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# A certain class of completely monotonic sequences

*Advances in Difference Equations*
**volume 2013**, Article number: 294 (2013)

## Abstract

In this article, we present some necessary conditions, a sufficient condition and a necessary and sufficient condition for sequences to be completely monotonic. One counterexample is also presented.

**MSC:**40A05, 26A45, 26A48, 39A60.

## 1 Introduction and the main results

We first recall some definitions and basic results on or related to completely monotonic sequences and completely monotonic functions.

**Definition 1** [1]

A sequence {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} is called a moment sequence if there exists a function \alpha (t) of bounded variation on the interval [0,1] such that

Here, in Definition 1 and throughout the paper,

and ℕ is the set of all positive integers.

**Definition 2** [1]

A sequence {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} is called completely monotonic if

where

and

Such a sequence is called *totally monotone* in [2].

From Definition 2, using mathematical induction, we can prove, for a completely monotonic sequence {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}}, that the sequence {\{{(-1)}^{m}{\mathrm{\Delta}}^{m}{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} is non-increasing for any fixed m\in {\mathbb{N}}_{0}, and that the sequence {\{{(-1)}^{m}{\mathrm{\Delta}}^{m}{\mu}_{n}\}}_{m=0}^{\mathrm{\infty}} is non-increasing for any fixed n\in {\mathbb{N}}_{0}. The difference equation (4) plays an important role in the proofs of these properties and our main results of this paper.

In [3], the authors showed that for a completely monotonic sequence {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}}, we always have

unless {\mu}_{n}=c, a constant for all n\in \mathbb{N}.

Let

It was shown (see [1]) as follows.

**Theorem 1**
*A sequence*
{\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}}
*is a moment sequence if and only if there exists a constant*
*L*
*such that*

*where in* (7), {\lambda}_{k,m} *is defined by* (6).

For completely monotonic sequences, the following is the well-known Hausdorff’s theorem (see [1]).

**Theorem 2** *A sequence* {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} *is completely monotonic if and only if there exists a non*-*decreasing and bounded function* \alpha (t) *on* [0,1] *such that*

From this theorem, we know (see [1]) that a completely monotonic sequence is a moment sequence and is as follows.

**Theorem 3** *A necessary and sufficient condition that the sequence* {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} *should be a moment sequence is that it should be the difference of two completely monotonic sequences*.

We also recall the following definition.

**Definition 3** [1]

A function *f* is said to be completely monotonic on an interval *I* if *f* is continuous on *I* has derivatives of all orders on {I}^{o} (the interior of *I*) and for all n\in {\mathbb{N}}_{0},

Some mathematicians use the terminology *completely monotone* instead of completely monotonic. The class of all completely monotonic functions on the interval *I* is denoted by CM(I).

The completely monotonic functions and completely monotonic sequences have remarkable applications in probability and statistics [4–10], physics [11, 12], numerical and asymptotic analysis [2], *etc*.

For the completely monotonic functions on the interval [0,\mathrm{\infty}), Widder proved (see [1]).

**Theorem 4** *A function* *f* *on the interval* [0,\mathrm{\infty}) *is completely monotonic if and only if there exists a bounded and non*-*decreasing function* \alpha (t) *on* [0,\mathrm{\infty}) *such that*

There is rich literature on completely monotonic functions. For more recent works, see, for example, [13–26].

There exists a close relationship between completely monotonic functions and completely monotonic sequences. For example, Widder [27] showed the following.

**Theorem 5** *Suppose that* f\in CM[a,\mathrm{\infty}), *then for any* \delta \geqq 0, *the sequence* {\{f(a+n\delta )\}}_{n=0}^{\mathrm{\infty}} *is completely monotonic*.

This result was generalized in [28] as follows.

**Theorem 6** *Suppose that* f\in CM[a,\mathrm{\infty}). *If the sequence* {\{\mathrm{\Delta}{x}_{k}\}}_{k=0}^{\mathrm{\infty}} *is completely monotonic and* {x}_{0}\geqq a, *then the sequence* {\{f({x}_{k})\}}_{k=0}^{\mathrm{\infty}} *is also completely monotonic*.

For the meaning of \mathrm{\Delta}{x}_{k}, k\in {\mathbb{N}}_{0} in Theorem 6, see (3) and (4).

Suppose that f\in CM[0,\mathrm{\infty}). By Theorem 5, we know that {\{f(n)\}}_{n=0}^{\mathrm{\infty}} is completely monotonic.

The following result was obtained in [16].

**Theorem 7** *Suppose that the sequence* {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} *is completely monotonic*, *then for any* \epsilon \in (0,1), *there exists a continuous interpolating function* f(x) *on the interval* [0,\mathrm{\infty}) *such that* f{|}_{[0,\epsilon ]} *and* f{|}_{[\epsilon ,\mathrm{\infty})} *are both completely monotonic and*

From this result or Theorem 2, we can get the following.

