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

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).