Ruiming Zhang
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.
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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 study the signed Bernoulli convolution $$ν_β^{(n)}=*_{j=1}^n \left (\frac12δ_{β^{-j}}-\frac12δ_{-β^{-j}}\right ),\ n\ge 1$$ where $β>1$ satisfies $$β^m=β^{m-1}+\cdots+β+1$$ for some integer $m\ge 2$. When $m$ is odd, we show that the variation $|ν_β^{(n)}|$ coincides the unsigned Bernoulli convolution $$μ_β^{(n)}=*_{j=1}^n \left (\frac12δ_{β^{-j}}+\frac12δ_{-β^{-j}}\right ).$$ When $m$ is even, we obtain the exact asymptotic of the total variation $\|ν_β^{(n)}\|$ as $n\rightarrow\infty$.
A counterexample to Abel-type asymptotics for scaled Volterra equations
本文通过构造特定核函数和函数g，证明缩放维塔尔方程的解序列在某些点不趋于零，推翻经典结果。
We will use a discrete analogue of the classical \emph{Laplace} method to show that for infinitely many positive integers $n$, the main term of the asymptotic expansions of scaled confluent basic hypergeometric functions, including the Ramanujan's entire function $A_{q}(z)$, could be expressed in $θ$-functions, and the error term depends on the ergodic property of certain real scaling parameter.
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 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 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.
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。
We consider the Bernoulli polynomials of the second kind, which can be related to the generalised Bernoulli polynomials $B_n^{(n)}(z)$. The asymptotic expansions of the scaled polynomials $B_n^{(n)}(nz)$ are obtained as $n\to\infty$ when (i) $z$ is real and (ii) $z$ is complex bounded away from $[0,1]$. These results complement recent work of Štampach [{\it J. Approx. Theory}, {\bf 262} (2021) 105517]. Numerical results are presented to illustrate the accuracy of the different expansions obtained.
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 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…
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 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.
In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel--Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered separately. We introduce the "branch cut" technique to express the error terms as integrals on the contour taken as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple error bounds in terms of elementary functions. We also provide recursive procedures for …
The manuscript establishes a series expansion of the core integral that relates changes in longitude and latitude along the geodetic line in oblate elliptical coordinates, and of a companion integral which is the path length along this line as a function of latitude. The expansion is a power series in the scaled (constant) altitude of the trajectory over the surface of the ellipsoid. Each term of this series is reduced to sums over inverse trigonometric functions, square roots and Elliptic Integrals. The aim is to avoid purely numerical means of integration.
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.
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。
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。
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