Some relatives of the Catalan sequence
Introduction
The Catalan sequence : (entry A000108 in OEIS [19]) plays an important role in mathematics. It counts several combinatorial objects, such as binary trees, noncrossing partitions, Dyck paths, dissections of a polygon into triangles and many others, as can be seen in the Stanley's book [20]. It also shows up as the moment sequence of the Marchenko-Pastur law and (in its aerated version) of the Wigner law, which in the free probability theory are the analogs of the Poisson and the normal law.
One of the generalizations are Fuss numbers , which count for example p-ary trees with leaves, see [6]. The generating function satisfies equality Lambert observed that the powers of admit quite nice Taylor expansion: The coefficients have also combinatorial interpretations, see [6], and are called generalized Fuss numbers or Raney numbers. Formulas (2), (3) remain true if the parameters are real.
It turns out that for the sequence is positive definite if and only if either or or , see [11], [12], [13], [5], [8]. Moreover, the corresponding probability distributions are interesting from the point of view of noncommutative probability, for example the free R-transform of is , , , where “⊠” is the multiplicative free convolution, and , where “▷” is the monotone convolution, see [14].
In this paper we are going to study sequences which are defined by real parameters and recurrence relation (4), with . If , then is the Catalan sequence. First we provide combinatorial motivations for such sequences, for example in terms of action of a finite family of operators on a product. Formula (4) implies equations (6) and (7) for the generating and the upshifted generating function , .
In Section 2 we study these functions, in particular we give a sufficient condition when the domain of and is contained in . Consequently, is a Pick function and then the sequence is positive definite. If this is the case then the corresponding probability distribution will be denoted . In Section 3 we find the free R-transform for which is a rational function, and study the convolution semigroup . We also prove that if is freely infinitely divisible then . Next we observe that the considered class of measures is closed under monotone convolution: , where the sequence a is given by (18).
In Section 6 we study in details the case . We give formulas for , free cumulants , the generating functions , and prove that the sequence is positive definite if and only if . Moreover, we show that for the distribution is freely infinitely divisible. For the case we provide sufficient conditions for positive definiteness (Theorem 7.2, Proposition 7.3). Then we consider the symmetric case, i.e. when whenever j is even.
In the final section we provide a record of several integer sequences from OEIS which are of the form and verify their positive definiteness.
Section snippets
Operators on a set
In this section we assume that , , is a fixed finite sequence of positive integers. Suppose that X is a set and that for we are given some j-ary operators on X, , . Denote by the number of all possible compositions of these operations applied to the product . For example, for we can apply either where , , so that . Each such composed
The generating functions and positive definiteness
Motivated by the previous section we are going to study sequences defined by the recurrence relation (4), with , where , , with . The generating function , given by (5), and the upshifted generating function , satisfy relations (6) and (7), with .
First we note the following homogeneity:
Proposition 2.1 If and then
Proof This is a direct consequence of (4) and (5). □
Proposition 2.2 The functions ,
R-transform and free additive convolution semigroups
For a sequence , with generating function (possibly a formal power series), the free R-transform of , or of , is defined by The coefficients in the Taylor expansion are called free cumulants of the sequence . If are moments of a probability distribution on , i.e. , , then R (resp ) is called the free R-transform (resp. the free cumulants) of μ. If are R-transforms of probability
Compositions and monotone convolution
For compactly supported probability distributions on , with the moment generating functions there exists unique compactly supported probability distribution on , called monotone convolution of and , such that its moment generating function satisfies For details we refer to [14]. In terms of the upshifted moment generating functions relation (16) becomes
The case : Catalan numbers
For the sake of completeness we briefly describe the case . If then which leads to the dilated Catalan numbers with the generating function It is well know that the Catalan sequence is positive definite, namely . Taking we can now prove this applying Theorem 2.6.
It is also known that (and hence , with ) is infinitely divisible
The case
In this section we will confine ourselves to . Put , . The corresponding sequence, polynomial and the generating functions will be denoted , , , .
The case
Put , with , . We have , , , , .
Proposition 7.1 If the sequence is positive definite then
Since inequality (34) implies that .
Proof Inequality (34) is equivalent to , which is nonnegativity of the second Hankel determinant. □
Symmetric case
In this section we assume that whenever j is even, i.e. that the polynomial is odd. Then is odd as well, is even and whenever n is odd. If in addition the sequence is positive definite then the corresponding probability distribution is symmetric, i.e. for every Borel set . The case was already studied in Subsection 6.6.
Examples
In this section we provide a record of examples of sequences for which we can verify positive definiteness. Some of them appear in OEIS with an additional 0 or 1 term at the beginning. We refer also to Table 5 in [9]. To the best of our knowledge, positive definiteness of the sequences listed in subsections 9.1 (except A048779 and A001700), 9.3 (except A006013), 9.4 (except A006632) and 9.5 is the result of this paper.
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W.M. is supported by the Polish National Science Center grant No. 2016/21/B/ST1/00628.