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

@article{Segura2011OnBF,
title={On bounds for solutions of monotonic first order difference-differential systems},
author={Javier Segura},
journal={Journal of Inequalities and Applications},
year={2011},
volume={2012},
pages={1-17}
}
• 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…
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## References

SHOWING 1-10 OF 30 REFERENCES

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.
• Mathematics
SIAM J. Numer. Anal.
• 2003
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.
• Computer Science, Mathematics
TOMS
• 2006
In an accompanying article, the precise domains for these methods are described and the Fortran 90 codes for the computation of these functions are presented.
• Computer Science
TOMS
• 2006
Fortran 90 programs for the computation of real parabolic cylinder functions are presented and the aimed relative accuracy for scaled functions is better than 5 10<sup>−14</sup>.
• Biology
• 2007
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.
• Mathematics
• 2007
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
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,