2,547 research outputs found
Identities on the k-ary Lyndon words related to a family of zeta functions
The main aim of this paper is to investigate and introduce relations between
the numbers of k-ary Lyndon words and unified zeta-type functions which was
defined by Ozden et al [15, p. 2785]. Finally, we give some identities on
generating functions for the numbers of k-ary Lyndon words and some special
numbers and polynomials such as the Apostol-Bernoulli numbers and polynomials,
Frobenius-Euler numbers, Euler numbers and Bernoulli numbers.Comment: 9 page
On The Properties Of -Bernstein-Type Polynomials
The aim of this paper is to give a new approach to modified -Bernstein
polynomials for functions of several variables. By using these polynomials, the
recurrence formulas and some new interesting identities related to the second
Stirling numbers and generalized Bernoulli polynomials are derived. Moreover,
the generating function, interpolation function of these polynomials of several
variables and also the derivatives of these polynomials and their generating
function are given. Finally, we get new interesting identities of modified
-Bernoulli numbers and -Euler numbers applying -adic -integral
representation on and -adic fermionic -invariant integral
on , respectively, to the inverse of -Bernstein polynomials.Comment: 17 pages, some theorems added to new versio
General Convolution Identities for Bernoulli and Euler Polynomials
Using general identities for difference operators, as well as a technique of
symbolic computation and tools from probability theory, we derive very general
kth order (k \ge 2) convolution identities for Bernoulli and Euler polynomials.
This is achieved by use of an elementary result on uniformly distributed random
variables. These identities depend on k positive real parameters, and as
special cases we obtain numerous known and new identities for these
polynomials. In particular we show that the well-known identities of Miki and
Matiyasevich for Bernoulli numbers are special cases of the same general
formula.Comment: 20 page
Some New Identities on the Bernoulli and Euler Numbers
We give some new identities on the Bernoulli and Euler numbers by using the bosonic p-adic integral on Zp and reflection symmetric properties of Bernoulli and Euler polynomials
Derivatives of tangent function and tangent numbers
In the paper, by induction, the Fa\`a 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.Comment: 17 page
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