We introduce some new infinite products, the simplest being(1−y)∏k=2∞∏j∈ϕk(1−ykqj)1/k=(1−y1−qy)1/(1−q),where ϕk is the set of positive integers less than and relatively prime to k, valid for |y|∧|qy| both less than unity, with q≠1. The idea of a q-analogue for the Euler totient function is suggested

ABSTRACT. About fifty new multivariate combinatorial identities are given, connected with partition theory, prime products, and Dirichlet series. Connections to Lattice Sums in Chemistry and Physics are alluded to also.

ABSTRACT. expressing We derive new classes of infinite products taken over the primes, for exampleas,an infinite produce of Riemann zeta functions, this product being taken over the set of rational numbers a/3 geater than zero with a relatively prime to 3

This paper introduces the concept of a D-analogue. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series, often called the basic hypergeometric series. The main result in this paper is a transform, based on an Euler product over the primes. Examples given are D-analogues of the q-binomial theorem and the q-Gauss summation.

ABSTRACT.We obtain infinite products related to the concept of visible from the origin point vectors. Among these is Z"