Infinite products over hyperpyramid lattices
type: Item
instance of: scholarly article
title: Infinite products over hyperpyramid lattices
zbMATH Open document ID: 0955.05009
DOI: 10.1155/S0161171200000764
author: Geoffrey B. Campbell
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2001-02-18T00:00:00Z
full work available at URL: https://eudml.org/doc/48634
review text: 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 \[ \exp\Biggl\{ \sum_{k\geq 1}{x^k_n\over k^{b_n}}\Biggr\} \prod_{\substack{ (a_1,a_2,\dots, a_n)= 1\\ a_1,a_2,\dots, a_{n-1}< a_n\\ a_1,a_2,\dots, a_{n-1}> 0\\ a_n> 1}} (1- x^{a_1}_1 x^{a_2}_2\dots x^{a_n}_n)^{- a^{b_1}_1\dots a^{b_n}_n}= \exp\Biggl\{ \sum^\infty_{k=1} \prod^{n- 1}_{i= 1} \Biggl(\sum^{k-1}_{j= 1} {x^j_i\over j^{b_i}}\Biggr) {x^k_n\over k^{b_n}}\Biggr\}. \]
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 1455547
zbMATH Keywords: infinite product identities
zbMATH Keywords: lattice point vectors
reviewed by: László A. Székely
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1455547