An infinitely divisible distribution involving modified Bessel functions
Authors:
Mourad E. H. Ismail and Kenneth S. Miller
Journal:
Proc. Amer. Math. Soc. 85 (1982), 233238
MSC:
Primary 60E07; Secondary 33A40
MathSciNet review:
652449
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We prove that the function is the Laplace transform of an infinitely divisible probability distribution when and . 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 lies in the right halfplane. We conjecture that also is the Laplace transform of an infinitely divisible probability distribution. It is known that in the limit as , the infinite divisibility property holds for both functions.
 [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
(29 #4914)
 [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 𝑞ultraspherical polynomials,
Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980),
Academic Press, New YorkLondon, 1980, pp. 175–182. MR 602713
(82h:42026)
 [4]
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. 2, McGrawHill, 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
(19,466a)
 [6]
I.
I. Hirschman and D.
V. Widder, The convolution transform, Princeton University
Press, Princeton, N. J., 1955. MR 0073746
(17,479c)
 [7]
Mourad
E. H. Ismail, Bessel functions and the infinite divisibility of the
Student 𝑡distribution, Ann. Probability 5
(1977), no. 4, 582–585. MR 0448480
(56 #6786)
 [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
(57 #3474)
 [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
(80k:33005), http://dx.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
(80d:33005), http://dx.doi.org/10.1017/S0305004100055912
 [11]
John
Kent, Some probabilistic properties of Bessel functions, Ann.
Probab. 6 (1978), no. 5, 760–770. MR 0501378
(58 #18750)
 [12]
Kenneth
S. Miller, Hypothesis testing with complex distributions,
Robert E. Krieger Publishing Co., Huntington, N.Y., 1980. Applied
Mathematics Series. MR 564654
(82b:62026)
 [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
(96i:33010)
 [14]
J.
G. Wendel, Hitting spheres with Brownian motion, Ann. Probab.
8 (1980), no. 1, 164–169. MR 556423
(80m:60085)
 [15]
J. Wimp, Orthogonal polynomials in the tabulation of Stieltjes transforms, J. Math. Anal. Appl. (to appear).
 [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 ultraspherical polynomials, Approximation Theory, edited by E. Cheney, Academic Press, New York, 1980, pp. 175182. MR 602713 (82h:42026)
 [4]
 A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. 2, McGrawHill, 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 distribution, Ann. Probab. 5 (1977), 582585. MR 0448480 (56:6786)
 [8]
 , Integral representations and complete monotonicity of various quotients of Bessel functions, Canad. J. Math. 29 (1977), 11981207. 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), 884901. 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), 453464. MR 520462 (80d:33005)
 [11]
 J. Kent, Some probabilistic properties of Bessel functions, Ann. Probab. 6 (1978), 760770. 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), 164169. MR 556423 (80m:60085)
 [15]
 J. Wimp, Orthogonal polynomials in the tabulation of Stieltjes transforms, J. Math. Anal. Appl. (to appear).
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
60E07,
33A40
Retrieve articles in all journals
with MSC:
60E07,
33A40
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198206524499
PII:
S 00029939(1982)06524499
Keywords:
Modified Bessel functions,
Laplace transform,
Stieltjes transform,
infinite divisibility
Article copyright:
© Copyright 1982
American Mathematical Society
