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

TL;DR

This paper derives highly accurate uniform bounds for ratios of Parabolic Cylinder functions, providing simple elementary approximations with proven monotonicity and sharpness across various parameter ranges.

Contribution

It introduces novel, sharp bounds for ratios of PCFs and their double ratios, with proofs of monotonicity and global accuracy, filling a gap in existing literature.

Findings

Bounds are sharp as x approaches infinity.

Bounds are sharp as n approaches infinity.

Elementary bounds achieve several digits of accuracy for moderate parameters.

Abstract

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 $n$.