A uniform asymptotic expansion of the single variable Bell polynomials

In this paper, we investigate the uniform asymptotic behavior of the single variable Bell polynomials on the negative real axis, to which all zeros belong. It is found that there exists an ascending sequence {Z_k}₁^∞(−e,0) such that the polynomials are represented by a finite sum of infinite asymptotic series, each in terms of the Airy function and its derivative, and the number of series under this sum is 1 in the interval (−∞,Z₁) and k+1 in [Z_k,Z_k+1), k1. Furthermore, it is shown that an asymptotic expansion, also in terms of Airy function and its derivative, completed with error bounds, holds uniformly in (−∞,−δ] for positive δ.     

Analogues of the Gauss-Vinogradov formula on the critical line

An asymptotic behavior of the sum for X is studied in the critical strip, where L(s, X_p) is the Dirichlet series with the quadratic character X_p modulo p, where p is a prime number; v=1 or 3. With the help of large seive estimates a formula for this sum is obtained with two asymptotic terms on the critical line of the variable s. As a corollary the asymptotic expansion of this sum at the point s=1/2 is presented. The asymptotic formula for the sum, where d runs over discriminants of quadratic fields, is also obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 109, pp. 41–82, 1981.     

Asymptotic expansion for log <Emphasis Type="Italic">n</Emphasis>! in terms of the reciprocal of a triangular number

Ramanujan suggested an expansion for the nth partial sum of the harmonic series which employs the reciprocal of the nth triangular number. This has been proved in 2006 by Villarino, who speculated that there might also exist a similar expansion for the logarithm of the factorial. This study shows that such an asymptotic expansion indeed exists and provides formulas for its generic coefficient and for the bounds on its errors.     

Colored graphs and matrix integrals

We discuss applications of generating functions for colored graphs to asymptotic expansions of matrix integrals. The described technique provides an asymptotic expansion of the Kontsevich integral. We prove that this expansion is a refinement of the Kontsevich expansion, which is the sum over the set of classes of isomorphic ribbon graphs. This yields a proof of Kontsevich’s results that is independent of the Feynman graph technique. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 8–24.     

Limit distribution of the sum and maximum from multivariate Gaussian sequences

In this paper we study the asymptotic joint behavior of the maximum and the partial sum of a multivariate Gaussian sequence. The multivariate maximum is defined to be the coordinatewise maximum. Results extend univariate results of McCormick and Qi. We show that, under regularity conditions, if the maximum has a limiting distribution it is asymptotically independent of the partial sum. We also prove that the maximum of a stationary sequence, when normalized in a special sense which includes subtracting the sample mean, is asymptotically independent of the partial sum (again, under regularity conditions). The limiting distributions are also obtained.     

On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain

The initial boundary value problem for the non-steady Stokes system is considered in bounded domains with the boundary having a peak-type singularity (power cusp singularity). The case of the boundary value with a nonzero time-dependent flow rate is studied. The formal asymptotic expansion of the solution near the singular point is constructed. This expansion contains both the outer asymptotic expansion and the boundary-layer-in-time corrector with the ‘fast time’ variable depending on the distance to the cusp point. The solution of the problem is constructed as the sum of the asymptotic expansion and the term with finite energy.     

An asymptotic expansion for the normalizing constant of the Conway–Maxwell–Poisson distribution

The Conway–Maxwell–Poisson distribution is a two-parameter generalization of the Poisson distribution that can be used to model data that are under- or over-dispersed relative to the Poisson distribution. The normalizing constant \(Z(\lambda ,\nu )\) is given by an infinite series that in general has no closed form, although several papers have derived approximations for this sum. In this work, we start by using probabilistic argument to obtain the leading term in the asymptotic expansion of \(Z(\lambda ,\nu )\) in the limit \(\lambda \rightarrow \infty \) that holds for all \(\nu >0\). We then use an integral representation to obtain the entire asymptotic series and give explicit formulas for the first eight coefficients. We apply this asymptotic series to obtain approximations for the mean, variance, cumulants, skewness, excess kurtosis and raw moments of CMP random variables. Numerical results confirm that these correction terms yield more accurate estimates than those obtained using just the leading-order term.

