The Applications Of Interpolation Formulas And Insertion Algorithms In Q-Series

Basic hypergeometric series, shortened as q-series, has developed rapidly during the past two decades and has diverse applications in Analysis, Number Theory, Com-binatorics, Physics, and Computer Algebra. Interpolations, by polynomials or other functions, are fundamental subjects in Approximation Theory and Numerical Analysis. Moreover, Constructive Partition Theory is a rich subject, with many classical and im-portant results which influenced the development of Enumerative Combinatorics in the twentieth century.The main results of this thesis consist of two approaches to basic hypergeometric series, including divided difference operators for interpolation formulas and insertion algorithm. This thesis is organized as follows.The first chapter is devoted to an introduction to basic hypergeometric series, in-cluding bilateral hypergeometric series and elliptic hypergeometric series. We will give a brief history and basic definitions, as well as some classical q-identities.In Chapter2, we shall give an overview of some well-known expansion formulas, including Newton and Lagrange interpolation formulas, q-Taylor formula, a Newton type rational interpolation formula,q-expansion formula of Liu and so on. We are also concerned with the applications of these expansion formulas, from which some important q-identities can be derived.In Chapter3, we shall give a2n-point interpolation formula, which allows us to recover many important classical q-identities, such as the terminating6φ5summation formula, Bailey’s q-analogues of Dixon’s theorem, q-identities related to divisor func-tions, finite forms of the quintuple product identity, as well as a bibasic hypergeometric identity.In Chapter4, we will give a new derivative operator, based on which a4n-point in-terpolation formula is derived. Then we generalize our result to elliptic hypergeometric series and give an elliptic analogue of the4n-point interpolation formula. Some appli-cations of these two interpolation formulas to basic hypergeometric series and elliptic hypergeometric series will also be discussed.In Chapter5, we will focus on the applications of insertion algorithm in proving q-identities. We recall an insertion algorithm, raised by Zeilberger and named Algo-rithm Z. Following the idea of Algorithm Z, we give a new insertion algorithm, called the variation of Algorithm Z. In the end, by means of the variation, we give a new combinatorial proof of Ramanujan’s1Ψ1summation formula, which is one of the most classical identities in q-series.