On bounds for solutions of monotonic first order difference-differential systems

TL;DR

This paper derives sharp bounds for solutions of monotonic first order difference-differential systems, connecting them to classical asymptotic methods and applying them to special functions like Bessel and Legendre functions.

Contribution

It introduces new bounds for solutions of monotonic systems, linking them to Liouville-Green approximation and asymptotic analysis, with applications to special functions and inequalities.

Findings

Bounds for ratios and derivatives of solutions are sharp and convergent.

Application to special functions yields new inequalities and zero bounds.

Analysis covers systems with positive and negative coefficient products.

Abstract

Many special functions are solutions of first order linear systems $y_n'(x)=a_n(x)y_n(x)+d_n(x)y_{n-1}(x)$, $y_{n-1}'(x)=b_n(x)y_{n-1}(x)+e_{n}(x)y_n(x)$. We obtain bounds for the ratios $y_n(x)/y_{n-1}(x)$ and the logarithmic derivatives of $y_n(x)$ for solutions of monotonic systems satisfying certain initial conditions. For the case $d_n(x)e_n(x)>0$, sequences of upper and lower bounds can be obtained by iterating the recurrence relation; for minimal solutions of the recurrence these are convergent sequences. The bounds are related to the Liouville-Green approximation for the associated second order ODEs as well as to the asymptotic behavior of the associated three-term recurrence relation as $n\rightarrow +\infty$; the bounds are sharp both as a function of $n$ and $x$. Many special functions are amenable to this analysis, and we give several examples of application: modified Bessel functions, parabolic cylinder functions, Legendre functions of imaginary variable and Laguerre functions. New Tur\'an-type inequalities are established from the function ratio bounds. Bounds for monotonic systems with $d_n(x)e_n(x)<0$ are also given, in particular for Hermite and Laguerre polynomials of real positive variable; in that case the bounds can be used for bounding the monotonic region (and then the extreme zeros).