相关论文: Uniform Asymptotic Expansions for the Discrete Che…

In this paper, we consider the discrete Laguerre polynomials $P_{n, N}(z)$ orthogonal with respect to the weight function $w(x) = x^{\alpha} e^{-N cx}$ supported on the infinite nodes $L_N = \{ x_{k,N} = \frac{k^2}{N^2}, k \in \mathbb{N}…

We derive asymptotic expansions of the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed. For both functions we consider $b/a\le 1$ and $b/a\ge 1$, with special attention for the case…

We derive optimal asymptotic and non-asymptotic lower bounds on the Widom factors for weighted Chebyshev and orthogonal polynomials on compact subsets of the real line. In the Chebyshev case we extend the optimal non-asymptotic lower bound…

A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of…

This article derives full asymptotic expansions for integrals of the form \[ \int_{0}^{1}f(u)(1+q\cdot u^{n})^{w/n}du \] as $n\rightarrow\infty$, with parameters real $w\neq 0$ and $q\in(-1,1]$, or positive $w$ for $q=-1$. We relate the…

Given a sequence of orthonormal polynomials on $\Bbb R$,$\{p_n\}_{n\geq 0}$, with $p_n$ of degree $n$, we define the discrete probability distribution $\Psi_n(x) = \left(\Psi_{n,1}(x), \dots \Psi_{n,n}(x) \right) $, with $\Psi_{n,j}(x) =…

We describe a fast, simple, and stable transform of Chebyshev expansion coefficients to Jacobi expansion coefficients and its inverse based on the numerical evaluation of Jacobi expansions at the Chebyshev--Lobatto points. This is achieved…

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and…

Recently presented explicit formulae for asymptotic expansions of Feynman diagrams in the Sudakov limit are applied to typical two-loop diagrams. For a diagram with one non-zero mass these formulae provide an algorithm for analytical…

A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…

We consider the asymptotic expansion of the Mathieu-Bessel series \[S_\nu(a,b)=\sum_{n=1}^\infty \frac{n^\gamma J_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu, b>0,\ \gamma, \nu\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed,…

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…

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree $n$ goes to $\infty$. These are defined on the interval $[-1,1]$ with weight function…

A new uniform asymptotic expansion for the incomplete gamma function $\Gamma(a,z)$ valid for large values of $z$ was given by the author in {\it J. Comput. Appl. Math.} {\bf 148} (2002) 323--339. This expansion contains a complementary…

Asymptotic expansions are derived for the tail distribution of the product of two correlated normal random variables with non-zero means and arbitrary variances, and more generally the sum of independent copies of such random variables.…

We construct asymptotic expansions for the normalised incomplete gamma function $Q(a,z)=\Gamma(a,z)/\Gamma(a)$ that are valid in the transition regions, including the case $z\approx a$, and have simple polynomial coefficients. For Bessel…

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation {equation*} P_{n+1}(x)-(A_{n}x+B_{n})P_{n}(x)+P_{n-1}(x)=0, {equation*} where $A_n$ and $B_n$ have asymptotic expansions of…

We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations $P_0(x)=1$, $P_1(x)=x$, $x P_n(x)=(1-c_n)P_{n+1}(x)+c_n P_{n-1}(x),$ $n\in\mathbb{N}$ with…

We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…

The real and complex zeros of the parabolic cylinder function $U(a,z)$ are studied. Asymptotic expansions for the zeros are derived, involving the zeros of Airy functions, and these are valid for $a$ positive or negative and large in…