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Proceedings of the American Mathematical Society

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

 

 

An infinitely divisible distribution involving modified Bessel functions


Authors: Mourad E. H. Ismail and Kenneth S. Miller
Journal: Proc. Amer. Math. Soc. 85 (1982), 233-238
MSC: Primary 60E07; Secondary 33A40
DOI: https://doi.org/10.1090/S0002-9939-1982-0652449-9
MathSciNet review: 652449
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Abstract: We prove that the function

$\displaystyle {\left( {\frac{b} {a}} \right)^{\mu - v}}\frac{{{K_\mu }(b{x^{1/2}}){K_v}(a{x^{1/2}})}} {{{K_\mu }(a{x^{1/2}}){K_v}(b{x^{1/2}})}}$

is the Laplace transform of an infinitely divisible probability distribution when $ v > \mu \geqslant 0$ and $ b > a > 0$. This implies the complete monotonic ity of the function. We also establish a representation as a Stieltjes transform, which implies in particular that the function has positive real part when $ x$ lies in the right half-plane. We conjecture that

$\displaystyle {\left( {\frac{b} {a}} \right)^{\mu - v}}\frac{{{I_\mu }(a{x^{1/2}}){I_v}(b{x^{1/2}})}} {{{I_\mu }(b{x^{1/2}}){I_v}(a{x^{1/2}})}}$

also is the Laplace transform of an infinitely divisible probability distribution. It is known that in the limit as $ v \to \infty $, the infinite divisibility property holds for both functions.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0652449-9
Keywords: Modified Bessel functions, Laplace transform, Stieltjes transform, infinite divisibility
Article copyright: © Copyright 1982 American Mathematical Society