2011/10/04 by Javier Segura, Segura, Javier
Mathematics · #26D20 #33CXX #34C10 #34C11 #39A06 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical functions and polynomials #Nonlinear Differential Equations Analysis #math.CA #msc:26D20 #msc:33CXX #msc:34C10 #msc:34C11 #msc:39A06
paper · pdf · doi:10.48550/arxiv.1110.0870
arxiv created 2011/10/04 · openalex publication_date 2011/10/04 · arxiv updated 2011/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Many special functions are solutions of first order linear systems yn'(x)=an(x)yn(x)+dn(x)yn-1(x), yn-1'(x)=bn(x)yn-1(x)+en(x)yn(x). We obtain bounds for the ratios yn(x)/yn-1(x) and the logarithmic derivatives of yn(x) for solutions of monotonic systems satisfying certain initial conditions. For the case dn(x)en(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→ +∞; 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 Turán-type inequalities are established from the function ratio bounds. Bounds for monotonic systems with dn(x)en(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).