Mihail Nikitin
We present an asymptotic evaluation unitary formula for large argument values existing for defined class of functions. The asymptotic evaluation is obtained using only power series expansion coefficients of a function, what is a new result in power series analytic continuation theory.
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We consider a generalized Mathieu series where the summands of the classical Mathieu series are multiplied by powers of a complex number. The Mellin transform of this series can be expressed by the polylogarithm or the Hurwitz zeta function. From this we derive a full asymptotic expansion, generalizing known expansions for alternating Mathieu series. Another asymptotic regime for trigonometric Mathieu series is also considered, to first order, by applying known results on the asymptotic behavior of trigonometric series.
Inequalities, asymptotics and, for some specific cases, asymptotical expansions were obtained for generalized Mathieu's series. A connection between inequalities for Mathieu's series and positive definite and completely monotonic functions.
We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics.
We show an asymptotic formula of the divergent bilateral basic hypergeometric series ${}_1ψ_0 (a;-;q,\cdot)$ with using the $q$-Borel-Laplace method. We also give the limit $q\to 1-0$ of our asymptotic formula.
We present a sufficient condition of existence of asymptotic expansion in negative power series for a function defined by Taylor series and unitary formulas for coefficients of this expansion. An example of computing scheme for arctangent function is represented.
The asymptotic power series of three kinds of sequences
本文提出一种通用方法，为三类序列建立渐近幂级数，并给出系数递推关系，进而推导出若干著名序列和含参数积分的近似公式。
In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\to1$. These $q$-series generalize Ramanujan function $A_{q}(z)$; Jackson's $q$-Bessel function $J_ν^{(2)}$(z;q), Ismail-Masson orthogonal polynomials($q^{-1}$-Hermite polynomials) $h_{n}(x|q)$, Stieltjes-Wigert orthogonal polynomials $S_{n}(x;q)$, $q$-Laguerre orthogonal polynomials $L_{n}^{(α)}(x;q)$ and confluent basic hypergeometric series.
We find the asymptotics of the series $\sum_{n=1}^\infty (-1)^n n^{-1} \exp(-t/n)$ as $t\to+\infty$. The answer is an oscillating function of $t$ dominated by $\exp(-(2πt)^{1/2})$. The intermediate step is to find the asymptotics of the two-dimensional Fourier transform $\hat F(ξ)$ of the function $F(x)=(1+\exp(\|x\|^2))^{-1}$ as $\|ξ\|\to\infty$.
In this paper we present a family of values of the parameters of the third Painlevé equation such that Puiseux series formally satisfying this equation -- considered as series of $z^{2/3}$ -- are series of exact Gevrey order one. We prove the divergence of these series and provide analytic functions which are approximated by them in sectors with the vertices at infinity.
In this note we study the convergence of recursively defined infinite series. We explore the role of the derivative of the defining function at the origin (if it exists), and develop a comparison test for such series which can be used even if the defining function of the series is not differentiable.
In this paper, modified gamma and beta functions containing generalized M-series in their kernel are defined. Also, modified Gauss and confluent hypergeometric functions are defined using the modified beta function. Then, some properties of these modified special functions are obtained. Finally, the relationships between generalized special functions in the literature with newly defined modified special functions are examined.
The present article is devoted to certain examples of functions whose argument represented in terms of Cantor series.
The aim of this work is to expose some asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. All proofs and the methods used for improving the classical inequalities announced in the final part of the first section are presented in an extended form in a paper submitted by the author to a journal for publication.
In this work we study complete asymptotic expansions for the q-series $\sum_{n=1}^{\infty}\frac{1}{n^{b}}q^{n^{a}}$ and $\sum_{n=1}^{\infty}\frac{σ_α(n)}{n^{b}}q^{n^{a}}$ in the scale function $(\log q)^{n}$ as $q\to1^{-}$, where $a>0,\ q\in(0,1),\,b,α\in\mathbb{C}$ and $σ_α(n)$ is the divisor function $σ_α(n)=\sum_{d\vert n}d^α$.
Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{ν=0}^{\infty} a_ν$ of infinite series $\sum_{ν=0}^{\infty} a_ν$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $Δr_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hyperge…
We consider the asymptotic expansion of the functional series \[S_μ^\pm(a;λ)=\sum_{n=0}^\infty \frac{(\pm 1)^n e^{-λn}}{(n^2+a^2)^μ}\] for $λ>0$ and $μ\geq0$ as $|a|\to \infty$ in the sector $|\arg\,a|<π/2$. The approach employed consists of expressing $S_μ^\pm(a;λ)$ as a contour integral combined with suitable deformation of the integration path. Numerical examples are provided to illustrate the accuracy of the various expansions obtained.
