Tangent Numbers Research Articles

Generalized (h,q)-tangent numbers and polynomials

In this paper we introduce the generalized (h,q)-tangent numbers T (h) n,χ,q and polynomials T (h) n,χ,q(x). Some interesting results and relationships are obtained.

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

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

On Carlitz's type twisted tangent numbers and polynomials associated with p-adic integral on Zp

On Carlitz's type twisted tangent numbers and polynomials associated with p-adic integral on Zp

Multiple tangent zeta function and tangent polynomials of higher order

In this paper, we introduce Tangent numbers T (k) n and Tangent polynomials T (k) n (x) of higher order. Finally, we obtain interesting properties of these numbers and polynomials.

On the twisted tangent numbers and polynomials of higher order associated with multiple twisted tangent zeta function

On the twisted tangent numbers and polynomials of higher order associated with multiple twisted tangent zeta function

Symmetric identities for the generalized tangent polynomials associated with p-adic integral on Zp

In this paper, we study the symmetry for the generalized tangent numbers Tn,χ and polynomials Tn,χ(x). We obtain some interesting identities of the power sums and the generalized polynomials Tn,χ(x) using the symmetric properties for the p-adic invariant integral on Zp. Mathematics Subject Classification: 11B68, 11S40, 11S80

Probabilistic Proofs of Euler Identities

Formulae for ζ(2n) andLχ4(2n+ 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2/ 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

Probabilistic Proofs of Euler Identities

Formulae for ζ(2n) andLχ4(2n+ 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2/ 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

On the analogues of tangent numbers and polynomials associated with p-adic integral on Zp

In this paper we introduce the twisted tangent numbers Tn,w and polynomials Tn,w(x). Some interesting results and relationships are obtained.

On the (h,q)-tangent numbers and polynomials

On the (h,q)-tangent numbers and polynomials

A note on the tangent numbers and polynomials

In this paper we introduce the tangent numbers Tn and polynomials Tn(x). Some interesting results and relationships are obtained. Mathematics Subject Classification: 11B68, 11S40, 11S80

A note on the symmetric properties for the tangent polynomials

In [4], we studied the tangent numbers and polynomials. By using these numbers and polynomials, we give some interesting relations between the power sums and the tangent polynomials.

Symmetric properties for the twisted tangent polynomials

In [4], we studied the twisted tangent numbers and polynomials. By using these numbers and polynomials, we give some interesting relations between the power sums and the the twisted tangent polynomials. Mathematics Subject Classification: 11B68, 11S40, 11S80

Numerical investigation on the structure of the zeros of the twisted tangent polynomials

In [3], we introduced the twisted tangent numbers Tn,w and polynomials Tn,w(x). In this paper, we observe the distribution of complex roots of the twisted tangent polynomials Tn,w(x), using numerical investigation. Finally, we give a table for the solutions of the twisted tangent polynomials Tn,w(x).

Generalized tangent numbers and polynomials associated with p-adic integral on Zp

In this paper we introduce the generalized tangent numbers Tn,χ and polynomials Tn,χ(x). Some interesting results and relationships are obtained.

Closed planar curves without inflections

We define a computable topological invariant μ ( γ ) \mu (\gamma ) for generic closed planar regular curves γ \gamma , which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant μ ( γ ) \mu (\gamma ) of a given γ \gamma .

A Note on Eulerian Polynomials

We study Genocchi, Euler, and tangent numbers. From those numbers we derive some identities on Eulerian polynomials in connection with Genocchi and tangent numbers.

Symmetry Fermionic -Adic -Integral on for Eulerian Polynomials

Kim et al. (2012) introduced an interestingp-adic analogue of the Eulerian polynomials. They studied some identities on the Eulerian polynomials in connection with the Genocchi, Euler, and tangent numbers. In this paper, by applying the symmetry of the fermionicp-adicq-integral on , defined by Kim (2008), we show a symmetric relation between theq-extension of the alternating sum of integer powers and the Eulerian polynomials.

Bilabelled increasing trees and hook-length formulae

We introduce two different kinds of increasing bilabellings of trees, for which we provide enumeration formulae. One of the bilabelled tree families considered is enumerated by the reduced tangent numbers and is in bijection with a tree family introduced by Poupard [11]. Both increasing bilabellings naturally lead to hook-length formulae for trees and forests; in particular, one construction gives a combinatorial interpretation of a formula for labelled unordered forests obtained recently by Chen et al. [1].

The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

This paper was motivated by a conjecture of Brändén [P. Brändén, Actions on permutations and unimodality of descent polynomials, European J. Combin. 29 (2) (2008) 514–531] about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a ( p , q ) -analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). The ( p , q ) -analogue unifies and generalizes our recent results [H. Shin, J. Zeng, The q -tangent and q -secant numbers via continued fractions, European J. Combin. 31 (7) (2010) 1689–1705] and that of Josuat-Vergès [M. Josuat-Vergés, A q -enumeration of alternating permutations, European J. Combin. 31 (7) (2010) 1892–1906].