**Theorem 8** *Suppose that the sequence* {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} *is completely monotonic*. *Then there exists a completely monotonic interpolating function* g(x) *on the interval* [1,\mathrm{\infty}) *such that*

It should be noted that (see [[1], Chapter IV]) under the condition of Theorem 8, we cannot guarantee that there exists a completely monotonic interpolating function g(x) on the interval [0,\mathrm{\infty}) such that

In this article, we shall further investigate the properties of the completely monotonic sequences. We shall give some necessary conditions, a sufficient condition and a necessary and sufficient condition for sequences to be completely monotonic. More precisely we have the following results.

**Theorem 9** *Suppose that the sequence* {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} *is completely monotonic*. *Then*, *for any* m\in {\mathbb{N}}_{0}, *the series*

*converges and*

**Corollary 1** *Suppose that the sequence* {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} *is completely monotonic*. *Then for* m,k\in {\mathbb{N}}_{0},

**Remark 1** Although from the complete monotonicity of the sequence {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}}, we can deduce that for any m\in {\mathbb{N}}_{0}, the series

converges, it cannot guarantee the convergence of the series

For example, let

Since the function

is completely monotonic on the interval [0,\mathrm{\infty}), by Theorem 5, we see that the sequence

is completely monotonic. This conclusion can also be obtained by setting

in Theorem 2.

We can verify that

Hence,

is the famous harmonic series, which is divergent.

**Theorem 10** *Suppose that the sequence* {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} *is completely monotonic*. *Then for any* k,m\in {\mathbb{N}}_{0},

**Theorem 11**
*Suppose that the sequence*
{\{{\mu}_{n}\}}_{n=1}^{\mathrm{\infty}}
*is completely monotonic and that the series*

*converges*. *Let* {\mu}_{0} *be such that*

*Then the sequence* {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} *is completely monotonic*.

**Theorem 12** *A necessary and sufficient condition for the sequence* {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} *to be completely monotonic is that the sequence* {\{{\mu}_{n}\}}_{n=1}^{\mathrm{\infty}} *is completely monotonic*, *the series*

*converges and*

## 2 Proofs of the main results

Now, we are in a position to prove the main results.

*Proof of Theorem 9* Since {\{{\mu}_{n}\}}_{n=0}^{\mathrm{\infty}} is completely monotonic, by Theorem 2, there exists a non-decreasing and bounded function \alpha (t) on the interval [0,1] such that

From (3), (4) and (13), we can prove that

Now, for k\in \mathbb{N}, we have

Hence, for k\in \mathbb{N},

Since

from (15), we get, for k\geqq 1,

From (16), we also know that

is a positive series. Then by (17), we obtain that

converges and that

The proof of Theorem 9 is thus completed. □

*Proof of Corollary 1* This corollary can be obtained from (15). □

*Proof of Theorem 10* Let *m* be a fixed non-negative integer.

From Theorem 9, we see that

which means that (12) is valid for k=0.

Suppose that (12) is valid for k=r. Then

which means that (12) is valid for k=r+1. Therefore, by mathematical induction, (12) is valid for all k\in {\mathbb{N}}_{0}. The proof of Theorem 10 is completed. □

*Proof of Theorem 11* By the definition of completely monotonic sequence, we only need to prove that

We first prove that

From the condition of Theorem 11, (22) is valid for k=0.

Suppose that (22) is valid for k=m. Then we have

which means that (22) is valid for k=m+1. Therefore, by mathematical induction, (22) is valid for all k\in {\mathbb{N}}_{0}.

Since

is a convergent positive series, we know that

From (22) and (24), we obtain that

The proof of Theorem 11 is completed. □

*Proof of Theorem 12* By Definition 2 and by setting m=0 in Theorem 9, we see that the condition is necessary. By Theorem 11, we know that the condition is sufficient. The proof of Theorem 12 is completed. □

## References

Widder DV:

*The Laplace Transform*. Princeton University Press, Princeton; 1946.Wimp J:

*Sequence Transformations and Their Applications*. Academic Press, New York; 1981.Lorch L, Moser L: A remark on completely monotonic sequences, with an application to summability.

*Can. Math. Bull.*1963, 6: 171-173. 10.4153/CMB-1963-016-3Feller W 2. In

*An Introduction to Probability Theory and Its Applications*. Wiley, New York; 1966.Kimberling CH: A probabilistic interpretation of complete monotonicity.

*Aequ. Math.*1974, 10: 152-164. 10.1007/BF01832852Kimberling CH: Exchangeable events and completely monotonic sequences.

*Rocky Mt. J. Math.*1973, 3: 565-574. 10.1216/RMJ-1973-3-4-565Kuk AYC: A litter-based approach to risk assessment in developmental toxicity studies via a power family of completely monotone functions.

