Results 11 to 20 of about 427 (155)
Acta Arithmetica, 2012 Let $c_q(n)$ denote the Ramanujan sum modulo $q$, and let $x$ and $y$ be large reals, with $x = o(y)$. We obtain asymptotic formulas for the sums $$\sum_{n \le y}(\sum_{q \le x} c_q(n))^k \qquad (k = 1, 2).$$
Chan, Tsz Ho, Kumchev, Angel V
openaire +2 more sources
Mellin transforms for some families of q-polynomials [PDF]
, 2003 By using Ramanujan's q-extension of the Euler integral representation for the gamma function, we derive the Mellin integral transforms for the families of the discrete q-Hermite II, the Al-Salam–Carlitz II, the big q-Laguerre, the big q-Legendre, the big
Atakishiyev Mektiyev, Natig +2 more
core +1 more source
Sums of products of Ramanujan sums [PDF]
ANNALI DELL'UNIVERSITA' DI FERRARA, 2011 The Ramanujan sum $c_n(k)$ is defined as the sum of $k$-th powers of the primitive $n$-th roots of unity. We investigate arithmetic functions of $r$ variables defined as certain sums of the products $c_{m_1}(g_1(k))...c_{m_r}(g_r(k))$, where $g_1,..., g_r$ are polynomials with integer coefficients.
openaire +3 more sources
International Journal of Mathematics and Mathematical Sciences, 2007 New simple nested-sum representations for powers of the arcsin function are given. This generalization of Ramanujan's work makes connections to finite binomial sums and polylogarithms.
Jonathan M. Borwein, Marc Chamberland
doaj +1 more source
An identity for a class of arithmetical functions of several variables
International Journal of Mathematics and Mathematical Sciences, 1993 Johnson [1] evaluated the sum ∑d|n|C(d;r)|, where C(n;r) denotes Ramanujan's trigonometric sum. This evaluation has been generalized to a wide class of arithmetical functions of two variables.
Pentti Haukkanen
doaj +1 more source
An identity for a class of arithmetical functions of two variables
International Journal of Mathematics and Mathematical Sciences, 1988 For a positive integer r, let r∗ denote the quotient of r by its largest squarefree divisor (1∗=1). Recently, K. R. Johnson proved that(∗)∑d|n|C(d,r)|=r∗∏pa‖nr∗p+r(a+1)∏pa‖nr∗p|r(a(p−1)+1) or 0according as r∗|n or not where C(n,r) is the well known ...
J. Chidambaraswamy, P. V. Krishnaiah
doaj +1 more source
Ramanujan sums via generalized Möbius functions and applications
International Journal of Mathematics and Mathematical Sciences, 2006 A generalized Ramanujan sum (GRS) is defined by replacing the usual Möbius function in the classical Ramanujan sum with the Souriau-Hsu-Möbius function.
Vichian Laohakosol +2 more
doaj +1 more source
A $q$-microscope for supercongruences [PDF]
, 2019 By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for
Guo, Victor J. W., Zudilin, Wadim
core +3 more sources
Ramanujan Sums for Image Pattern Analysis
International Journal of Wavelets, Multiresolution and Information Processing, 2013 <p>Ramanujan sums (RS) have been found to be very successful in signal processing recently. However, as far as we know, the RS have not been applied to image analysis. In this paper, we propose two novel algorithms for image analysis, including moment invariants and pattern recognition.
Chen, Guangyi +2 more
openaire +3 more sources
Arithmetic properties of �-regular overpartition pairs [PDF]
, 2017 In this paper, we investigate the arithmetic properties of â -regular overpartition pairs. Let Bâ (n) denote the number of â -regular overpartition pairs of n.
Naika, M.S.M., Shivashankar, C.
core +1 more source