Uniform (very) sharp bounds for ratios of Parabolic Cylinder functions

Javier Segura

Published 2020-10-21Version 1

Parabolic Cylinder functions (PCFs) are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and bounds for these functions. We obtain very sharp bounds for the ratio $\Phi_n(x)=U(n-1,x)/U(n,x)$ and the double ratio $\Phi_n(x)/\Phi_{n+1}(x)$ in terms of elementary functions (algebraic or trigonometric) and prove the monotonicity of these ratios; bounds for $U(n,z)/U(n,y)$ are also made available. The bounds are very sharp as $x\rightarrow \pm \infty$ and $n\rightarrow +\infty$, and this simultaneous sharpness in three different directions explains their remarkable global accuracy. Upper and lower elementary bounds are obtained which are able to produce several digits of accuracy for moderately large $|x|$ and/or $n$.