An Infinitely Divisible Distribution Involving Modified Bessel Functions

Mourad E. H. Ismail, Kenneth S. Miller
1982 Proceedings of the American Mathematical Society  
We prove that the function~v Kli(bxl'*)Ku(axll'1) is the Laplace transform of an infinitely divisible probability distribution when v > ß > 0 and b > a > 0. This implies the complete monotonie 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 F(x ;/i>I/) = G) ATM(oi1/2)iCl/(6i1/2)
doi:10.2307/2044288 fatcat:wdowj2qofjbqxadc5afsaaq7be