Hausdorff Moment Sequences Research Articles

On an Involution in the Class of Non-negative Hermitian Matrix Measures on a Compact Interval

We study a particular involution in the class {mathcal {M}^succcurlyeq _{q}({[alpha ,beta ]})} of all Non-negative Hermitian {qtimes q} measures on a compact interval {[alpha ,beta ]} of mathbb {R}. Several connections to the matricial version of the Hausdorff moment problem on {[alpha ,beta ]} are investigated in terms of {[alpha ,beta ]}-Stieltjes transforms as well as in the language of matricial Hausdorff moment sequences and associated matricial canonical moments.

Wall’s continued-fraction characterization of Hausdorff moment sequences: A conceptual proof

I give an elementary proof of Wall’s continued-fraction characterization of Hausdorff moment sequences.

Catalan-like number sequences and Hausdorff moment sequences

Many well-known Catalan-like sequences turn out to be Stieltjes moment sequences (Liang et al. (2016)). However, a Stieltjes moment sequence is in general not determinate; Liang et al. suggested a further analysis about whether these moment sequences are determinate and how to obtain the associated measures. In this paper we find necessary conditions for a Catalan-like sequence to be a Hausdorff moment sequence. As a consequence, we will see that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers, are Hausdorff moment sequences. We can also identify the smallest interval including the support of the unique representing measure. Since Hausdorff moment sequences are determinate and a representing measure for above mentioned sequences are already known, we could almost complete the analysis raised by Liang et al. In addition, subsequences of Catalan-like number sequences are also considered; we will see a necessary and sufficient condition for subsequences of Stieltjes Catalan-like number sequences to be Stieltjes Catalan-like number sequences. We will also study a representing measure for a linear combination of consecutive terms in Catalan-like number sequences.

Schur analysis of matricial Hausdorff moment sequences

We develop the algebraic instance of an algorithmic approach to the matricial Hausdorff moment problem on a compact interval [α,β] of the real axis. Our considerations are along the lines of the classical Schur algorithm and the treatment of the Hamburger moment problem on the real axis by Nevanlinna. More precisely, a transformation of matrix sequences is constructed, which transforms Hausdorff moment sequences into Hausdorff moment sequences reduced by 1 in length. It is shown that this transformation corresponds essentially to the left shift of the associated sequences of canonical moments. As an application, we show that a matricial version of the arcsine distribution can be used to characterize a certain centrality property of non-negative Hermitian measures on [α,β].

Hausdorff Moment Sequences Induced by Rational Functions

We study the Hausdorff moment problem for a class of sequences, namely $(r(n))_{n\in\mathbb Z_+},$ where $r$ is a rational function in the complex plane. We obtain a necessary condition for such sequence to be a Hausdorff moment sequence. We found an interesting connection between Hausdorff moment problem for this class of sequences with finite divided differences and convolution of complex exponential functions. We provide a sufficient condition on the zeros and poles of a rational function $r$ so that $(r(n))_{n\in\mathbb Z_+}$ is a Hausdorff moment sequence. G. Misra asked whether the module tensor product of a subnormal module with the Hardy module over the polynomial ring is again a subnormal module or not. Using our necessary condition we answer the question of G. Misra in negative. Finally, we obtain a characterization of all real polynomials $p$ of degree up to $4$ and a certain class of real polynomials of degree $5$ for which the sequence $(1/p(n))_{n\in\mathbb Z_+}$ is a Hausdorff moment sequence.

A one-parameter family of Pick functions defined by the Gamma function and related to the volume of the unit ball in $n$-space

We show that can be considered as a Pick function when a > 1, i.e. extends to a holomorphic function mapping the upper half-plane into itself. We also consider the function and show that In f(x+1) is a Stieltjes function and that f(x + 1) is completely monotonic on ]0, ∞[. In particular, f(n) = Ω 1/(n ln n) n , n ≥ 2, is a Hausdorff moment sequence. Here Ω n is the volume of the unit ball in Euclidean n-space.

Hausdorff means and moment sequences

We prove Schur’s theorem on the complete symmetric functions by using Hausdorff means. This approach leads to a more general result and to some new properties of moment sequences.

The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function

We study the fixed point for a non-linear transformation in the set of Hausdorff moment sequences, defined by the formula: $T((a_n))_n=1/(a_0+\cdots +a_n)$. We determine the corresponding measure $\mu$, which has an increasing and convex density on $\mathopen]0,1\mathclose[$, and we study some analytic functions related to it. The Mellin transform $F$ of $\mu$ extends to a meromorphic function in the whole complex plane. It can be characterized in analogy with the Gamma function as the unique log-convex function on $\mathopen]-1,\infty\mathclose[$ satisfying $F(0)=1$ and the functional equation $1/F(s)=1/F(s+1)-F(s+1)$, $s>-1$.

