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Special functions, infinite divisibility and transcendental equations

  • Mourad E. H. Ismail (a1) and C. Ping May (a2)

We establish integral representations for quotients of Tricomi ψ functions and of several quotients of modified Bessel functions and of linear combinations of them. These integral representations are used to prove the complete monotonicity of the functions considered and to prove the infinite divisibility of a three parameter probability distribution. Limiting cases of this distribution are the hitting time distributions considered recently by Kent and Wendel. We also derive explicit formulas for the Kent–Wendel probability density functions.

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(2) L. Bondesson A general result in infinite divisibility. Ann. Prob. 7 (1979) (To appear.)

(3) H. Buchholz The confluent hypergeometric function (New York, Springer-Verlag, 1969).

(4) L. Durand Nicholson-type integrals for products of Jacobi functions I. SIAM J. Math. Anal. 9, (1978), 7686.

(9) E. Grosswald The Student t-distribution of any degrees of freedom is infinitely divisible. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36, (1976), 103109.

(11) M. E. H. Ismail Bessel functions and the infinite divisibility of the Student t-distribution. Ann. Prob. 5, (1977), 582585.

(12) M. E. H. Ismail Integral representations and complete monotonicity of various quotients of Bessel functions. Canadian J. Math. 29, (1977), 11981207.

(13) M. E. H. Ismail and D. H. Kelker Special functions, Stieltjes transforms and infinite divisibility. SIAM J. Math. Anal. 10 (1979) (To appear.)

(14) J. Kent Some probabilistic properties of Bessel functions. Ann. Prob. 6, (1978), 760770.

(17) G. E. Siewert and E. E. Burniston An exact analytical solution of x coth x = αx2 + 1. J. Comp. Appl. Math. 2 (1976), 1921.

(19) O. Thorin On the infinite divisibility of the Pareto distribution. Scand. Actuarial J. (1977), 3140.

(20) O. Thorin On the infinite divisibility of the log normal distribution. Scand. Actuarial J. (1977), 121148.

(21) F. G. Tricomi Über die Abzählung der Nullstellen der konfluenten hypergeometrischen Funktionen. Math. Zeit. 52, (1950), 669675.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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