On powers of the Catalan number sequence

The Catalan number sequence is one of the most famous number sequences in combinatorics and is well studied in the literature. In this paper we further investigate its fundamental properties related to the moment problem and prove for the first time that it is an infinitely divisible Stieltjes moment sequence in the sense of S.-G.  Tyan. Besides, any positive real power of the sequence is still a Stieltjes determinate sequence. Some more cases including (a) the central binomial coefficient sequence (related to the Catalan sequence), (b) a double factorial number sequence and (c) the generalized Catalan (or Fuss–Catalan) sequence are also investigated. Finally, we pose two conjectures including the determinacy equivalence between powers of nonnegative random variables and powers of their moment sequences, which is supported by some existing results.     

Log-convex and Stieltjes moment sequences

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.     

On powers of Stieltjes moment sequences,II

We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, and prove an integral representation of the logarithm of the moment sequence in analogy to the Lévy–Khintchine representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an application we construct a positive generating function for the orthonormal Hermite polynomials.     

Sigma function solution of the initial value problem for Somos 5 sequences

The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we prove that the two subsequences of odd/even index terms each satisfy a Somos 4 (fourth order) recurrence. This leads directly to the explicit solution of the initial value problem for the Somos 5 sequences in terms of the Weierstrass sigma function for an associated elliptic curve.

     

Stieltjes moment problem and fractional moments

Stieltjes moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, a determinate Stieltjes moment problem, whose corresponding Hamburger moment problem is determinate too, is investigated in the setup of Maximum Entropy. Condition number in entropy calculation is provided endowing both Stieltjes moment problem existence conditions and Hamburger moment problem determinacy conditions by a geometric meaning. Then the resorting to fractional moments is considered; numerical aspects are investigated and a stable algorithm for calculating fractional moments from integer moments is proposed.     

Inequalities for linear combinations of monomials in p-Newton sequences

The partial order on monomials that corresponds to domination when evaluated at positive Newton sequences is fully understood. Here we take up the corresponding partial order on linear combinations of monomials. In part using analysis based upon the cone structure of the exponents in p-Newton sequences, an array of conditions is given for this new partial order. It appears that a characterization in general will be difficult. Within the case in which all coefficients are 1, the situation in which, for general sequence length, there are two monomials, each of length two and nonnegative integer exponents, the partial order is fully characterized. The characterization is combinatorial, in terms of indices in the monomials, and, already here there is much more than term-wise domination.     

On Almost Convergent and Statistically Convergent Subsequences

There are two well-known non-matrix summability methods which we will consider, namely “almost convergence” and “statistical convergence”. The results presented in this paper will be of two types, dealing with Lebesgue measure and Baire category. Establishing a one-to-one correspondence between the interval (0; 1] and the collection of all subsequences of a given sequence s = (s _n), we will examine the measure and category of the set of all almost convergent subsequences of (s _n). Similar questions for statistical and lacunary statistical convergence are considered. Results on rearrangements of sequences are also presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

The set of stable switching sequences for discrete-time linear switched systems

In this paper we study the characterization of the asymptotical stability for discrete-time switched linear systems. We first translate the system dynamics into a symbolic setting under the framework of symbolic topology. Then by using the ergodic measure theory, a lower bound estimate of Hausdorff dimension of the set of asymptotically stable sequences is obtained. We show that the Hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence. The obtained result reveals an underlying fundamental principle: a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them, provided that the switching is arbitrary. We also develop frequency and density indexes to identify those asymptotically stable switching sequences of the system.     

From Asymptotics to Spectral Measures: Determinate Versus Indeterminate Moment Problems

In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the spectral measure is under control of the polynomials asymptotics. The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are to be constructed using Nevanlinna parametrization. Nevertheless it is interesting to observe that some spectral measures can still be obtained from weaker forms of the Markov theorem. The exposition will be illustrated by orthogonal polynomials related to elliptic functions: in the determinate case by examples due to Stieltjes and some of their generalizations and in the indeterminate case by more recent examples.     