     

Regularized Trace for Sturm–Liouville Differential Operator on a Star-Shaped Graph

The sum of the eigenvalues {λ _n } of an operator is usually called its trace. For the eigenvalues λ _n of an differential operator, the series ${\sum_n \lambda_n}$ , generally speaking, diverges; however, it can be regularized by subtracting from λ _n the first terms of the asymptotic expansion, which interfere with the convergence of the series. The sum of such a regularized series is called the trace. In this work, we consider the spectral problem for Sturm–Liouville differential operator on d-star-type graph with a Kirchhoff-type condition in the internal vertex, where the integer d ≥ 2. Regularized trace formula of this operator is established with residue techniques in complex analysis.     

On singular solutions of time-periodic and steady Stokes problems in a power cusp domain

The time-periodic and steady Stokes problems with the boundary value having a nonzero flux are considered in the power cusp domains. The asymptotic expansion near the singularity point is constructed in order to reduce the problem to the case where the energy solution exists. The solution of the problem is found then as the sum of the asymptotic expansion and the term with finite dissipation of energy.     

Short-wave asymptotic behavior of space-time creeping waves in elasticity theory

We consider the elastic space-time (ST) wave on an unstressed convex surface in a deep shadow zone. The uniform high-frequency asymptotic expansion of the wave field is constructed as the sum of the caustic expansion for the longitudinal (transverse) wave containing the Airy function and the space-time ray series for the transverse (longitudinal) wave. The contribution of the ray expansion with the transverse eikonal is comparable to the contribution of the longitudinal creeping wave to the wave field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 176–189, 1985.I would like to thank V. M. Babich for suggesting the topic and for discussion of results.     

On the asymptotics of some partial theta functions

Using the Euler–Maclaurin sum formula, we develop an asymptotic expansion for a fairly general sum of exponentials, which when specialized includes some common partial theta functions. Some conjectured asymptotic expansions for relevant integrals are given. We give a simple proof of a theorem by Bruce Berndt and Byungchan Kim generalizing a result found in Ramanujan’s second notebook.     

Multiple solutions for the laminar flow in a porous pipe with suction at slowly expanding or contracting wall

In this paper, the asymptotic solution for the similarity equation of the laminar flow in a porous pipe with suction at expanding and contracting wall has been obtained using the singular perturbation method. However, this solution neglects exponentially small terms in the matching process. To take into account these exponentially small terms, a method involving the inclusion of exponentially small terms in a perturbation series was used to find the two solutions analytically. The series involving the exponentially small terms and expansion ratio predicts dual solutions. Furthermore, the result indicates that the expansion ratio has much important influence on the solutions. When the expansion ratio is zero, it is a special case that Terrill has discussed.     

On Gevrey solutions of threefold singular nonlinear partial differential equations

We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the Borel–Laplace summation method, we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their Gevrey asymptotic expansion in the perturbation parameter. This result rests on the Malgrange–Sibuya theorem, and it requires to prove that the difference between two neighboring solutions is exponentially small, what in this case involves an asymptotic estimate for a particular Dirichlet-like series.     

二次有限元解的渐近展开与外推

在Poisson方程的求解域Ω存在一致的三角剖分,并且相邻两初始单元构成平行四边形的假设下,证明了若Poisson方程的解u属于H6(Ω),那么二次有限元的误差有h4的渐近展开.基于误差的渐近展开,可以利用h4-Richardson外推进一步提高数值解的精度阶,并且能够得到一个后验误差估计.最后,一个数值算例验证了理论分析.     

AN ASYMPTOTIC EXPANSION FORMULA OF KERNEL FUNCTION FOR QUASI FOURIER-LEGENDRE SERIES AND ITS APPLICATION

Is this paper we shall give cm asymptotic expansion formula of the kernel functim for the Quasi Faurier-Legendre series on an ellipse, whose error is 0(1/n2) and then applying it we shall sham an analogue of an exact result in trigonometric series.     

A shifted sum for the congruent number problem

The Ramanujan Journal - We introduce a shifted convolution sum that is parametrized by the squarefree natural number t. The asymptotic growth of this series depends explicitly on whether or not t...     

On the stability of a class of convergence acceleration methods for power series

In this paper we study the problem of evaluating the sum of a power series whose terms are given numerically with a moderate accuracy. For a large class of divergent series a sum may be defined using analytic continuation. This sum may be estimated using the values of a finite number of terms. However, it is established here that the accuracy of this estimate will generally deteriorate if we use an ever-growing number of terms. A result on the stability of product quadrature is also obtained as a corollary of our main stability theorem.Dedicated to professor Germund Dahlquist, on the occasion of his 60th birthday