A semi-constructive approach to evidently positivepartition generating functions

In this work, we start with the celebrated Rogers–Ramanujan identities, whichare fundamental results in the theory of integer partitions. Then, we continue bypresenting certain combinatorial generalizations related to these theorems, calledRogers-Ramanujan type identities. These identities involve two types of constraints:modulus constraints and difference constraints. Although generating functions onthe modulus side are relatively straightforward to construct, manipulate, and interpret,addressing the difference side requires a more nuanced approach.To this end, we introduce a general framework, termed the moves framework, forinterpreting evidently positive series arising from a specific form of two-variable generatingfunctions. This framework is applicable under certain algebraic conditionson the exponents of the generating functions. For cases where these conditions arenot satisfied, we propose an alternative method. This involves deriving a systemof functional equations satisfied by the series and translating this information intoa recursive combinatorial construction, which allows us to provide a combinatorialinterpretation of the series.The thesis concludes with a discussion of potential directions for future research,highlighting open problems and areas for further exploration.