Higher Order Thorin–Bernstein Functions

We investigate subclasses of generalized Bernstein functions related to complete Bernstein and Thorin–Bernstein functions. Representations in terms of incomplete beta and gamma as well as hypergeometric functions are presented. Several special cases and examples are discussed.


Introduction
In this paper subclasses of the so-called generalized Bernstein functions of positive order are investigated.A non-negative function f defined on (0, ∞) having derivatives of all orders is a generalized Bernstein function of order λ > 0 if f (x)x 1−λ is a completely monotonic function.This is equivalent to f admitting an integral representation of the form where a and b are non-negative numbers, and μ is a positive measure on (0, ∞) making the integral converge for all x > 0. Here, γ denotes the incomplete gamma function γ(λ, t) = t 0 e −u u λ−1 du.
Research supported by grant DFF-1026-00267B from The Danish Council for Independent Research | Natural Sciences.

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Page 2 of 22 S. Koumandos and H. L. Pedersen Results Math The convergence of the integral in ( 1) is equivalent to The class of generalized Bernstein functions of order λ is denoted by B λ and was studied in [12].
A function g is called a generalized Stieltjes function of order λ if there exist a positive measure ν on [0, ∞) and a non-negative constant c such that for x > 0. The class of these functions, denoted by S λ , can also be characterized in terms of the Laplace transform L: g ∈ S λ if and only if This relation will be used throughout the paper.Several properties of generalized Stieltjes functions are given in [13] and [14].(For λ = 1 the class is the class of Stieltjes functions, denoted by S.) The classes B λ and S λ are closely related: in fact, the map Φ : B λ → S λ defined as is a bijection.See [12,Theorem 3.1].
Let us also recall the definition of a completely monotonic function of order α.A function f : (0, ∞) → [0, ∞) is completely monotonic of order α if x α f (x) is completely monotonic.This class was introduced and characterized in [11].Let us remark, that α in this definition can be any real number.The class of completely monotonic functions of order α is denoted by C α .
The functions studied in this paper correspond to putting some extra conditions on the measure μ in the representation (1).Definition 1.1.We say that a function f : (0, ∞) → R is a higher order Thorin-Bernstein function, if there exist λ > 0 and α < λ + 1 such that where a and b are non-negative numbers, and ϕ is a completely monotonic function of order α.
For given numbers λ and α the class of functions satisfying (2) is denoted by T λ,α , and they are called (λ, α)-Thorin-Bernstein functions.
The higher order Thorin-Bernstein functions are closely related to the incomplete Beta function B, defined for x ∈ [0, 1), a > 0, and b ∈ R as In fact, letting ϕ(t) = t −α e −ct = t −α L( c )(t), for c > 0, where c denotes the point mass at c, it follows that which is a special case of Proposition 2.10.See also Example 2.12.Moreover, the incomplete Beta function is the main building block of the classes of higher order Thorin-Bernstein functions.See Corollary 3.2.
Bondesson introduced in [4, p. 150] the classes T 1,β and they also appear in [16,Chapter 8].The higher order Thorin-Bernstein functions defined above play similar roles but in the context of generalized Bernstein functions of positive order λ.
In this paper we shall relate these functions to, among other things, generalized Stieltjes functions and find representations in terms of hypergeometric functions.Proposition 2.10 gives an equivalent integral representation of the functions f ∈ T λ,α in terms of the measure representing the completely monotonic function t α ϕ(t) as a Laplace transform.This leads to a characterization in Theorem 2.14 of the class T λ,α in terms of generalized Stieltjes functions of positive order: Based on the above result we prove that any generalized Bernstein function of order λ > 0 is the pointwise limit of a sequence of functions from ∪ α<λ+1 T λ,α .This result generalizes the corresponding result for Bernstein functions, obtained by Bondesson.The motivation for studying these classes comes from specific examples including the incomplete Beta function.The incomplete Beta function plays, as the incomplete Gamma functions, an extensive role in probability and mathematical statistics.It also appears in e.g.Monte Carlo sampling in statistical mechanics (see [9,10]).For additional applications we refer to [5] and the references given therein.
In this paper all measures are supposed to be positive Radon measures.

