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

  title={On bounds for solutions of monotonic first order difference-differential systems},
  author={Javier Segura},
  journal={Journal of Inequalities and Applications},
  • J. Segura
  • Published 4 October 2011
  • Mathematics
  • Journal of Inequalities and Applications
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… 

Uniform (very) sharp bounds for ratios of parabolic cylinder functions

  • J. Segura
  • Mathematics
    Studies in Applied Mathematics
  • 2021
Parabolic cylinder functions are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and

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The Sturm comparison theorems for second order ODEs are classical results from which information on the properties of the zeros of special functions can be obtained. Sturm separation and comparison

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  • Robert E. Gaunt
  • Mathematics
    Journal of Mathematical Analysis and Applications
  • 2020

On bounds for Kummer's function ratio

Summary: In this paper we present lower and upper bounds for Kummer’s function ratios of the form M ( a,b,z ) ′ M ( a,b,z ) when 0 < a < b . The derived bounds are asymptotically precise,

Comments on the paper "Universal bounds and monotonicity properties of ratios of Hermite and Parabolic Cylinder functions".

In the abstract of [1] we read: "We obtain so far unproved properties of a ratio involving a classof Hermite and parabolic cylinder functions." However, we explain how some of the main results in

Remarks on the paper: "Bounds for functions involving ratios of modified Bessel functions"

A recent paper by C.G. Kokologiannaki published in J. Math. Anal. Appl. \cite{Kolo:2012:BFI} gives some properties for ratios of modified Bessel functions and, in particular, some bounds. These

Universal bounds and monotonicity properties of ratios of Hermite and parabolic cylinder functions

  • T. Koch
  • Mathematics
    Proceedings of the American Mathematical Society
  • 2020
We obtain so far unproved properties of a ratio involving a class of Hermite and parabolic cylinder functions. Those ratios are shown to be strictly decreasing and bounded by universal constants.

Turán Type Inequalities for Tricomi Confluent Hypergeometric Functions

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The Zeros of Special Functions from a Fixed Point Method

The structure of the first order difference-differential equations (DDEs) is studied to set global bounds on the differences between adjacent zeros of functions of consecutive orders and to find iteration steps which guarantee that all the zeros inside a given interval can be found with certainty.

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Two fixed point methods are introduced that provide algorithms for the efficient computation of the zeros and turning points of a broad family of special functions, includinghypergeometric and confluent hypergeometric functions of real parameters and variables, Bessel, Airy, Coulomb, and conical functions, among others.

Computing the real parabolic cylinder functions U(a, x), V(a, x)

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Algorithm 850: Real parabolic cylinder functions U(a, x), V(a, x)

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Numerical methods for special functions

This book provides an up-to-date overview of methods for computing special functions and discusses when to use them in standard parameter domains, as well as in large and complex domains.

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Monotonicity Properties of Determinants of Special Functions

AbstractWe prove the absolute monotonicity or complete monotonicity of some determinant functions whose entries involve $\psi^{(m)}(x)=({d^m}/{dx^m}) [\Gamma'(x)/\Gamma(x)],$ modified Bessel

Bounds on iterated coerror functions and their ratios

Upper and lower bounds on yn = in erfc(x) and rn = Yn/Yn-i, n > 1, X 1, expressing monotone decreasing behavior of rn(x) = x)lyn-,(x) n > 1, in both n and x,