The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to partially fill this gap by finding the asymptotics of trigonometric series in several variables with the terms, having a form of `one minus the cosine' up to a decreasing power-like factor: \[ \sum_{z\i…
We consider the asymptotic expansion of the Mathieu-Bessel series \[S_ν(a,b)=\sum_{n=1}^\infty \frac{n^γJ_ν(nb/a)}{(n^2+a^2)^μ}, \qquad (μ, b>0,\ γ, ν\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed, where $J_ν(x)$ is the Bessel function of the first kind of order $ν$. A special case arises when $γ+ν$ is a positive even integer, where the expansion comprises finite algebraic terms together with an exponentially small expansion. Numerical examples are presented to illustrate the accuracy of the various expansions. The expansion of the alternating variant of $S_ν(a,b)$ is con…
We consider the asymptotic expansion of the Mathieu-Bessel series \[S_{ν,γ}^μ(a,b)=\sum_{n=1}^\infty \frac{n^γK_ν(nb/a)}{(n^2+a^2)^μ}, \qquad (μ>0, ν\geq 0, b>0, γ\in {\bf R})\] as $|a|\to\infty$ in $|\arg\,a|<π/2$ with the other parameters held fixed, where $K_ν(x)$ is the modified Bessel function of the second kind of order $ν$. We employ a Mellin transform approach to determine an integral representation for $S_{ν,γ}^μ(a,b)$ involving the Riemann zeta function. Asymptotic evaluation of this integral involves appropriate residue calculations. Numerical examples are presented to illustrate th…
Convergence of sub-series' and sub-signed series' in terms of the asymptotic $ψ$-density
本文研究发散非负级数在子集限制下的收敛与发散问题，利用渐近ψ-密度更精确地度量子集大小，并进一步讨论取值于{-1,0,1}的子符号级数的收敛性。
Lerch $Φ$ asymptotics
本文利用 Mellin-Barnes 积分表示研究 Lerch 超越函数的大 $z$ 渐近近似，通过求和与最优截断揭示隐藏在发散尾部中的收敛级数。
Asymptotic expansions of series $\sum_{k=0}^\infty ε^k(k+a)^γe^{-(k+a)^αx}$ and $\sum_{k=0}^\infty ε^k(k+a)^γ/ (x(k+a)^α+1)^μ}$ in powers of $x$ as $x\to+0$ are found, where $ε=1$ or $ε=-1$. These expansions are applied to obtain precise inequalities for Mathieu series. Keywords: Asymptotic expansion, residues, generalized Mathieu series, inequalities.
We consider the asymptotic expansion of the sum \[S_p(a;w)=\sum_{n=1}^\infty n^{-w}\e^{-an^p}\] as $a\rightarrow 0$ in $|\arg\,a|<π/2$ for arbitrary finite $p>$ and $w>0$. Our attention is concentrated mainly on the case when $p$ and $w$ are both even integers, where the expansion consists of a {\it finite} algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for $S_p(a;w)$ analogous to the well-known Poisson-Jacobi transformation for the sum with $p=2$ and $w=0$. Numerical results are give…
The paper concerns asymptotic studies for the sixth Painlevé transcendent as independent variable tends to infinity. The primary tool is averaging and the Whitham method. Elliptic ansatz, appropriate modulation equation and asymptotics for some partial solution to PVI are found.
We consider the asymptotic expansion of the functional series \[S_{μ,γ}(a;λ)=\sum_{n=1}^\infty \frac{n^γe^{-λn^2/a^2}}{(n^2+a^2)^μ}\] for real values of the parameters $γ$, $λ>0$ and $μ\geq0$ as $|a|\to \infty$ in the sector $|\arg\,a|<π/4$. For general values of $γ$ the expansion is of algebraic type with terms involving the Riemann zeta function and a terminating confluent hypergeometric function. Of principal interest in this study is the case corresponding to even integer values of $γ$, where the algebraic-type expansion consists of a finite number of terms together with a contribution com…
The aim of this paper is to investigate in detail the known large argument asymptotic series of the Lommel function by Stieltjes transform representations. We obtain a number of properties of this asymptotic expansion, including explicit and realistic error bounds, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. An interesting consequence related to the large argument asymptotic series of the Struve function is also proved.
We consider the integral $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$ as a function of the positive integer $n$. We show that there exists an asymptotic series in $\frac{1}{n}$ and compute the first terms of this series together with an explicit error bound.
We study some series expansions for the Lambert $W$ function. We show that known asymptotic series converge in both real and complex domains. We establish the precise domains of convergence and other properties of the series, including asymptotic expressions for the expansion coefficients. We introduce an new invariant transformation of the series. The transformation contains a parameter whose effect on the domain and rate of convergence is studied theoretically and numerically. We also give alternate representations of the expansion coefficients, which imply a number of combinatorial identiti…
We obtain asymptotic estimates for the best approximations by trigonometric polynomials in the metric space $C$ $(L_p)$ of classes of periodic functions that can be represented as a convolution of kernels $Ψ_β$, which Fourier coefficients tend to zero faster than any power sequence, with functions $φ\in C (φ\in L_p),$ which moduli of continuity do not exceed a fixed majorant $ω(t)$. It is proved that in the spaces $C$ and $L_1$ the obtained estimates are asymptotically exact for convex moduli of continuity $ω(t)$.