Fundamental Results
As mentioned above the classes T λ,α for fixed λ > 0 are nested.Their intersection ∩ α<λ+1 T λ,α turns out to be the functions {ax λ + b | a, b ≥ 0}.This can be seen from Theorem 2.14 as follows: if f belongs to the intersection then x 1−λ f (x) belongs to ∩ β>0 S β .This intersection, however, is equal to the constant functions (see [17]).The class T λ,α is closed under pointwise limits: Proof.Since T λ,α is a subclass of B λ and B λ is closed under pointwise limits then f ∈ B λ and furthermore, f n → f .See [12,Proposition 2.4].This entails x 1−λ f n (x) → x 1−λ f (x).From Theorem 2.14 we also have x 1−λ f n (x) ∈ S λ+1−α .Since the Stieltjes classes are also closed under pointwise limits (see [8,Theorem 10]), x 1−λ f (x) ∈ S λ+1−α .Using again Theorem 2.14 we see that Since B λ is closed under pointwise limits the limit of any pointwise convergent sequence of functions from ∪ α<λ+1 T λ,α belongs to B λ .Conversely, we have the following proposition.Proposition 2.2.Let f ∈ B λ .Then there is a sequence from ∪ α<λ+1 T λ,α that converges pointwise to f .The proof of Proposition 2.2 relies on the fact that any completely monotonic function is the pointwise limit of a sequence of generalized Stieltjes functions of positive order.See [8].For the reader's convenience and in order to have a basis for proving Proposition 2.2 we describe briefly a slightly different approach for proving this result from [8].
If f is completely monotonic then by Bernstein's theorem, for some positive measure σ on [0, ∞).Then take σ n = σ| [0,n] and let These functions converge pointwise to f (and even uniformly on [r, ∞) for any r > 0) and they are bounded by σ([0, n]).
Next define It can be shown that {h k } is a non-negative and decreasing sequence of functions and furthermore that (In fact, one may argue that h k has a unique maximum on [0, ∞) and that the maximal value does not exceed e/k.) Now, given a completely monotonic function f we construct {f n } as above and then we choose, for any n so large that σ([0, n]) > 0, k n such that Next we put g n (x) = n 0 (1 + tx/k n ) −kn dσ(t) and notice that g n ∈ S kn and also that g n − f n converges uniformly to 0, indeed: This gives as n → ∞.
Proof of Proposition 2.2.Let f ∈ B λ .Then x 1−λ f (x) is completely monotonic, and thus of the form Indeed, the integrand tends pointwise to 0 and is an integrable majorant so that Lebesgue's theorem on dominated convergence can be applied. Furthermore, The first term tends to 0, and the second term tends, by monotone convergence, to Thus, defining From Theorem 2.14 it now follows that F n ∈ T λ,λ+1−kn .This completes the proof.
By inspection of the construction preceeding the proof of Proposition 2.2 one sees that any bounded completely monotonic function is the uniform limit of a sequence of bounded generalized Stieltjes functions.This observation can be used to obtain the next corollary, which refines [12,Proposition 3.5].
Corollary 2.3.For any finite Borel measure μ on [0, ∞) there exists a sequence of bounded generalized Bernstein functions b n of positive order and a sequence of non-negative numbers c n such that Proof.Given μ we notice that f = L(μ) is a bounded completely monotonic function.As remarked above there is a sequence , where and therefore σ n → μ weakly.Defining This proves the assertion.
It should be noted that bounded generalized Bernstein functions have been characterized in [12,Proposition 3.9].
Indeed, we have according to [12,Proposition 2.4] that Also, using Theorem 2.14 and this gives Next we develop the theory aiming, among other things, at the proof of Theorem 2.14.The first lemma is a simple consequence of Fubini's theorem.Lemma 2.5.For any non-negative Borel measurable function g and any positive measure μ we have The next result is about Laplace transforms and convolution measures.For the reader's convenience let us mention that the convolution measure μ * ν of two measures μ and ν is given as the image measure of the product measure under the map τ (x, t) = x + t.Lemma 2.6.Let μ and ν be two measures on [0, ∞) and let β > 0. Then Proof.This follows from Lemma 2.5 and the fact that Remark 2.7.Because of positivity of the function and measure in Lemma 2.6 interchanging the order of integration is permitted.It may of course happen that the integrals equal ∞.For values of t near 0, L(μ)(t)L(ν)(t) is in general bounded from below by some positive constant.Hence, in order for the integral on the left hand side to be convergent, β must be positive.
Corollary 2.8.Let λ > 0 and α < λ + 1.For f ∈ B λ and any measure μ on [0, ∞) the following relation holds: where ω is the measure in the Bernstein representation of the completely monotonic function t −λ f (t).
Taking μ to be the point mass x at x the corollary yields the following.Corollary 2.9.Let f ∈ B λ and suppose that α < λ + 1.Then where L(ω)(t) = t −λ f (t).In particular, the Laplace transform maps the class Next, let us show some consequences of these results.The incomplete gamma function is, due to the relation (1), a fundamental building block in constructing functions in B λ .
Letting α = 0 and μ be the point mass at s in Proposition 2.10 we record the following elementary relation (see also [6, 6.451.1]): Let us mention a couple of examples: Again letting μ be the point mass at s in Proposition 2.10 we obtain the assertion in the motivating example mentioned in the introduction.
Example 2.12.For s > 0 the function Example 2.13.For a positive integer n and c > 0 the function where E n is the generalized exponential integral (see [5, 8.19.24] and the end of Sect.5).Also the function (where μ is any positive measure making this multiple integral converge) belongs to T λ,n .(See Proposition 4.4 with β = 0.) Proof.Assume that f ∈ T λ,α .In Definition 1.1 the function t α ϕ(t) is the Laplace transform of a positive measure μ, and using Proposition 2.10, f can be written in the form where ϕ is completely monotonic and c ≥ 0. Since f is increasing, integration of this relation yields By definition, s −α ϕ(s) is completely monotonic of order α and so f ∈ T λ,α .This completes the proof.

