Catalan-like numbers and Stieltjes moment sequences

Abstract

We provide sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. As applications, we show that many well-known counting coefficients, including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers, the large and little Schr\"oder numbers, are Stieltjes moment sequences in a unified approach.Comment: 7 page

Total positivity of Narayana matrices

“…The following is a classic characterization for Stieltjes moment sequences (see [18,Theorem 4.4] for instance). Many well-known counting coefficients are Stieltjes moment sequences, see [15]. For example, the sequence (n!)…”

Log-convex and Stieltjes moment sequences

“…Stieltjes moment sequences are much better behaved than infinitely logconvex sequences and there are various approaches to show that a sequence is a Stieltjes moment sequence. For example, Liang et al [16] showed that many Catalan-like numbers form Stieltjes moment sequences via the total positivity of the corresponding Aigner's recursive matrices [1], including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers and the large Schröder numbers (see also Corollary 3.6). These Catalan-like numbers are therefore infinitely log-convex, which, in particular, settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers [12,Conjecture 5.4].…”

“…It is well known that many counting coefficients form Stieltjes moment sequences, including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers, the Schröder numbers. See [16] for details. Boros and Moll [4, p. 157] introduced the concept of the infinite logconcavity.…”

Catalan-like number sequences and Hausdorff moment sequences

“…Liang at al. [13] showed that these types of sequences are Stieltjes moment sequences in unified setting. Wang [25] showed that Stieltjes moment sequences are infinitely log-convex.…”

“…For more recent work, see [24,14]. [13] remarked that it is questionable if many well-know sequences are determinate. To check whether a given moment sequence is determinate is an important problem, but very difficult.…”