On certain non-unique solutions of the Stieltjes moment problem

We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${\rho}_{1}^{(r)}(n)=(2rn)!$ and ${\rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,...$, $n=0,1,2,...$, \textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $\int_{0}^{\infty}x^{n}W^{(r)}_{1,2}(x)dx = {\rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${\rho}_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing ${\rho}_{1,2}^{(r)}(n)$, such as the product ${\rho}_{1}^{(r)}(n)\cdot{\rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,...$.


Introduction
This paper concerns solutions of the Stieltjes moment problem [1,2], i.e. positive functions W (x) which satisfy the infinite set of equations We previously met this problem when considering the properties of the so-called coherent states (CS) of Quantum Mechanics [3].These CS, as well as their many generalizations, should satisfy the property of Resolution of Unity which is essentially equivalent to Eq. ( 1) [4,5].Standard CS lead to ρ(n) = n! and W (x) = e −x .Generalized CS lead to ρ(n)'s other than n! [6] and the solutions of Eq.( 1), if any, must be studied in each individual case separately [7,8,9,10].It quickly occurred to us that an efficient approach to this problem was to use the inverse Mellin transform method which allows one to establish many solutions of Eq. ( 1), either by analytic methods [11] or by extensive use of available tables [12,13].As a byproduct of this method we have established that, for a large number of combinatorial sequences such as Bell and Catalan numbers, etc., the corresponding sequences ρ(n) are solutions of the moment problem Eq. ( 1) [14,15].Likewise, sequences arising in theory of ordering of differential operators [16] solve appropriate moment problems too.This admixture of quantum-mechanical, analytical and combinatorial features deserves a deeper study which we intend to pursue.
Our paper is partly expository in character, and has the following structure: first we establish a link between the Mellin transform and the moment problem; next, in Section 3, we provide principal solutions to two moment problems, termed toy models.In Section 4 we discuss criteria for the uniqueness of solutions of the moment problem.Section 5 is devoted to the explicit construction of non-unique solutions of the toy-models.In Section 5 some generalizations of model sequences, together with their solutions, are reviewed.Section 6 is devoted to a discussion and conclusions.
We dedicate this paper to Philippe Flajolet on the occasion of his 60th birthday.His pioneering applications of Mellin transform asymptotics to the analysis of combinatorial structures [17,18,19] have been a source of inspiration for us.

The Mellin transform versus moment problem
The Mellin transform of a function f (x) of the real variable x is defined for complex s by the following relation ( in this definition f * (s) is not the complex conjugate of f (s) ! ), and its inverse M −1 is defined by See Ref. [20] for a discussion of the dependence of f (x) on the real constant c.Among the many relations satisfied by the Mellin transform we shall mainly use the following which is called the Mellin convolution property.Note that if in Eq. ( 5) both f (x) and g(x) are positive dt t is also positive for x > 0. This means that the Mellin convolution preserves positivity, an essential property when considering the moment problem.
Eq.( 1), if rewritten for n = s − 1 as is, if W (x) > 0, a solution of a Stieltjes moment problem.Thus according to Eq. ( 6) one can solve the Stieltjes moment problem by performing the inverse Mellin transform on the moment sequence and checking if the resulting function is positive.All the solutions in the sequel have been obtained using Eqs.( 6), ( 4) and (5).Note that W (x) obtained via Eq.( 6) may not be the only solution of Eq.( 1).We call W (x) > 0 obtained via Eq.( 6) from Eq. ( 1) the principal solution.

Principal solutions of the moment problems
Let us consider two sequences of integers given by ρ (r) In the following we shall obtain the solutions of the Stieltjes moment problem for the moment sequences given by Eqs. ( 7) and ( 8), i.e. the functions From now on we shall refer to the model problems ρ (r) 1,2 (n) as toy models TM1 and TM2, respectively.a) TM1: we begin by obtaining W (1) ).We now apply Eq.( 4) with a = 1, b = − 1 2 and h = 1 2 .We subsequently use In the same spirit we observe that (2rn . Upon using Eq. ( 4) but now with a = 1, b = − 2r−1 2r and h = 1 2r one obtains We call W b) TM2: we begin by deriving W (1) x] and employ the Mellin convolution Eq. (5).
By using the Sommerfeld representation of the modified Bessel function of second kind K 0 (x) [21] one obtains Subsequently note that ] 2 and again apply Eq.( 4) Since K 0 (t) > 0 for t > 0, W (r) 2 (x) > 0 is the principal solution of the moment problem Eq. ( 13).

