Infinite products over hyperpyramid lattices (Q1567221)

The author continues his research on infinite product identities. His identities are based on the following observation: considering an infinite region raying out of the origin in an Euclidean vector space, the set of all lattice point vectors from the origin in that region is precisely the set of positive integer multiples of the visible (from the origin) point vectors in that region. One of the two identities proved in the paper is as follows: If \(i= 1,2,\dots, n\), \(\sum_i b_i= 1\), \(|x_n|< 1\), \(|x_nx_{n-1}|< 1\), \(|x_nx_{n-1}\dots x_1|< 1\), then \[ \exp\Biggl\{ \sum_{k\geq 1}{x^k_n\over k^{b_n}}\Biggr\} \prod_{\substack{ (a_1,a_2,\dots, a_n)= 1\\ a_1,a_2,\dots, a_{n-1}< a_n\\ a_1,a_2,\dots, a_{n-1}> 0\\ a_n> 1}} (1- x^{a_1}_1 x^{a_2}_2\dots x^{a_n}_n)^{- a^{b_1}_1\dots a^{b_n}_n}= \exp\Biggl\{ \sum^\infty_{k=1} \prod^{n- 1}_{i= 1} \Biggl(\sum^{k-1}_{j= 1} {x^j_i\over j^{b_i}}\Biggr) {x^k_n\over k^{b_n}}\Biggr\}. \]