Representation Via Hypergeometric Functions
In this section we prove that the (λ, α)-Thorin-Bernstein functions admit an integral representation in terms of the hypergeometric function 2 F 1 .
Theorem 3.1.A function f belongs to T λ,α if and only if there are non-negative constants a and b and a positive measure μ such that In the affirmative case, μ is the measure such that t α ϕ(t) = L(μ)(t), ϕ being the function in Definition 1.1.
Proof.In the first relation in Proposition 2.10 we perform the change of variable v = u/x.This gives us Next, a combination of Euler's integral representation of the 2 F 1 and Euler's transformation (see [1, Theorem 2.2.1 and Theorem 2.2.5]) yields This proves the asserted formula.
We record the following equivalent characterizations, obtained by using Pfaff's transformation ([1, Theorem 2.2.5]),Proposition 2.10 and the identity see [5, 8.17 (a) f ∈ T λ,α , (b) f can be represented as Remark 3.3.As was shown in Proposition 2.1, if f n ∈ T λ,α converges pointwise to f then f also belongs to T λ,α .Letting f n correspond to the triple (a n , b n , μ n ) in (c) of Corollary 3.2 and f to (a, b, μ), then μ n → μ vaguely on (0, ∞).To see this we notice which is in accordance with the representation of ordinary Thorin-Bernstein functions.See [16,Theorem 8.2].See also Proposition 4.2.

