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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Complete monotonicity of modified Bessel functions


Author: Mourad E. H. Ismail
Journal: Proc. Amer. Math. Soc. 108 (1990), 353-361
MSC: Primary 33A40; Secondary 35S99, 60E10
DOI: https://doi.org/10.1090/S0002-9939-1990-0993753-9
MathSciNet review: 993753
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Abstract: We prove that if $ \nu > 1/2$, then $ {2^{\nu - 1}}\Gamma (\nu )/[{x^{\nu /2}}{e^{\sqrt x }}{K_\nu }(\sqrt x )]$ is the Laplace transform of a selfdecomposable probability distribution while $ {2^\nu }\Gamma \left( {\nu + 1} \right){x^{ - \nu /2}}{e^{ - \sqrt x }}{I_\nu }\left( {\sqrt x } \right)$ is the Laplace transform of an infinitely divisible distribution. The former result is used to show that an estimate of $ {\text{M}}$. Wong [13] is sharp. We also prove that the roots of the equations

$\displaystyle {b^3}{l_{\nu - 1}}\left( {a\sqrt z } \right)/{I_\nu }\left( {a\sq... ...a^3}{I_{\nu - 1}}\left( {b\sqrt z } \right)/{I_\nu }\left( {b\sqrt z } \right),$

and

$\displaystyle {b^3}{K_{\nu + 1}}\left( {a\sqrt z } \right)/{K_\nu }\left( {a\sq... ...}\left( {b\sqrt z } \right)/{K_\nu }\left( {b\sqrt z } \right),\nu > 0,z \ne 0,$

lie in a certain sector contained in the open left half plane. This proves and extends a conjecture of $ {\text{H}}$. Hattori arising from his work in partial differential equations.

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DOI: https://doi.org/10.1090/S0002-9939-1990-0993753-9
Article copyright: © Copyright 1990 American Mathematical Society