A Note on Bernoulli-Goss Polynomials

1984 ◽  
Vol 27 (2) ◽  
pp. 179-184 ◽  
Author(s):  
K. Ireland ◽  
D. Small
Keyword(s):  
Closed Form ◽  
Existence Theorem ◽  
Function Fields ◽  
Bernoulli Polynomials ◽  
Bernoulli Polynomial ◽  
Computer Search ◽  
Quadratic Polynomials ◽  
Fermat's Equation

AbstractIn an important series of papers ([3], [4], [5]), (see also Rosen and Galovich [1], [2]), D. Goss has developed the arithmetic of cyclotomic function fields. In particular, he has introduced Bernoulli polynomials and proved a non-existence theorem for an analogue to Fermat’s equation for regular “exponent”. For each odd prime p and integer n, l ≤ n ≤ p2-2 we derive a closed form for the nth Bernoulli polynomial. Using this result a computer search for regular quadratic polynomials of the form x2-a was made. For primes less than or equal to 269 regular quadratics exist for p= 3, 5, 7, 13, 31.

Bernoulli and poly-Bernoulli polynomial convolutions and identities of p-adic Arakawa–Kaneko zeta functions

2016 ◽  
Vol 12 (05) ◽  
pp. 1295-1309 ◽  
Author(s):  
Paul Thomas Young
Keyword(s):  
Closed Form ◽  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Bernoulli Polynomial ◽  
Sum Formula

We evaluate the ordinary convolution of Bernoulli polynomials in closed form in terms of poly-Bernoulli polynomials. As applications we derive identities for [Formula: see text]-adic Arakawa–Kaneko zeta functions, including a [Formula: see text]-adic analogue of Ohno’s sum formula. These [Formula: see text]-adic identities serve to illustrate the relationships between real periods and their [Formula: see text]-adic analogues.

CLOSED FORMS FOR DEGENERATE BERNOULLI POLYNOMIALS

2020 ◽  
Vol 101 (2) ◽  
pp. 207-217 ◽  
Author(s):  
LEI DAI ◽  
HAO PAN
Keyword(s):  
Number Theory ◽  
Closed Form ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Closed Form Expression ◽  
Form Expression

Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.

scholarly journals Sums of Products Involving Power Sums of φ(n) Integers

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jitender Singh
Keyword(s):  
Closed Form ◽  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Rational Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Power Sums ◽  
Power Sum ◽  
Mobius Function ◽  
Sums Of Products

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums Ψk(x,n):=∑d|n‍μ(d)dkSkx/d,  n∈ℤ+ which are defined via the Möbius function μ and the usual power sum Sk(x) of a real or complex variable x. The power sum Sk(x) is expressible in terms of the well-known Bernoulli polynomials by Sk(x):=(Bk+1(x+1)-Bk+1(1))/(k+1).

scholarly journals Bernoulli polynomials collocation for weakly singular Volterra integro-differential equations of fractional order

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3623-3635 ◽  
Author(s):  
Haman Azodi ◽  
Mohammad Yaghouti
Keyword(s):  
Differential Equations ◽  
Numerical Procedure ◽  
Bernoulli Polynomials ◽  
Bernoulli Polynomial ◽  
Matrix Form ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Weakly Singular ◽  
The Matrix ◽  
Singular Kernels

This paper is concerned with a numerical procedure for fractional Volterra integro-differential equations with weakly singular kernels. The fractional derivative is in the Caputo sense. In this study, Bernoulli polynomial of first kind is used and its matrix form is given. Then, the matrix form based on the collocation points is constructed for each term of the problem. Hence, the proposed scheme simplifies the problem to a system of algebraic equations. Error analysis is also investigated. Numerical examples are announced to demonstrate the validity of the method.

scholarly journals On the Existence of Unramified Separable Infinite Solvable Extensions of Function Fields over Finite Fields

1958 ◽  
Vol 13 ◽  
pp. 95-100
Author(s):  
Hisasi Morikawa
Keyword(s):  
Finite Field ◽  
Existence Theorem ◽  
Finite Fields ◽  
Present Note ◽  
Function Fields ◽  
Regular Extension ◽  
Solvable Extension ◽  
Dimension One

In the present note, using the results in the previous paper, we shall prove the following existence theorem:THEOREM. Let k be a finite field with q elements and K/k be a regular extension of dimension one over k. Then, if q ≧ 11 and the genus gK of K/k is greater than one, there exists an unramified separable infinite solvable extension of K ivhich is regular over k.

The Pintz–Steiger–Szemerédi estimate for intersective quadratic polynomials in function fields

2021 ◽  
pp. 1-50
Author(s):  
Guoquan Li
Keyword(s):  
Finite Field ◽  
Natural Number ◽  
Polynomial Ring ◽  
Function Fields ◽  
Difference Set ◽  
Quadratic Polynomial ◽  
Quadratic Polynomials ◽  
Number Formula ◽  
The Difference

Let [Formula: see text] be the polynomial ring over the finite field [Formula: see text] of [Formula: see text] elements. For a natural number [Formula: see text] let [Formula: see text] be the set of all polynomials in [Formula: see text] of degree less than [Formula: see text] Let [Formula: see text] be a quadratic polynomial over [Formula: see text] Suppose that [Formula: see text] is intersective, that is, which satisfies [Formula: see text] for any [Formula: see text] with [Formula: see text] where [Formula: see text] denotes the difference set of [Formula: see text] Let [Formula: see text] Suppose that [Formula: see text] and that the characteristic of [Formula: see text] is not divisible by 2. It is proved that [Formula: see text] for any [Formula: see text] where [Formula: see text] is a constant depending only on [Formula: see text] and [Formula: see text]

scholarly journals RADEMACHER CHAOSES AND BERNOULLI POLYNOMIALS

2017 ◽  
Vol 18 (3.1) ◽  
pp. 66-73
Author(s):  
R.S. Sukhanov
Keyword(s):  
Bernoulli Polynomials ◽  
Convergent Series ◽  
Bernoulli Polynomial ◽  
Odd Order ◽  
Series Of Functions

In this paper we prove that any Bernoulli polynomial of even (odd) order is an absolutely convergent series of functions from some Rademacher chaoses, each of them is of even (odd) order

Machine Learning of Motor Vehicle Accident Categories from Narrative Data

1996 ◽  
Vol 35 (04/05) ◽  
pp. 309-316 ◽  
Author(s):  
M. R. Lehto ◽  
G. S. Sorock
Keyword(s):  
Machine Learning ◽  
Bayesian Model ◽  
Keyword Search ◽  
Motor Vehicle ◽  
Motor Vehicle Accident ◽  
Computer Search ◽  
Vehicle Accident ◽  
Learning Technique ◽  
Expert Ratings ◽  
Keyword Searches

Abstract:Bayesian inferencing as a machine learning technique was evaluated for identifying pre-crash activity and crash type from accident narratives describing 3,686 motor vehicle crashes. It was hypothesized that a Bayesian model could learn from a computer search for 63 keywords related to accident categories. Learning was described in terms of the ability to accurately classify previously unclassifiable narratives not containing the original keywords. When narratives contained keywords, the results obtained using both the Bayesian model and keyword search corresponded closely to expert ratings (P(detection)≥0.9, and P(false positive)≤0.05). For narratives not containing keywords, when the threshold used by the Bayesian model was varied between p>0.5 and p>0.9, the overall probability of detecting a category assigned by the expert varied between 67% and 12%. False positives correspondingly varied between 32% and 3%. These latter results demonstrated that the Bayesian system learned from the results of the keyword searches.

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