Criteria of uniqueness and non-uniqueness of the Stieltjes moment problem
It was realized from the very beginning of the history of the moment problem that its solutions may not be unique; i.e. for a given moment sequence there may exist more than one solution.Stieltjes himself gave an example of a non-unique solution of a problem leading to what turned out to be the lognormal distribution [22].This example, being about the only one available, was quoted repeatedly in the literature.As recently as twenty years ago new non-unique solutions of other types of problems have been constructed.Of special interest were Stieltjes moment problems arising in probability theory, related to investigation of probability distributions not determined by their moments.Consequently, the subject was further developed and systematized, largely due to the comprehensive work of Berg [23], Berg and Pedersen [24], Lin [25], Stoyanov [26,27], Pakes [28], Gut [29] and others.From the practical point of view one needs criteria to decide whether the pursuit of non-unique solutions is reasonable.Such criteria are either based on the ρ(n)'s alone or on the solution W (x) alone, or on both ρ(n) and W (x), see below.From now on we assume that all the ρ(n)'s are finite and that W (x) is continuous.We now give , in a somewhat condensed form, a list of such criteria.
C1 Carleman uniqueness criterion (T.Carleman, 1922, [1]) This is based on the properties of the ρ(n)'s alone and is: This criterion does not imply that if S < ∞ then the solution is non-unique.In fact it is possible to construct models for which S < ∞ and solutions are still unique [26,30].

C2 Krein's non-uniqueness criterion (M. G. Krein, around 1950, [31])
This is based entirely on the solution W (x) and does not involve the moments.
dx < ∞, i.e., the so-called Krein integral exists, then the solution is non-unique.

Construction of non-unique solutions for TM1 and TM2
We first apply the criterion C1 to sequences ρ (r) 1,2 (n) (the logarithmic test of divergence of the series S is conclusive) and conclude that for r = 1 both solutions W (1) x and W (1) is convergent.Application of criterion C2 gives convergence of the Krein integral, showing that the solutions of TM1 and TM2 are non-unique.These findings are confirmed by use of criterion C3: convexity of ψ (r) 1,2 (e x )] is proved as it is equivalent to e x/r > 0 for x > 0, (TM1) and K 1 (2e x/r ) − K 0 (2e x/r ) > 0 for x > 0, (TM2).The quest for non-unique solutions for r > 1 is then well founded.
We know of no general method to construct such solutions.However we have proposed a procedure based on the application of inverse Mellin transform which generates the required solutions [8,32] which we now briefly expose.
The first step is to construct, within the framework of a given set of ρ(n)'s, a family of functions ω k (x), parametrized by a constant k (to be defined below), such that all their moments vanish, i.e.
The Mellin transform of ω k (x), ω ⋆ k (s), vanishes for s = 1, 2, . . . .Such functions are orthogonal to all polynomials and play an important role in the study of integral transforms [33].For our purposes we choose a particular method of producing the functions ω ⋆ k (x): or equivalently where h k (s) is any holomorphic function vanishing for s = 1, 2, . . . .Among an infinity of possible choices the simplest one is h k (s) = sin (πk(s + 1)) and it defines a discrete parameter k = ±1, ±2, ±3, . ... The function ω k (x) acquires new parameters now and is formally obtained by calculating the inverse Mellin transform It turns our that for both ρ (r) 1,2 (s) the integration in Eq.( 17) can be performed: a) TM1: for ρ (r) 1,k (x) is a special case of earlier evaluation [8] and reads: In Eq.( 18) we notice a pleasant factorization of W (r) 2,k (x) has to be, in the first place, represented as ω (r) which can be conceived as another case of Mellin convolution.A little thought gives the two partners to be convoluted as (see Eq.( 18)) and (compare Eq.( 18)).Thus, the integral form of ω (r) whose evaluation requires a number of changes of variables as well as the use of formula 2.5.37.2, p. 453 of vol.1 Ref. [12], but is essentially elementary.The final result is where we note a "near" factorization of W for real ε, |ε| < 1. TM2: for r > 2|k|.
As 1 + γ ) is an oscillating function of bounded variation, a constant γ = γ(k, r) can be always found to assure the overall positivity of W (r) 2 (γ, k, x).The above technique for obtaining non-unique solutions can be readily extended to moment sequences more general than ρ (r) 1,2 (n).We shall simply mention two such extensions without entering into details.
For ρ (r) 3 we begin with the sequence (n!) 3 for which the solution is where we use a convenient and self-explanatory notation for Meijer's G-function borrowed from that of computer algebra systems.The extension, via Eq.( 4), leads to the principal solution The integrand in Eq. ( 27) is a positive function for x > 0 which cannot be represented by any other known special function.It possesses an infinite series representation in terms of polygamma functions, which we will not quote here.The Carleman sum S is convergent but the criterion C2 is not conclusive.Only the criterion C3 permits to ascertain the non-uniqueness.The corresponding function ω 1,k (x), see Eqs. ( 18) and ( 23), respectively.
As a final example consider the sequence ρ which is non-unique by C3 only, as C2 remains inconlusive.

Discussion and Conclusion
We have demonstrated a methodology for obtaining unique and non-unique solutions of the Stieltjes moment problem using the Mellin convolution method.Although the initial moment sequences were simple and classical, one is rapidly forced to leave the realm of standard special functions, as the resulting solutions are special cases of Meijer G-functions.In most cases they resist the check for non-uniqueness via both the Carleman criterion C1 and the Krein criterion C2, and criterion C3 appears to be the only tool to decide this question.We have generated parametrized families of non-unique solutions exemplified here by Eqs. ( 18) and (23).Such functions are now called Stieltjes classes [34,35] and their properties are investigated.