# Completely monotonic functions

@article{Miller2001CompletelyMF, title={Completely monotonic functions}, author={K. S. Miller and Stefan G. Samko}, journal={Integral Transforms and Special Functions}, year={2001}, volume={12}, pages={389 - 402} }

In this expository article we survey some properties of completely monotonic functions and give various examples, including some famous special functions. Such function are useful, for example, in probability theory. It is known, [1, p.450], for example, that a function w is the Laplace transform of an infinitely divisible probability distribution on (0,∞), if and only if w = e-h , where the derivative of h is completely monotonic and h(0+) = 0.

## 287 Citations

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## References

SHOWING 1-10 OF 46 REFERENCES

Integral Representations and Complete Monotonicity of Various Quotients of Bessel Functions

- MathematicsCanadian Journal of Mathematics
- 1977

Complete monotonicity of functions, Definition 3.1, is often proved by showing that their inverse Laplace transforms are nonnegative. There are relatively few simple functions whose inverse Laplace…

An infinitely divisible distribution involving modified Bessel functions

- Mathematics
- 1982

We prove that the function (yb h` K/(bx 1/2)K,(axl/2) Va Kj,(ax1/2)K,,(bx1/2) is the Laplace transform of an infinitely divisible probability distribution when v > IA > 0 and b > a > 0. This implies…

The completely monotonic character of the Mittag-Leffler function $E_a \left( { - x} \right)$

- Mathematics
- 1948

where L consists of three parts as follows: &: the line y= —(tan \f/)x from x— + °° to # = p , p > 0 . C2: an arc of circle \z\ = p sec \p> — ̂ ^ a r g z^\//. Cz\ the reflection of G in the x-axis.…

On mittag-leffler type function, fractional calculas operators and solutions of integral equations

- Mathematics
- 1996

The special entire function of the form with is introduced, where α>0, m>0 and α(im+1)+1≠ 0,−1, −2,.....for i=0,1,2,....For m = 1, Eα1,l(z) coincides with the Mittag-Leffler function Eα,α+1 ,with…

The student t-distribution of any degree of freedom is infinitely divisible

- Mathematics
- 1976

values of n and conjectured that this is always the case. This conjecture is proved here by a twofold application of Bernstein's theorem and the use of some special properties of the zeros of the…

On Mittag-Leffler type function and applications

- Mathematics
- 1998

This is the continuation of the paper [4] which was devoted to introduce a special entire function named a Mittag -Leffler type function E α,m,l (z) to discuss its connections with the Riemnn…

Special Functions, Stieltjes Transforms and Infinite Divisibility

- Mathematics
- 1979

We establish the complete monotonicity of several quotients of Whittaker (Tricomi) functions and of parabolic cylinder functions. These results are used to show that the F distribution of any…

Some Integral Equations Involving Hypergeometric Functions

- Mathematics
- 1967

An integral equation of the first kind, with kernel involving a hypergeometric function, is discussed. Conditions sufficient for uniqueness of solutions are given, then conditions necessary for…

Complete monotonicity of modified Bessel functions

- Mathematics
- 1990

We prove that if ν>1/2, then 2 ν−1 Γ(ν)/[x ν/2 e √ x K ν (√ x )] is the Laplace transform of a selfdecomposable probability distribution while 2 ν Γ(ν+1)x −ν/2 e − √ x I ν (√x) is the Laplace…

Tables of Integrals

- Mathematics
- 1962

By Herbert Bristol Dwight New York. London: The Macmillan Company. Pp. x + 336. Price 26s. A deservedly popular work of reference, first published almost thirty years ago, this book now appears in a…