Diophantine properties for $q$-analogues of Dirichlet’s beta function at positive integers
نویسندگان
چکیده
منابع مشابه
q-ANALOGUES OF THE SUMS OF POWERS OF CONSECUTIVE INTEGERS
Let n, k be the positive integers (k > 1), and let Sn,q(k) be the sums of the n-th powers of positive q-integers up to k − 1: Sn,q(k) = ∑k−1 l=0 ql. Following an idea due to J. Bernoulli, we explore a formula for Sn,q(k).
متن کاملAdditive and Multiplicative Properties of Beta - Integers
| To each number > 1 there corresponds a discrete countable set of numbers denoted by Z and named set of beta-integers. The set Z is precisely the set of real numbers which are polynomial in when they are written in \basis ", and Z = Zwhen 2 N. We prove here a list of arithmetic properties of Z : addition, multiplication, relation with integers, when is a quadratic Pisot-Vijayaraghavan unit (qu...
متن کاملDiophantine approximation by conjugate algebraic integers
Building on the work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel’fond’s transcendence criterion which provides a sufficient condition for a complex or p-adic number ξ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at ξ together with most of their derivatives. The second one, which follows from this crit...
متن کاملq-Analogues of the Sums of Consecutive Integers, Squares, Cubes, Quarts and Quints
We first show how a special case of Jackson’s 8φ7 summation immediately gives Warnaar’s q-analogue of the sum of the first n cubes, as well as q-analogues of the sums of the first n integers and first n squares. Similarly, by appropriately specializing Bailey’s terminating very-well-poised balanced 10φ9 transformation and applying the terminating very-well-poised 6φ5 summation, we find q-analog...
متن کاملDIOPHANTINE PROBLEMS FOR q-ZETA VALUES 1
hold. The above defined q-zeta values (1) present several new interesting problems in the theory of diophantine approximations and transcendental numbers; these problems are extensions of the corresponding problems for ordinary zeta values and we state some of them in Section 3 of this note. Our nearest aim is to demonstrate how some recent contributions to the arithmetic study of the numbers ζ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2011
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-2010-05236-5