We give a systematic study of q-algebraic equations. The questions of existence, uniqueness and regularity of the solutions are solved in the space of grid-based Hahn series. The regularity is understood in terms of asymptotic behavior of coefficients, and is the main focus of this work. The results and algorithms are illustrated by many examples.
In this work we investigate the asymptotics for Euler's $q$-Exponential $E_{q}(z)$, $q$-Gamma function $Γ_{q}(z)$, Ramanujan's function $A_{q}(z)$, Jackson's $q$-Bessel function $J_ν^{(2)}$(z;q) of second kind, Stieltjes-Wigert orthogonal polynomials $S_{n}(x;q)$ and $q$-Laguerre polynomials $L_{n}^{(α)}(x;q)$ as $q$ approaching 1.
We study the precise asymptotic behavior of a non-trivial solution that converges to zero, as time tends to infinity, of dissipative systems of nonlinear ordinary differential equations. The nonlinear term of the equations may not possess a Taylor series expansion about the origin. This absence technically cripples previous proofs in establishing an asymptotic expansion, as an infinite series, for such a decaying solution. In the current paper, we overcome this limitation and obtain an infinite series asymptotic expansion, as time goes to infinity. This series expansion provides large time app…
Integral representations and asymptotic expansions for the second type Neumann series of Bessel functions of the first kind
本文针对一类含参数的第一类贝塞尔函数第二类诺伊曼级数，推导出完全显式的积分表示，并给出带显式项的渐近展开，从而揭示贝塞尔函数导数级数的渐近行为。
In this work we study the Plancherel-Rotach type asymptotics for selected $q$-series and $q$-orthogonal polynomials with complex scalings. The $q$-series we cover are Euler's $q$-exponential, Ramanujan function, Jackson's $q$-Bessel function of second kind, Ismail-Masson orthogonal polynomials, Stieltjes-Wigert polynomials and $q$-Laguerre polynomials. For a fixed $q$ with $0<q<1$, in each case the main term of the asymptotic formulas may contain Ramanujan function or theta function depending on the value of scaling parameter. Furthemore, when the scaling parameter is in certain strip of the c…
An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $ϕ(X)$ for large $λ$ where $X$ is a Poisson random variable with mean $λ$ and $ϕ$ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of $ϕ(X)$ when the distribution of $X$ belongs to a suitable family indexed by a convolution parameter. Examples include the problem of inverse moments for distribution families such as the binomial or the negative binomial.
Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study
本文通过等单值形变形方法，将退化 Painlevé III 方程（DP3E）解的跨级数渐近用相关线性辅助问题的单值性数据参数化，并获得了相关哈密顿量、辅助函数及 sigma 形式解的渐近，同时推导了 DP3E 的李点对称性作用。
Global Rational Approximations of Functions With Factorially Divergent Asymptotic Series
该研究提出新型数值高效的dyadic展开方法，实现阶乘发散渐近级数函数的全局几何收敛有理逼近，适用特殊函数与Écalle再生函数，还可推广至算子预解式表示。
We provide new formulae for the degenerations of the bilateral basic hypergeometric function ${}_1ψ_1 ( a; b; q, z )$ with using the $q$-Borel-Laplace transformation. These are thought of as the first step to construct connection formulae of $q$-difference equation for ${}_1ψ_1 ( a; b; q, z )$. Moreover, we show that our formulae have the $q \to 1 - 0$ limit.
We produce an estimate for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin θ$ for a fixed parameter $0\leq θ\leq π/2$ or $t= y \cosh μ$ for a fixed parameter $μ>0$. In particular, we compute the dominant term of the asymptotic expansion of $K_{r + i t}(y)$ as $y \rightarrow \infty$. When $t$ and $y$ are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $E_0^{(j)}(z, r+it)$ for each inequivalen…
Theta functions and transformations of bilateral basic hypergeometric series
本文通过theta函数和双侧基本超几何级数建立新变换公式，构造伴生q级数，证明McIntosh关于q级数渐近变换的猜想，并扩展了Ramanujan和McIntosh的一些恒等式。
We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for $\lnΓ(z+1/2)$, and for the Riemann-Siegel theta function, and make some historical remarks.
In this paper asymptotic formulas are given for the Lebesgue constants generated by three special approximation processes related to the $\ell_1$-partial sums of Fourier series. In particular, we consider the Lagrange interpolation polynomials based on the Lissajous-Chebyshev node points, the partial sums of the Fourier series generated by the anisotropically dilated rhombus, and the corresponding discrete partial sums.
We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of the corresponding implicitly defined formal power series solution. The main theorem can therefore be rephrased as an Implicit Function Theorem for Borel summable power series. As an application, we give a diagonal or Jordan decomposition for holomorphic matrices in Gevrey asymptotic families.
We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is an LE-series as defined by van den Dries et al. As these series generally have support of order type larger than that of the set of natural numbers, the notion of asymptotic expansion itself needs to be generalized.
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