Partial Factorizations of Products of Binomial Coefficients

Abstract

Let Gn = n k=0 n k , the product of the elements of the n-th row of Pascal's triangle. This paper studies the partial factorizations of Gn given by the product G(n, x) of all prime factors p of Gn having p ≤ x, counted with multiplicity. It shows log G(n, αn) ∼ f G (α)n 2 as n → ∞ for a limit function f G (α) defined for 0 ≤ α ≤ 1. The main results are deduced from study of functions A(n, x), B(n, x), that encode statistics of the base p radix expansions of the integer n (and smaller integers), where the base p ranges over primes p ≤ x. Asymptotics of A(n, x) and B(n, x) are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.

Partial Factorizations of Generalized Binomial Products

“…parallel to those in [13]. The main results of this paper determine the growth rate of integer sequence G n and more generally the growth behavior of log G(n, x) for all n ≥ 1.…”

“…where d p (n) is the sum of the base p digits of n and S p (n) := n−1 j=1 d p (j). (See [13,Theorem 5.1].) The left side of (1.4) is a nonnegative integer, while examples show the two terms on the right side are sometimes not integers.…”

“…The left side of (1.4) is a nonnegative integer, while examples show the two terms on the right side are sometimes not integers. In the paper [13] the first and second authors studied the sizes of partial factorizations of G n : G(n, x) := p≤x p νp(Gn) , (1.5) where 1 ≤ x ≤ n, deriving estimates for the size of log G(n, x). Theorem 1 of [13] obtained for 0 < α ≤ 1 and for all n ≥ 2 the estimate log G(n, αn) = f G (α)…”

“…The main results of this paper determine the growth rate of integer sequence G n and more generally the growth behavior of log G(n, x) for all n ≥ 1. The overall approach of the proofs have parallels to that in [13] but have some significant differences, as given in Section 1.2.…”

“…This result is proved in Section 4. Unlike [13] the definition (1.8) has no known interpretation as a product of ratios of factorials; however, see the discussion in Section 1.4. Thus we do not have Stirling's formula available to directly estimate the size of G n , nor do we have combinatorial identities and recursion formulas available in dealing with binomial coefficients.…”