Partial Factorizations of Generalized Binomial Products

Abstract

This paper studies an integer sequence Gn analogous to the product Gn = n k=0 n k , the product of the elements of the n-th row of Pascal's triangle. It is known that Gn = p≤n p νp(Gn) with νp(Gn) being computable from the base p expansions of integers up to n. These radix statistics make sense for all bases b ≥ 2, and we define the generalized binomial product Gn = 2≤b≤n b ν(n,b) and show it is an integer. The statistic b ν(n,b) is not the same as the maximal power of b dividing Gn. This paper studies the partial products G(n, x) = 2≤b≤x b ν(n,b) , which are also integers, and estimates their size. It shows log G(n, αn) is well approximated by f G (α)n 2 log n + g G (α)n 2 as n → ∞ for limit functions f G (α) and g G (α) defined for 0 ≤ α ≤ 1. The remainder term has a power saving in n. The main results are deduced from study of functions A(n, x) and B(n, x) that encode statistics of the base b radix expansions of the integer n (and smaller integers), where the base b ranges over all integers 2 ≤ b ≤ x. Unconditional estimates of A(n, x) and B(n, x) are derived.