free space (called “free common tangents”) among the splinegons. Our techniques also improve several previous results: (1) For the polygon case (i.e., when all splinegons are polygons), the shortest path problem was previously solved in O ( n log n ) time, or in O ( n + h 2 log n ) time. Thus, our algorithm improves the O ( n + h 2 log n ) time result, and is faster than the O ( n log n ) time solution for sufficiently small h , for example, h = o (√ n ,log n . (2) Our techniques produce an optimal output-sensitive algorithm for a basic visibility problem of computing all free common tangents among h pairwise disjoint convex splinegons with a total of n vertices. Our algorithm runs in O ( n + h log h + k ) time and O ( n ) working space, where k is the number of all free common tangents. Note that k = O ( h 2 ). Even for the special case where all splinegons are convex polygons , the previously best algorithm for this visibility problem takes O ( n + h 2 log n ) time. (3) We improve the previous work for computing the shortest path between two points among convex pseudodisks of O (1) complexity each. In addition, a by-product of our techniques is an optimal O ( n + h log h ) time and O ( n ) space algorithm for computing the Voronoi diagram of a set of h pairwise disjoint convex splinegons with a total of n vertices.

Carlitz's type twisted (h,q)-tangent numbers and polynomials

In this paper we dene the Carlitz’s type twisted ( h;q)-tangent numbers T (h) n;q;w and polynomials T (h) n;q;w(x). By using Carlitz’s type twisted (h;q)-tangent numbers and polynomials, twisted (h;q)-tangent zeta function and Hurwitz twisted (h;q)-tangent zeta functions are dened. Mathematics Subject Classication: 11B68, 11S40, 11S80

Q-Extension of tangent numbers and polynomials associated with the p-adic q-integral on Zp

q-Extension of tangent numbers and polynomials associated with the p-adic q-integral on Zp

Symmetric properties for the degenerate tangent polynomials associated with p-adic integral on Zp

In [7], we studied the degenerate tangent numbers and polynomials. By using these numbers and polynomials, we give some interesting relations between the generalized falling factorial sums and the degenerate tangent polynomials. Mathematics Subject Classication: 11B68, 11S40, 11S80

Numerical investigation of the zeros of q-extension of tangent polynomials

In [4], we introduced the q-extension of tangent numbers Tn;q and polynomials Tn;q(x). In this paper, we observe the distribution of complex roots of the q-extension of tangent polynomials Tn;q(x), using numerical investigation. Finally, we give a table for the solutions of the q-extension of tangent polynomials Tn;q(x). Mathematics Subject Classication: 11B68, 11S40, 11S80

A note on q-extension of twisted tangent numbers and polynomials associated with the p-adic q-integral on Zp

A note on q-extension of twisted tangent numbers and polynomials associated with the p-adic q-integral on Zp

Automatic differentiation for stress and consistent tangent computation

In the field of nonlinear finite element analysis, the linearization within the Newton–Raphson and the Multilevel-Newton–Raphson algorithm requires the computation of the stresses and the so-called consistent tangent operator. There are several possibilities for coding the derivatives either by deriving analytical expressions, by using numerical differentiation schemes, or by applying the concept of automatic differentiation. In this article, the three possibilities are compared for three different kinds of constitutive models in order to ascertain particular differences. These models are a finite strain hyperelasticity relation, a more sophisticated finite strain viscoplasticity model, and a small strain von Mises-type thermo-viscoplasticity model. Especially for the last example, it is required to provide a number of tangents to be derived within a monolithic, fully coupled finite element computation. Numerical examples investigate the advantages and disadvantages of analytical, numerical, and automatic differentiation. It is shown that the latter one is a very helpful tool in the developing stage of constitutive modeling.

A note on generalized twisted (h; q)-tangent numbers and polynomials

In this paper we construct the generalized twisted (h;q)-tangent numbers T (h) n;;q;w and polynomials T (h) n;;q;w (x). Some interesting results and relationships are obtained.

A note on the Carlitz's type (h,q)-tangent numbers and polynomials associated with p-adic integral on Zp

In this paper we construct the Carlitz’s type (h;q)-tangent numbers T (h) n;q and polynomials T (h) n;q (x). From these numbers and polynomials,