A note on generating functions for Hausdorff moment sequences

For functions f f whose Taylor coefficients at the origin form a Hausdorff moment sequence we study the behaviour of w ( y ) := | f ( γ + i y ) | w(y):=|f(\gamma +iy)| for y > 0 y>0 ( γ ≤ 1 \gamma \leq 1 fixed).

Iteration of the rational function z-1/z and a Hausdorff moment sequence

In a previous paper we considered a positive function f , uniquely determined for s > 0 by the requirements f ( 1 ) = 1 , log ( 1 / f ) is convex and the functional equation f ( s ) = ψ ( f ( s + 1 ) ) , with ψ ( s ) = s - 1 / s . Denoting ψ ∘ 1 ( z ) = ψ ( z ) , ψ ∘ n ( z ) = ψ ( ψ ∘ ( n - 1 ) ( z ) ) , n ⩾ 2 , we prove that the meromorphic extension of f to the whole complex plane is given by the formula f ( z ) = lim n → ∞ ψ ∘ n ( λ n ( λ n + 1 / λ n ) z ) , where the numbers λ n are defined by λ 0 = 0 and the recursion λ n + 1 = ( 1 / 2 ) ( λ n + λ n 2 + 4 ) . The numbers m n = 1 / λ n + 1 form a Hausdorff moment sequence of a probability measure μ such that ∫ t z - 1 d μ ( t ) = 1 / f ( z ) .

A q-analogue of complete monotonicity

The aim of this paper is to give a $q$-analogue for complete monotonicity. We apply a classical characterization of Hausdorff moment sequences in terms of positive definiteness and complete monotonicity, adapted to the $q$-situation. The method due to Mas

Extinction and explosion of nonlinear Markov branching processes

AbstractThis paper concerns a generalization of the Markov branching process that preserves the random walk jump chain, but admits arbitrary positive jump rates. Necessary and sufficient conditions are found for regularity, including a generalization of the Harris-Dynkin integral condition when the jump rates are reciprocals of a Hausdorff moment sequence. Behaviour of the expected time to extinction is found, and some asymptotic properties of the explosion time are given for the case where extinction cannot occur. Existence of a unique invariant measure is shown, and conditions found for unique solution of the Forward equations. The ergodicity of a resurrected version is investigated.

Some Transformations of Hausdorff Moment Sequences and Harmonic Numbers

AbstractWe introduce some non-linear transformations from the set of Hausdorff moment sequences into itself; among them is the one defined by the formula: . We give some examples of Hausdorff moment sequences arising from the transformations and provide the corresponding measures: one of these sequences is the reciprocal of the harmonic numbers .

A transformation from Hausdorff to Stieltjes moment sequences

We introduce a non-linear injective transformation τ from the set of non-vanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula T[(an) ${}_{n=1}^{∞}$ ]n = 1/a1 ... an. Special cases of this transformation have appeared in various papers on exponential functionals of Lévy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to q-series.

On the nonequivalence of conservative Hausdorff methods and Hausdorff moment sequences

On the nonequivalence of conservative Hausdorff methods and Hausdorff moment sequences

Characterization of Regular Hausdorff Moment Sequences

Characterization of Regular Hausdorff Moment Sequences

Quadratic forms and chain sequences

It has been shown that a monotone Hausdorff moment sequence can be characterized by Stieltjes type quadratic forms. This was a result of the theory of chain sequences [4]. In the present paper we shall characterize an extended monotone Hausdorff moment sequence by Jacobi type quadratic forms. This will be done by extending the properties of ordinary chain sequences. The paper is in two sections. In the first section a relation between chain sequences and Jacobi type quadratic forms will be established. The connection between the extended monotone Hausdorff moment problem and chain sequences will be discussed in the second section.

Characterization of regular Hausdorff moment sequences

for each/ in C[0, 1], the class of continuous functions from [0, l] to the complex numbers. Indeed, we find that this proposal is precisely the assertion that the sequence p, pn = l/2n (re = 0, 1, • • • ),is a regular Hausdorff mom,ent sequence with limit 0. The latter assertion is well known: for relevant ideas, one may consult [4, Chapters 14 and 16] and, especially, [4, p. 309] regarding the Euler-Knopp mean (E, 1/2) associated with the aforementioned sequence p. The connecting link is the following result.