Products of extended binomial coefficients and their partial factorizations

Abstract This paper studies properties of the integer sequence $$\overline{\overline{G}}_n=\prod _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) _{\mathbb {Z,N}}$$ G ¯ ¯ n = ∏ k = 0 n n k Z , N which is analogous to $$\overline{G}_n=\prod _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) $$ G ¯ n = ∏ k = 0 n n k , the product of the elements of the n -th row of Pascal’s triangle. Here $$\left( {\begin{array}{c}n\\ k\end{array}}\right) _{\mathbb {Z,N}}$$ n k Z , N is an extended binomial coefficient, defined in the paper, constructed using an extended version of M. Bhargava’s theory of generalized factorials. In 1996 M. Bhargava introduced a generalization of the factorial function, $$n!_S=\prod _p\nu _n(S,p)$$ n ! S = ∏ p ν n ( S , p ) in terms of their prime factorization, and defines associated binomial coefficients. The last two authors extended Bhargava’s invariants further to define such invariants attached to each integer $$b\ge 2$$ b ≥ 2 . One obtains extended factorials and extended binomial coefficients, and the maximal extension defines extended factorials $$n!_{\mathbb {Z,N}}=\prod _{b\ge 2}b^{\alpha _n(\mathbb {Z},b)}$$ n ! Z , N = ∏ b ≥ 2 b α n ( Z , b ) including all $$b\ge 2$$ b ≥ 2 , with associated extended binomial coefficients