Integer Values of the Parameters
We find in Proposition 4.2 another form of the representation of functions from T λ,α in the case where λ = α = n, n being a positive integer.First, a lemma.Lemma 4.1.For z ∈ (−1, 1) \ {0} we have Proof.The first equality holds by definition.Furthermore, and this proves the lemma.

Proposition 4.2. Suppose that n is a positive integer. A function f belongs to T n,n if and only if
Proof.Corollary 3.2 gives us and Lemma 4.1 yields (for x = 0) The integral in the right-hand side of this relation can be rewritten using Taylor's formula with integral remainder: x/s In this way the proposition is proved.
Remark 4.3.In the case where α = 1 and λ is a positive rational number, the integrand in the representation in Corollary 3.2(b) can be summed.Indeed, writing λ = n + p/q, where n, p and q are non-negative integers and p < q we have The infinite sum on the right hand side can be rewritten as This formula can be found in [15].We indicate how to prove a more general formula below.The representation thus takes the form In particular, for λ = n + 1/2, using 2 tanh −1 (s) = log((1 + s)/(1 − s)) for |s| < 1, The summation formula alluded to: Suppose that g(z) = ∞ n=1 a n z n converges for |z| < 1.Then, interchanging the order of summation and using elementary properties of the roots of unity it follows that q−1 k=0 e −2πikp/q g z 1/q e 2πik/q = qz p/q ∞ k=0 a p+kq z k .
The formula needed above corresponds to a n = 1/n.
In the next proposition we examine in more detail the representation of functions in T λ,α taking into account both the integer part and the fractional part of the order α of complete monotonicity of ϕ.Proposition 4.4.Suppose that ϕ is a completely monotonic function of order α = n + β, n ∈ N, β ∈ [0, 1).There exists a positive measure μ such that, with Proof.According to [11] we may write ϕ as ϕ(t) = t −β L(ξ n )(t) where ξ n is the fractional integral of positive integer order The first equality now follows from Proposition 2.10.
As it is well known, we have and this gives the second equality.Let us complete this section by relating our results with earlier results for generalized complete Bernstein functions and generalized Thorin-Bernstein functions presented in [12].Remark 4.6.Proposition 4.4 extends two results from [12]: (a) A function f belongs to T λ,0 if and only if (See also [12,Proposition 3.12].)(b) A function f belongs to T λ,1 if and only if where h = ξ 1 is non-negative and increasing.(See also [12,Proposition 4.1].)

Additional Results and Comments
For non-negative α, it is easily seen that (−1) n ϕ (n) is completely monotonic of order α if ϕ is completely monotonic of order α.Indeed, this follows by writing ϕ(t) = t −α L(σ)(t) and using Leibniz' formula: Denoting by m r the measure on (0, ∞) having density s r−1 /Γ(r) w.r.t.Lebesgue measure (for r > 0) we notice that L(m r )(t) = t −r .Thus the relation above can be written as Definition 5.1.For α ∈ [0, λ) and a non-negative integer n, T λ,α is the subclass of T λ,α consisting of the functions of the form where a and b are non-negative numbers, and ϕ is a completely monotonic function of order α.
Conversely, if a, b ≥ 0 and σ is any positive measure for which the Laplace transform converges then defining the measure σ n by (5), f given by (4) belongs to T (n) λ,α .The corresponding function ϕ in (3) is given by ϕ(t) = t −α L(σ)(t) and the measure σ n is related to ϕ in the following way: Proof.When ϕ is a completely monotonic function of order α ≥ 0 we write t α ϕ(t) = L(σ)(t), ϕ(t) = L(μ)(t), and notice that (−1) n ϕ (n) (t) = L(s n dμ(s))(t).This gives, using Lemma 2.5 and Corollary 2.11, where σ n is the positive measure given in (5) and thus satisfies The proof of the other direction follows by retracing these steps.This concludes the proof of the proposition.
A C ∞ -function f : (0, ∞) → (0, ∞) is said to be logarithmically completely monotonic if −(log f (x)) = −f (x)/f (x) is completely monotonic.A measure or function on the positive half line is called infinitely divisible if its Laplace transform is a logarithmically completely monotonic function.For more details see [3].
Corollary 5.3.Let σ and σ n be as in the proposition above.Then the function

