Elliptic hypergeometric combinatorics

The aim of this project Elliptic hypergeometric combinatorics is to study certain combinatorial aspects of elliptic hypergeometric series and related objects, with the overall goal to push forward the relatively new theory of elliptic hypergeometric series. Elliptic hypergeometric series (EHS) are series which involve elliptic functions and form a natural generalization of hypergeometric series. EHS were discovered relatively late, namely only in 1987. It took ten more years until in 1997 the first explicit formulae for these series were found. The interest of the scientific community rapidly increased, and EHS has now become a hot topic of current research. Over the last years, a substantial amount of the existing theory for hypergeometric series has been extended to the elliptic setting. A lot of questions about EHS remain open, their connection to other areas in mathematics and physics needs to be better understood. In this project various combinatorial aspects of EHS are to be investigated; this focus is unique. In particular the project deals with combinatorial enumeration involving elliptic weights; one of the objectives is to find and thoroughly study elliptic extensions of special combinatorial numbers. Other topics of investigation include elliptic commuting variables, combinatorics of theta function identities, the study of analytic properties of generalized combinatorial sequences, and the further development of powerful tools for deriving identities. To achieve the specific objectives, a variety of methods will be used, ranging from combinatorial machinery to tools from algebra and classical analysis. The duration of this project, which is led by Michael J. Schlosser from the Faculty of Mathematics of the University of Vienna, is four years. In addition the team will consist of a doctoral and a postdoctoral student (with employment for four, respectively three, years).

The goal of this project "Elliptic hypergeometric combinatorics" was to study certain combinatorial aspects of elliptic hypergeometric series and related objects, with the overall goal to push forward the relatively new theory of elliptic hypergeometric series. Elliptic hypergeometric series (EHS) are series which involve elliptic functions and form a natural generalization of hypergeometric series. EHS were discovered relatively late, namely only in 1987. It took ten more years until in 1997 the first explicit formulae for these series were found. The interest of the scientific community rapidly increased, and EHS has already become a hot topic of current research. Over the last years, a substantial amount of the existing theory for hypergeometric series has been extended to the elliptic setting. Some questions about EHS have remained open, in particular concerning their connection to other areas in mathematics and physics. In this project various combinatorial aspects of EHS were investigated - the focus of the project being in the international research landscape unique. One of the main accomplished objectives of the project concerned the enumeration of combinatorial objects using elliptic weights; among the project's achievements are the are the construction of elliptic extensions of special combinatorial numbers. Other topics of investigation include elliptic commuting variables and the combinatorics of corresponding identities. Further, areas closely related to elliptic functions, namely theta functions, bilateral series, and Rogers-Ramanujan type identities, were investigated and remarkable results were achieved (including, for the first time, bilateral identities of the Rogers-Ramanujan type). To achieve the specific objectives, a variety of methods were used, ranging from combinatorial machinery (such as lattice paths) to tools from algebra and classical analysis. The duration of this project, which was led by Michael J. Schlosser from the Faculty of Mathematics of the University of Vienna, was, including a granted extension, five years. In addition the team consisted of a doctoral student, who successfully completed his doctoral studies with the completion of a doctoral thesis, and three postdoctoral students with different durations of employment. The research findings were published in quite respected journals (such as the Transactions of the American Mathematical Society), and several talks were given at conferences.