Asymptotic Analysis of Sums of Powers of Multinomial Coefficients: A Saddle Point Approach

When we first encountered these integer sequences, we assumed that their asymptotic developments had been well studied, but after extensively checking the literature, we believe that only special cases of the asymptotics have been analyzed. Due to the fundamental nature of these integer sequences, we decided to make a comprehensive characterization of the asymptotic growth of these integer sequences, as n→∞, for any (fixed) positive values m and k. These integer sequences are intimately connected with hypergeometric functions, as seen in equation (1). These integer sequences have also been of interest for a long time. The general family am,k(n) appears, for instance, in Barrucand [10]. The asymptotic growth of a2,k(n) has been known for (almost) a century [14], and perhaps longer. In the case k = 1, the values of am,k(n) are simply the powers of m, namely, am,1(n) = m n. The k = 2 case has myriad interpretations. They are used in Proposition 1 of Borwein et al. [3] and in the discussion and remarks after the proposition is proved. They use the notation “Wm(2n)” for our sequences am,2(n). Borwein and his co-authors point out that am,2(n) is the number of abelian squares of length 2n constructed from an alphabet that has m letters (i.e., strings of the form x1, . . . , xn, xσ(1), . . . , xσ(n) where σ is a permutation of 1, . . . , n); also see [15] for more details about such abelian squares. For another application of the family of sequences am,2(n), in number theory, see [2, p. 108]. The integers a4,2(n) are known as the Domb numbers; they enumerate the number of 2n-step polygons on a diamond lattice; see [6] and also OEIS #A002895. The sequences am,2(n) are also a key object of study in [17]. We close our discussion of the k = 2 case by noting ∗Université Libre de Bruxelles, Département d’Informatique, CP 212, Boulevard du Triomphe, B-1050 Bruxelles, Belgium, email: louchard@ulb.ac.be †Purdue University, Department of Statistics, 150 North University Street, West Lafayette, IN, USA, email: mdw@purdue.edu