Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Apery’s constant. (1)
- Bernoulli number. (1)
- Bernoulli polynomial (1)
- Completely multiplicative function (1)
- Dirichlet series (1)
-
- Euler polynomial (1)
- Euler’s formula (1)
- Multiplicative function (1)
- Multivariate Riordan array pair (1)
- Multivariate Sheffer-type differential operators (1)
- Multivariate Sheffer-type polynomials (1)
- Multivariate exponential polynomials. (1)
- Multivariate formal power series (1)
- Multivariate weighted Stirling numbers (1)
- M¨obius inversion (1)
- Riemann Zeta function (1)
- Riemann zeta function (1)
- Selberg multiplicative functions. (1)
Articles 1 - 6 of 6
Full-Text Articles in Entire DC Network
An Euler-Type Formula For Ζ(2k +1), Tian-Xiao He, Michael Dancs
An Euler-Type Formula For Ζ(2k +1), Tian-Xiao He, Michael Dancs
Scholarship
In this short paper, we give several new formulas for ζ(n) when n is an odd positive integer. The method is based on a recent proof, due to H. Tsumura, of Euler’s classical result for even n. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult.
Numerical Approximation To Ζ(2n+1), Tian-Xiao He, Michael Dancs
Numerical Approximation To Ζ(2n+1), Tian-Xiao He, Michael Dancs
Scholarship
In this short paper, we establish a family of rapidly converging series expansions ζ(2n +1) by discretizing an integral representation given by Cvijovic and Klinowski [3] in Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math. 142 (2002) 435–439. The proofs are elementary, using basic properties of the Bernoulli polynomials.
Multivariate Expansion Associated With Sheffer-Type Polynomials And Operators, Tian-Xiao He, Leetsch Hsu, Peter Shiue
Multivariate Expansion Associated With Sheffer-Type Polynomials And Operators, Tian-Xiao He, Leetsch Hsu, Peter Shiue
Scholarship
With the aid of multivariate Sheffer-type polynomials and differential operators, this paper provides two kinds of general expansion formulas, called respectively the first expansion formula and the second expansion formula, that yield a constructive solution to the problem of the expansion of A(ˆt)f([g(t)) (a composition of any given formal power series) and the expansion of the multivariate entire functions in terms of multivariate Sheffer-type polynomials, which may be considered an application of the first expansion formula and the Sheffer-type operators. The results are applicable to combinatorics and special function theory.
On The Convergence Of The Summation Formulas Constructed By Using A Symbolic Operator Approach, Tian-Xiao He, Leetsch Hsu, Peter Shiue
On The Convergence Of The Summation Formulas Constructed By Using A Symbolic Operator Approach, Tian-Xiao He, Leetsch Hsu, Peter Shiue
Scholarship
This paper deals with the convergence of the summation of power series of the form Σa ≤ k ≤ bf(k)xk, where 0 ≤ a ≤ b < ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) a differentiable function defined on [a, b). Here, the summation is found by using the symbolic operator approach shown in [1]. We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Several examples such as the generalized Euler's transformation series will also be given. In addition, we will compare the convergence of the given series transforms.
On The Generalized Möbius Inversion Formulas, Tian-Xiao He, Peter Shiue3, Leetsch Hsu
On The Generalized Möbius Inversion Formulas, Tian-Xiao He, Peter Shiue3, Leetsch Hsu
Scholarship
We provide a wide class of M¨obius inversion formulas in terms of the generalized M¨obius functions and its application to the setting of the Selberg multiplicative functions.
Construction Of Rational Points On Elliptic Curves Over Finite Fields, Andrew Shallue, Christiaan Van De Woestijne
Construction Of Rational Points On Elliptic Curves Over Finite Fields, Andrew Shallue, Christiaan Van De Woestijne
Scholarship
We give a deterministic polynomial-time algorithm that computes a nontrivial rational point on an elliptic curve over a finite field, given a Weierstrass equation for the curve. For this, we reduce the problem to the task of finding a rational point on a curve of genus zero.