论文标题
Gegenbauer多项式和相关功能的均匀渐近膨胀通过具有简单极的微分方程
Uniform asymptotic expansions for Gegenbauer polynomials and related functions via differential equations having a simple pole
论文作者
论文摘要
对于Gegenbauer（超球）多项式的渐近扩展，用于大订单$ n $，对于参数$ z $的无限复杂值均匀有效，包括实际的间隔$ 0 \ leq z \ leq 1 $，右半平面中的Zeros在其中：symmetry位于左侧的一半平面上。近似值来自这些多项式满足的微分方程，也考虑了其他独立的解决方案。对于大$ n $，此方程的特征是具有简单的极点，并且在此奇点上有效的扩展涉及贝塞尔功能和缓慢变化的系数函数。这些功能的扩展比以前的近似更简单，尤其是可以高度准确的计算。得出了仅涉及基本功能的简单显式误差界，从而简化了与具有较大参数和简单极点的微分方程相关的先前扩展和误差界。
Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in the right half plane are located: symmetry extends the results to the left half plane. The approximations are derived from the differential equation satisfied by these polynomials, and other independent solutions are also considered. For large $n$ this equation is characterized by having a simple pole, and expansions valid at this singularity involve Bessel functions and slowly varying coefficient functions. The expansions for these functions are simpler than previous approximations, in particular being computable to a high degree of accuracy. Simple explicit error bounds are derived which only involve elementary functions, and thereby provide a simplification of previous expansions and error bounds associated with differential equations having a large parameter and simple pole.
