Analysis and Applications

Let

In this paper, we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls [Hyperasymptotics for integrals with finite endpoints, Proc. Roy. Soc. London Ser. A 439 (1992) 373–396]. Using these representations, we obtain a number of properties of the asymptotic expansions of the incomplete gamma function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.

For $n\ge 1$ and $\alpha \in (-1,1)$, let $H^{\alpha ,2}$ be the space of harmonic functions $u$ on the upper half-space ${ℝ}_{+}^{n+1}$ satisfying

In this paper, we develop the Riemann–Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi–Carlitz polynomials $f_n^{(\alpha )}(z)$ where $\alpha$ is a positive parameter. Uniform Plancherel–Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [W. M. Y. Goh and J. Wimp, On the asymptotics of the Tricomi–Carlitz polynomials and their zero distribution. I, SIAM J. Math. Anal. 25 (1994) 420–428] and in [K. F. Lee and R. Wong, Uniform asymptotic expansions of the Tricomi–Carlitz polynomials, Proc. Amer. Math. Soc. 138 (2010) 2513–2519].

The Cauchy problem of the Navier–Stokes–Poisson equations in $n$ dimensions ($n\ge 3$) is considered. Based on some new observations on the system and a detailed analysis of Green’s function for linearized system, a refined pointwise result of the solution $(\rho ,u)$ is given when it is a small perturbation of the constant state $(\bar{\rho },u_{\ast })$ with $\bar{\rho }>0$. In particular, we show that $|D_x^{\alpha }(\rho (x,t)-\bar{\rho })|\le C{(1+t)}^{-\frac{n+1+\left|\alpha \right|}{2}}{(1+\frac{|x-u_{\ast }t{|}^2}{1+t})}^{-r}$ with $r>\frac{n}{2}$ and $|D_x^{\alpha }(u(x,t)-u_{\ast })|\le C{(1+t)}^{-\frac{n+\left|\alpha \right|}{2}}{(1+\frac{|x-u_{\ast }t{|}^2}{1+t})}^{-\frac{n}{2}}$. Furthermore, we obtain optimal $L^p({ℝ}^n)$-decay rates of the solution: $\parallel D_x^{\alpha }(\rho (x,t)-\bar{\rho }){\parallel }_{L^p({ℝ}^n)}\le C{(1+t)}^{-\frac{n}{2}(1-\frac{1}{p})-\frac{1}{2}-\frac{\left|\alpha \right|}{2}}$ with $p\in [1,\infty ]$ and $\parallel D_x^{\alpha }(u(x,t)-u_{\ast }){\parallel }_{L^p({ℝ}^n)}\le C{(1+t)}^{-\frac{n}{2}(1-\frac{1}{p})-\frac{\left|\alpha \right|}{2}}$ with $p\in (1,\infty ]$.