184 research outputs found
On irregular prime power divisors of the Bernoulli numbers
Let () denote the usual -th Bernoulli number. Let
be a positive even integer where or . It is well known
that the numerator of the reduced quotient is a product of powers of
irregular primes. Let be an irregular pair with B_l/l \not\equiv
B_{l+p-1}/(l+p-1) \modp{p^2}. We show that for every the congruence
B_{m_r}/m_r \equiv 0 \modp{p^r} has a unique solution where m_r \equiv
l \modp{p-1} and . The sequence
defines a -adic integer which is a zero of a certain
-adic zeta function originally defined by T. Kubota and H. W.
Leopoldt. We show some properties of these functions and give some
applications. Subsequently we give several computations of the (truncated)
-adic expansion of for irregular pairs with
below 1000.Comment: 42 pages; final accepted paper, slightly revised and extended, to
appear in Math. Com
Structure of the cuspidal rational torsion subgroup of J_1(p^n)
In this article, we determine the structure of the -primary subgroup of
the cuspidal rational torsion subgroup of the Jacobian of the
modular curve for a regular prime .Comment: 26 page
A conjecture about numerators of Bernoulli numbers related to Integer Sequence A092291
In this paper we disprove a conjecture about numerators of divided Bernoulli
numbers and which was suggested by Roland Bacher. We give
some counterexamples. Finally, we extend the results to the general case.Comment: 11 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
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