193 research outputs found
A -linear analogue of the plane wave expansion
We obtain a -linear analogue of Gegenbauer's expansion of the plane wave.
It is expanded in terms of the little -Gegenbauer polynomials and the
\textit{third} Jackson -Bessel function. The result is obtained by using a
method based on bilinear biorthogonal expansions.Comment: 12 pages, to appear in Adv. in Appl. Math. arXiv admin note: text
overlap with arXiv:0909.006
An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics
We present a three-point iterative method without memory for solving
nonlinear equations in one variable. The proposed method provides convergence
order eight with four function evaluations per iteration. Hence, it possesses a
very high computational efficiency and supports Kung and Traub's conjecture.
The construction, the convergence analysis, and the numerical implementation of
the method will be presented. Using several test problems, the proposed method
will be compared with existing methods of convergence order eight concerning
accuracy and basin of attraction. Furthermore, some measures are used to judge
methods with respect to their performance in finding the basin of attraction.Comment: arXiv admin note: substantial text overlap with arXiv:1508.0174
Winnerless competition in coupled Lotka-Volterra maps
Winnerless competition is analyzed in coupled maps with discrete temporal evolution of the Lotka-Volterra type of arbitrary dimension. Necessary and sufficient conditions for the appearance of structurally stable heteroclinic cycles as a function of the model parameters are deduced. It is shown that under such conditions winnerless competition dynamics is fully exhibited. Based on these conditions different cases characterizing low, intermediate, and high dimensions are therefore computationally recreated. An analytical expression for the residence times valid in the N-dimensional case is deduced and successfully compared with the simulations.J.L.C. and E.D.G. acknowledge support from IVIC-141, L.A.G.-D. acknowledges support from IVIC-1089 and P.V. acknowledges support from MINECO TIN2012-30883
Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials
We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials
in detail. The starting point is their Fourier
series on which, it is shown, remains valid as an asymptotic expansion
over compact subsets of the complex plane. This is used to determine explicit
estimates on the constants in the approximation, and also to analyze
oscillatory phenomena which arise in certain cases.
These results are transferred to the Apostol-Euler polynomials
via a simple relation linking them to the
Apostol-Bernoulli polynomials.Comment: 16 page
Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials
summary:One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the âmain familyâ those given by and as an âexceptional familyâ both of these for
The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials
AbstractHurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials
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