*J. R. Stat. Soc. C*2004, 53: 369-386. 10.1046/j.1467-9876.2003.05369.xSandhya E, Satheesh S: On distribution functions with completely monotone derivative.

*Stat. Probab. Lett.*1996, 27: 127-129. 10.1016/0167-7152(95)00053-4Satheesh S, Sandhya E: Distributions with completely monotone probability sequences.

*Far East J. Theor. Stat.*1997, 1: 69-75.Trimble SY, Wells J, Wright FT: Superadditive functions and a statistical application.

*SIAM J. Math. Anal.*1989, 20: 1255-1259. 10.1137/0520082Day WA: On monotonicity of the relaxation functions of viscoelastic materials.

*Proc. Camb. Philos. Soc.*1970, 67: 503-508. 10.1017/S0305004100045771Franosch T, Voigtmann T: Completely monotone solutions of the mode-coupling theory for mixtures.

*J. Stat. Phys.*2002, 109: 237-259. 10.1023/A:1019991729106Alzer H, Batir N: Monotonicity properties of the gamma function.

*Appl. Math. Lett.*2007, 20: 778-781. 10.1016/j.aml.2006.08.026Batir N: On some properties of the gamma function.

*Expo. Math.*2008, 26: 187-196. 10.1016/j.exmath.2007.10.001Guo S: Logarithmically completely monotonic functions and applications.

*Appl. Math. Comput.*2013, 221: 169-176.Guo S: Some properties of completely monotonic sequences and related interpolation.

*Appl. Math. Comput.*2013, 219: 4958-4962. 10.1016/j.amc.2012.11.073Guo S, Qi F: A class of logarithmically completely monotonic functions associated with the gamma function.

*J. Comput. Appl. Math.*2009, 224: 127-132. 10.1016/j.cam.2008.04.028Guo S, Qi F, Srivastava HM: A class of logarithmically completely monotonic functions related to the gamma function with applications.

*Integral Transforms Spec. Funct.*2012, 23: 557-566. 10.1080/10652469.2011.611331Guo S, Qi F, Srivastava HM: Supplements to a class of logarithmically completely monotonic functions associated with the gamma function.

*Appl. Math. Comput.*2008, 197: 768-774. 10.1016/j.amc.2007.08.011Guo S, Qi F, Srivastava HM: Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic.

*Integral Transforms Spec. Funct.*2007, 18: 819-826. 10.1080/10652460701528933Guo S, Srivastava HM: A certain function class related to the class of logarithmically completely monotonic functions.

*Math. Comput. Model.*2009, 49: 2073-2079. 10.1016/j.mcm.2009.01.002Guo S, Srivastava HM: A class of logarithmically completely monotonic functions.

*Appl. Math. Lett.*2008, 21: 1134-1141. 10.1016/j.aml.2007.10.028Guo S, Xu J-G, Qi F: Some exact constants for the approximation of the quantity in the Wallis’ formula.

*J. Inequal. Appl.*2013., 2013: Article ID 67Qi F, Guo S, Guo B-N: Complete monotonicity of some functions involving polygamma functions.

*J. Comput. Appl. Math.*2010, 233: 2149-2160. 10.1016/j.cam.2009.09.044Sevli H, Batir N: Complete monotonicity results for some functions involving the gamma and polygamma functions.

*Math. Comput. Model.*2011, 53: 1771-1775. 10.1016/j.mcm.2010.12.055Srivastava HM, Guo S, Qi F: Some properties of a class of functions related to completely monotonic functions.

*Comput. Math. Appl.*2012, 64: 1649-1654. 10.1016/j.camwa.2012.01.016Widder DV: Necessary and sufficient conditions for the representation of a function as a Laplace integral.

*Trans. Am. Math. Soc.*1931, 33: 851-892. 10.1090/S0002-9947-1931-1501621-6Lorch L, Newman DJ: On the composition of completely monotonic functions, completely monotonic sequences and related questions.

*J. Lond. Math. Soc.*1983, 28: 31-45.

## Acknowledgements

The authors thank the editor and the referees, one of whom brought our attention to the reference [6], for their valuable suggestions to improve the quality of this paper. The present investigation was supported, in part, by the Natural Science Foundation of Henan Province of China under Grant 112300410022.

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Guo, S., Srivastava, H.M. & Batir, N. A certain class of completely monotonic sequences.
*Adv Differ Equ* **2013**, 294 (2013). https://doi.org/10.1186/1687-1847-2013-294

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DOI: https://doi.org/10.1186/1687-1847-2013-294

### Keywords

- necessary condition
- sufficient condition
- necessary and sufficient condition
- difference equation
- moment sequence
- completely monotonic sequence
- completely monotonic function
- bounded variation
- Stieltjes integral