Proof. It follows from the relation
in the proof above that the Laplace transform of t λ−α−1 L(σ n )(t) belongs to S λ−α and thus to S 2 .Therefore it is logarithmically completely monotonic by a result of Kristiansen.See [7] and [3].
Let us end this section with a few additional observations on infinite divisibility.
First of all, if f ∈ C α for some α ≥ −1 then f is infinitely divisible.This is well-known for α ≥ 0 and for α ∈ [−1, 0) it follows from the relation The next results deal with products of completely monotonic functions of given real orders.The following is a consequence of Lemma 2.6.
Proof.From Proposition 5.4 we see that L(fg) belongs to S 1−α−β , which is contained in S 2 , and Kristiansen's result can thus be used.
The next corollary deals also with the distribution function of a so-called randomized Lomax distribution, defined in terms of the complementary incomplete Gamma function, See [12,Example 3.11], where it was shown that F λ ∈ B λ .A standard computation shows that from which it is immediate that F λ even belongs to T λ,0 .
Proof.Of course t −α belongs to C α and f belongs to C −λ .The result now follows from Corollary 5.5.
Next we investigate the image of the subclasses T λ,α under the Laplace transform.
Proof of Proposition 5.7.Let f ∈ T λ,α and write its representation as [12,Proposition 2.2]) and some positive measure μ.This gives, using Corollary 2.11 and Fubini's theorem, and thus In this last integral we make a change of variable v = (x/s)u and get by Fubini's theorem We remark that in general not all functions from S λ+1−α are of the form x α L(f )(x), for some f ∈ T λ,α .For example, since T λ,1 B λ it follows that Φ(T λ,1 ) Φ(B λ ) = S λ (Φ as in the introduction).Below an example is given when λ = α ∈ {1, 2, 3, . ..}.
Example 5.9.Let n ≥ 1.To see that {x n L(g)(x) | g ∈ T n,n } is not all of S let ϕ be a completely monotonic function and suppose that there is g ∈ T n,n such that x n L(g)(x) = L(ϕ)(x).This entails According to Theorem 2.14, g belongs to T n,n if and only if x 1−n g (x) belongs to S. We obtain x 1−n g (x) = (n − 1) (1 − t) n−2 ϕ(xt) dt, which is not in general a Stieltjes function.As an example, take ϕ(s) = e −s .In this case, x 1−n g (x) is the Laplace transform of the non-completely monotonic function (n − 1)(1 − t) n−2 1 (0,1) (t).
Let us conclude the paper by giving some examples related to the generalized exponential integral, E p , defined for p > 0, as Proposition 9.5].Restriction to the open half line establishes the result.

Remark 4 . 5 .
When n = 0 the contents of Proposition 4.4 are described in Proposition 2.10.
t t p dt. and this yields that E p is a completely monotonic function for any p > 0. We observe that0 ≤ tE p (t) ≤ ∞ t e −s ds = e −t ,so E p (t) decays exponentially for t tending to ∞ and is bounded by 1/t as t tends to 0.Example 5.10.We have, using Proposition 2.10, The change of variable t = x/(x + s) transforms the right-hand side into Γ(λ)x p B(λ + p, −p; x/(x + 1)),(6)which is thus an example of a function in T λ,0 , for any p > 0. Notice that it is infinitely divisible for λ ≤ 1.This gives, applying Proposition 2.10 followed by the change of variable u = 1 − 1/t p ∈ C 1−p with E p (x) = x p−1 L(h p )(x), where h p (t) = 1 (1,∞) (t) (t − 1) p−1 Γ(p)t .