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

Locally-generic Boolean algebras and cardinal sequences

Given a sequence of cardinals of length less than , with each cardinal in the sequence being either or , we construct a -poset (see Defnition 1 below) which, with a natural topology, becomes a locally-compact, Hausdorff, scattered space with cardinal sequence . The algebra of the clopen subsets of its one-point compactification yields, in turn, a superatomic Boolean algebra with as its cardinal sequence. The posets are locallygeneric, that is, they are constructed generically over countable sets. This gives them additional chain properties, specially under Under Martin's Axiom, the construction allows any cardinals in the sequence, provided it has length Finally, we modify a forcing argument of Baumgartner-Shelah [B-S], to build -posets for any given cardinal sequence of length with each cardinal in the sequence being either or . Received September 2, 1998; accepted in final form September 13, 2001.     

On the characterization of periodic generalized Horadam sequences

The Horadam sequence is a direct generalization of the Fibonacci numbers in the complex plane, which depends on a family of four complex parameters: two recurrence coefficients and two initial conditions. In this article a computational matrix-based method is developed to formulate necessary and sufficient conditions for the periodicity of generalized complex Horadam sequences, which are generated by higher order recurrences for arbitrary initial conditions. The asymptotic behaviour of generalized Horadam sequences generated by roots of unity is also examined, along with upper boundaries for the disc containing periodic orbits. Some applications are suggested, along with a number of future research directions.     

行为NSD随机变量阵列加权和的q阶矩完全收敛性

利用Hoffmann-Jφrgensen型概率不等式和截尾法,获得了行为NSD随机变量阵列加权和的q阶矩完全收敛性的充分条件.利用这些充分条件,不仅推广和深化梁汉营等(2010)和郭明乐等(2014)的结论,而且使他们的证明过程得到了极大地简化.     

Algorithms for enumeration of single-transition serial sequences

Sets of n-valued single-transition serial sequences consisting of two serial subsequences (an increasing one and a decreasing one) determined by constraints on the number of the series and on their lengths and heights are considered. Enumeration problems for sets of finite sequences in which the difference in height between the neighboring series is not less than some given value are solved. Algorithms that assign smaller numbers to lexicographically lower-order sequences and smaller numbers to lexicographically higher-order sequences are obtained.     

Tribonacci-like sequences and generalized Pascal's pyramids

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid. The results also extend similar properties of Fibonacci-like sequences.     

Specker sequences revisited

Specker sequences are constructive, increasing, bounded sequences of rationals that do not converge to any constructive real. A sequence is said to be a strong Specker sequence if it is Specker and eventually bounded away from every constructive real. Within Bishop's constructive mathematics we investigate non‐decreasing, bounded sequences of rationals that eventually avoid sets that are unions of (countable) sequences of intervals with rational endpoints. This yields surprisingly straightforward proofs of certain basic results fromconstructive mathematics. Within Russian constructivism, we show how to use this general method to generate Specker sequences. Furthermore, we show that any nonvoid subset of the constructive reals that has no isolated points contains a strictly increasing sequence that is eventually bounded away from every constructive real. If every neighborhood of every point in the subset contains a rational number different from that point, the subset contains a strong Specker sequence. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

NA列与两两NQD列的L^p收敛性

设{Xn,n≥1）是NA列或两两NQD列,{ank;1≤k≤n,n∈N）是实数阵列．利用矩不等式和截尾方法,研究了∑k=1^n ankXk的L^p收敛性,所获结论推广和改进了前人的相应结果．     

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 formulaT[(a _n ) _n=1 ^∞ ] _n = 1/a₁ ...a _n . 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 toq-series.     

Fibonacci numbers, alternating parity sequences and faces of the tridiagonal Birkhoff polytope

We determine the number of alternating parity sequences that are subsequences of an increasing m-tuple of integers. For this and other related counting problems we find formulas that are combinations of Fibonacci numbers. These results are applied to determine, among other things, the number of vertices of any face of the polytope of tridiagonal doubly stochastic matrices.