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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
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.

References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Appl. Math. Series No. 55, Nat. Bur. Standards, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642 (29:4914)
  • [2] R. Askey and M. E. H. Ismail, Recurrence relations, continued fractions and orthogonal polynomials (in preparation).
  • [3] R. Askey and M. E. H. Ismail, The Rogers $ q$-ultraspherical polynomials, Approximation Theory, edited by E. Cheney, Academic Press, New York, 1980, pp. 175-182. MR 602713 (82h:42026)
  • [4] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. 2, McGraw-Hill, New York, 1953.
  • [5] W. Feller, An introduction to probability theory and its applications, vol. II, 2nd ed., Wiley, New York, 1970. MR 0088081 (19:466a)
  • [6] I.I. Hirschman and D. V. Widder, The convolution transform, Princeton Univ. Press, Princeton, N.J., 1955. MR 0073746 (17:479c)
  • [7] M. E. H. Ismail, Bessel functions and the infinite divisibility of the Student $ t$-distribution, Ann. Probab. 5 (1977), 582-585. MR 0448480 (56:6786)
  • [8] -, Integral representations and complete monotonicity of various quotients of Bessel functions, Canad. J. Math. 29 (1977), 1198-1207. MR 0463527 (57:3474)
  • [9] M. E. H. Ismail and D. H. Kelker, Special functions, Stieltjes transforms and infinite divisibility, SIAM J. Math. Anal. 10 (1979), 884-901. MR 541088 (80k:33005)
  • [10] M. E. H. Ismail and C. P. May, Special functions, infinite divisibility and transcendental equations, Math. Proc. Cambridge Philos. Soc. 85 (1979), 453-464. MR 520462 (80d:33005)
  • [11] J. Kent, Some probabilistic properties of Bessel functions, Ann. Probab. 6 (1978), 760-770. MR 0501378 (58:18750)
  • [12] K. S. Miller, Hypothesis testing with complex distributions, Krieger, Huntington, N.Y., 1980. MR 564654 (82b:62026)
  • [13] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Univ. Press, Cambridge, 1966. MR 1349110 (96i:33010)
  • [14] J. Wendel, Hitting spheres with Brownian motion, Ann. Probab. 8 (1980), 164-169. MR 556423 (80m:60085)
  • [15] J. Wimp, Orthogonal polynomials in the tabulation of Stieltjes transforms, J. Math. Anal. Appl. (to appear).

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Keywords: Modified Bessel functions, Laplace transform, Stieltjes transform, infinite divisibility
Article copyright: © Copyright 1982 American Mathematical Society

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