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 | References | Similar Articles | Additional Information

Abstract: We prove that the function

**[1]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[2]**R. Askey and M. E. H. Ismail,*Recurrence relations, continued fractions and orthogonal polynomials*(in preparation).**[3]**Richard A. Askey and Mourad E. H. Ismail,*The Rogers 𝑞-ultraspherical polynomials*, Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), Academic Press, New York-London, 1980, pp. 175–182. MR**602713****[4]**A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,*Higher transcendental functions*, vol. 2, McGraw-Hill, New York, 1953.**[5]**William Feller,*An introduction to probability theory and its applications. Vol. I*, John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1957. 2nd ed. MR**0088081****[6]**I. I. Hirschman and D. V. Widder,*The convolution transform*, Princeton University Press, Princeton, N. J., 1955. MR**0073746****[7]**Mourad E. H. Ismail,*Bessel functions and the infinite divisibility of the Student 𝑡-distribution*, Ann. Probability**5**(1977), no. 4, 582–585. MR**0448480****[8]**Mourad E. H. Ismail,*Integral representations and complete monotonicity of various quotients of Bessel functions*, Canad. J. Math.**29**(1977), no. 6, 1198–1207. MR**0463527**, https://doi.org/10.4153/CJM-1977-119-5**[9]**Mourad E. H. Ismail and Douglas H. Kelker,*Special functions, Stieltjes transforms and infinite divisibility*, SIAM J. Math. Anal.**10**(1979), no. 5, 884–901. MR**541088**, https://doi.org/10.1137/0510083**[10]**Mourad E. H. Ismail and C. Ping May,*Special functions, infinite divisibility and transcendental equations*, Math. Proc. Cambridge Philos. Soc.**85**(1979), no. 3, 453–464. MR**520462**, https://doi.org/10.1017/S0305004100055912**[11]**John Kent,*Some probabilistic properties of Bessel functions*, Ann. Probab.**6**(1978), no. 5, 760–770. MR**0501378****[12]**Kenneth S. Miller,*Hypothesis testing with complex distributions*, Robert E. Krieger Publishing Co., Huntington, N.Y., 1980. Applied Mathematics Series. MR**564654****[13]**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110****[14]**J. G. Wendel,*Hitting spheres with Brownian motion*, Ann. Probab.**8**(1980), no. 1, 164–169. MR**556423****[15]**J. Wimp,*Orthogonal polynomials in the tabulation of Stieltjes transforms*, J. Math. Anal. Appl. (to appear).

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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