A note on the moments of the Kesten distribution
Introduction
Throughout this paper, we assume that p and q are positive real numbers with . Let denote a probability density function defined by (see, e.g., (6.30) on p.92 of [12]), which is called the Kesten distribution (see [8], [7], [12] for more details). For the case where and , the distribution is called, in the graph theoretic literature, the Kesten-McKay distribution. In fact, this distribution appears as the spectral distribution of the adjacency operator of the -regular tree (see [3]), and the limit distribution of the eigenvalues of the adjacency matrices of a sequence of -regular graphs , such that the girth of tends to infinity as (see [9]).
For , let denote the kth moment with respect to : It can easily be seen that , and holds for odd numbers k. Therefore, we are interested in . The Stieltjes transform of is given as (see, e.g., (4.8) on p.107 of [7]). Thus, we deduce the moment generating function of as follows:
Moment sequences with respect to probability density functions are sometimes related to combinatorial sequences. For example, it is well-known that the moment for is related to the Catalan number , namely, the identity holds. Since the explicit descriptions of moments can be obtained, to investigate such connections is important. The Catalan's triangles and the Shapiro's Catalan triangles are famous generalizations of , respectively. According to the above considerations, it is natural to ask the relations among the moments for and the numbers , . The present paper has an answer for it. That is, are described in terms of and , respectively.
The identities for the numbers and have been discussed. In 1979, Eplett [4] showed that the identity holds. In 2010, Miana and Romero [10] obtained some identities including . In 2017, Miana, Ohtsuka and Romero [11] presented some identities for the sums and (), respectively. (In [11], the symbol is used, instead of the symbol . These are related as follows: for .)
Arenas Longoria and Mingo [1] determined the moments with respect to the Kesten-McKay distribution . This is a special case of our result in the present paper. For details, see Remark 2.5 below.
Szabłowski studied the moments for classes including the Kesten distribution (see [15] for conditional q-normal distributions, and see [16] for multi-variable Kesten-McKay distributions, respectively). In [15, Proposition 3 (i)], the moments in terms of the q-Hermite polynomials and sums are given. In the proof, he calculated directly using the integral and an expansion . In this study, we present explicit expressions of the moments using the moment generating function and some identities for sums of binomial coefficients. For details about the relations between Szabłowski's results and our results, see Remark 2.11 below.
This paper is organized as detailed below. In Section 2, we provide two expressions for , one in terms of the Catalan's triangle and the other in terms of the Shapiro's Catalan triangle . As a corollary, we obtain an identity between and , which is a generalization of the identity proved by Eplett [4]. In Section 3, we present an asymptotic formula for as .
Section snippets
Main results
In this section, we first express as a polynomial in p and q. Theorem 2.1 Let p and q be positive real numbers with , and let denote the moment of the distribution defined by (1). Then, it follows that where is Catalan's triangle (see, e.g., [2] or [14, A009766]) defined by for integers .
Proof First, we consider the case where . It is well-known that is the probability density function of the Wigner semicircle distribution
Asymptotics of moments
As an application, we obtain an asymptotic formula on :
Corollary 3.1 If , we obtain Proof Based on the Stirling formula as , we have and then this implies that as . From the above asymptotics, the inequality , and Theorem 2.3, we have as . The proof ends here. □
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors are very grateful to the referees for the careful reading of the paper and the helpful comments. Takehiro Hasegawa was partially supported by JSPS KAKENHI (grant number 19K03400). Seiken Saito was partially supported by JSPS KAKENHI (grant number 19K03608).
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