13,995 research outputs found
Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function
In this paper, we present some completely monotonic functions and asymptotic expansions related to the gamma function. Based on the obtained expansions, we provide new bounds for Î(x + 1)/Î(x + 1/2) and Î(x + 1/2)
Lacunary formal power series and the Stern-Brocot sequence
Let be a real
lacunary formal power series, where and
. It is known that the denominators of
the convergents of its continued fraction expansion are polynomials with
coefficients , and that the number of nonzero terms in is
the th term of the Stern-Brocot sequence. We show that replacing the index
by any 2-adic integer makes sense. We prove that
is a polynomial if and only if . In all the other cases
is an infinite formal power series, the algebraic properties of
which we discuss in the special case .Comment: to appear in Acta Arithmetic
A model of heart rate kinetics in response to exercise
We present a mathematical model, in the form of two coupled ordinary differential equations, for the heart rate kinetics in response to exercise. Our heart rate model is an adaptation of the model of oxygen uptake kinetics of Stirling: a physiological justification for this adaptation, as well as the physiological basis of our heart rate model is provided. We also present the optimal fit of the heart rate model to a set of raw un averaged data for multiple constant intensity exercises for an individual at a particular level of fitness
Crossings, Motzkin paths and Moments
Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain
-analogues of Laguerre and Charlier polynomials. The moments of these
orthogonal polynomials have combinatorial models in terms of crossings in
permutations and set partitions. The aim of this article is to prove simple
formulas for the moments of the -Laguerre and the -Charlier polynomials,
in the style of the Touchard-Riordan formula (which gives the moments of some
-Hermite polynomials, and also the distribution of crossings in matchings).
Our method mainly consists in the enumeration of weighted Motzkin paths, which
are naturally associated with the moments. Some steps are bijective, in
particular we describe a decomposition of paths which generalises a previous
construction of Penaud for the case of the Touchard-Riordan formula. There are
also some non-bijective steps using basic hypergeometric series, and continued
fractions or, alternatively, functional equations.Comment: 21 page
Combinatorics and Boson normal ordering: A gentle introduction
We discuss a general combinatorial framework for operator ordering problems
by applying it to the normal ordering of the powers and exponential of the
boson number operator. The solution of the problem is given in terms of Bell
and Stirling numbers enumerating partitions of a set. This framework reveals
several inherent relations between ordering problems and combinatorial objects,
and displays the analytical background to Wick's theorem. The methodology can
be straightforwardly generalized from the simple example given herein to a wide
class of operators.Comment: 8 pages, 1 figur
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