In this work, we give an identity that leads to establishing the operator . Also, introduce polynomials addition, provide Operator proof for generating function with its extension and Rogers formula The bivariate Rogers-Szegö are deduced. allows obtain inverse linearization , which deduce A solution a q-difference equation is introduced expressed in terms of operators method used recover

We focus on writing double sum representations of the generating functions for number partitions satisfying some gap conditions. Some example sets to be considered are into distinct parts and that satisfy conditions Rogers–Ramanujan, Göllnitz–Gordon, little Göllnitz theorems. refine our by imposing a bound largest part find finite analogues these new representations. These refinements lead many...