96 research outputs found
Asymptotics of a polynomial associated with the Catalan-Larcombe-French sequence
The large behaviour of the hypergeometric polynomial
\FFF{-n}{\sfrac12}{\sfrac12}{\sfrac12-n}{\sfrac12-n}{-1} is considered by
using integral representations of this polynomial. This polynomial is
associated with the Catalan-Larcombe-French sequence. Several other
representations are mentioned, with references to the literature, and another
asymptotic method is described by using a generating function of the sequence.
The results are similar to those obtained by Clark (2004) who used a binomial
sum for obtaining an asymptotic expansion.Comment: 10 pages, 1 figure. Accepted for publication in {\em Analysis and
Applications
Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters
We consider the asymptotic behavior of the incomplete gamma functions
gamma(-a,-z) and Gamma(-a,-z) as a goes to infinity. Uniform expansions are
needed to describe the transition area z~a in which case error functions are
used as main approximants. We use integral representations of the incomplete
gamma functions and derive a uniform equation by applying techniques used for
the existing uniform expansions for gamma(a,z) and Gamma(a,z). The result is
compared with Olver's uniform expansion for the generalized exponential
integral. A numerical verification of the expansion is given in a final
section
Remarks on Slater's asymptotic expansions of Kummer functions for large values of the parameter
In Slater's 1960 standard work on confluent hypergeometric functions, also
called Kummer functions, a number of asymptotic expansions of these functions
can be found. We summarize expansions derived from a differential equation for
large values of the parameter. We show how similar expansions can be
derived by using integral representations, and we observe discrepancies with
Slater's expansions.Comment: To appear in Advances in Dynamical Systems and Applications.
Proceedings of the International Conference on Differential Equations,
Difference Equations and Special Functions, Patras, Greece, September 3 - 9,
2012, dedicated to the memory of Panayiotis D. Siafarikas. 13 page
Numerical and Asymptotic Aspects of Parabolic Cylinder Functions
Several uniform asymptotics expansions of the Weber parabolic cylinder
functions are considered, one group in terms of elementary functions, another
group in terms of Airy functions. Starting point for the discussion are
asymptotic expansions given earlier by F.W.J. Olver. Some of his results are
modified to improve the asymptotic properties and to enlarge the intervals for
using the expansions in numerical algorithms. Olver's results are obtained from
the differential equation of the parabolic cylinder functions; we mention how
modified expansions can be obtained from integral representations. Numerical
tests are given for three expansions in terms of elementary functions. In this
paper only real values of the parameters will be considered.Comment: 16 pages, 1 figur
The tail of the maximum of Brownian motion minus a parabola
We analyze the tail behavior of the maximum N of Brownian motion minus a
parabola and give an asymptotic expansion for P(N>x) as x tends to infinity.
This extends a first order result on the tail behavior, which can be deduced
from Huesler and Piterbarg (1999). We also point out the relation between
certain results in Groeneboom (2010) and Janson, Louchard and Martin-L\"of
(2010).Comment: 12 pages, submitted to the Electronic Communications in Probabilit
Convergent Asymptotic Expansions of Charlier, Laguerre and Jacobi Polynomials
Convergent expansions are derived for three types of orthogonal polynomials:
Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for
large values of the degree. The expansions are given in terms of functions that
are special cases of the given polynomials. The method is based on expanding
integrals in one or two points of the complex plane, these points being saddle
points of the phase functions of the integrands.Comment: 20 pages, 5 figures. Keywords: Charlier polynomials, Laguerre
polynomials, Jacobi polynomials, asymptotic expansions, saddle point methods,
two-points Taylor expansion
Multi-point Taylor Expansions of Analytic Functions
Taylor expansions of analytic functions are considered with respect to
several points, allowing confluence of any of them. Cauchy-type formulas are
given for coefficients and remainders in the expansions, and the regions of
convergence are indicated. It is explained how these expansions can be used in
deriving uniform asymptotic expansions of integrals. The method is also used
for obtaining Laurent expansions in several points as well as Taylor-Laurent
expansions.Comment: 20 pages, 7 figures. Keywords: multi-point Taylor expansions,
Cauchy's theorem, analytic functions, multi-point Laurent expansions, uniform
asymptotic expansions of integral
New Series Expansions of the Gauss Hypergeometric Function
The Gauss hypergeometric function can be computed by using
the power series in powers of . With
these expansions is not completely computable for all
complex values of . As pointed out in Gil, {\it et al.} [2007, \S2.3], the
points are always excluded from the domains of convergence
of these expansions. B\"uhring [1987] has given a power series expansion that
allows computation at and near these points. But, when is an integer, the
coefficients of that expansion become indeterminate and its computation
requires a nontrivial limiting process. Moreover, the convergence becomes
slower and slower in that case. In this paper we obtain new expansions of the
Gauss hypergeometric function in terms of rational functions of for which
the points are well inside their domains of convergence . In
addition, these expansion are well defined when is an integer and no
limits are needed in that case. Numerical computations show that these
expansions converge faster than B\"uhring's expansion for in the
neighborhood of the points , especially when is close to
an integer number.Comment: 18 pages, 6 figures, 4 tables. In Advances in Computational
Mathematics, 2012 Second version with corrected typos in equations (18) and
(19
Two-point Taylor Expansions of Analytic Functions
Taylor expansions of analytic functions are considered with respect to two
points. Cauchy-type formulas are given for coefficients and remainders in the
expansions, and the regions of convergence are indicated. It is explained how
these expansions can be used in deriving uniform asymptotic expansions of
integrals. The method is also used for obtaining Laurent expansions in two
points.Comment: 14 pages, 10 figure
Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters
We derive asymptotic expansions of the Kummer functions and
for large positive values of and , with fixed. For both
functions we consider and , with special attention for the
case . We use a uniform method to handle all cases of these
parameters.Comment: 17 pages, 2 figure
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