Study On Some Problems About Theta-function Products And Umbral Calculus

This paper is to study a few problems about theta functions and umbral calculus.It is composed of four chapters, as follows.Chapter one is mainly concerned with products of finitely many theta functions.As further study of the so-called ECS(exact covering system) due to Z. Cao [10,11], weput forward a new approach to this problem, i.e.,Theorem1.4.1, by using of the spaceof solutions of linear congruence equations associated with inversible integer matrix.Some applications to two and cubic theta-function products are discussed.In Chapter two, using a general expansion of Ramanujan theta-function productpresented recently in [28], we further show in a uniform way some formulas of threeJacobi theta-function products given by Chen [38]. Our argument seems simple thanothers. Furthermore, the meaning of our proofs implies to certain degree potentialapplication of the expansion formula on Ramanujan theta-function products built onthe t-coefcient method [28].In Chapter three, we present a new proof of the transfer formula of umbral op-erator and establish its general form. Our argument in turn asserts that the so-calledtransfer formulas associated with the Shefer sequences are in essence equivalent to theclassical Lagrange expansion theorem. As applications, we use the generalized transferformula to reprove two combinational inversions originally given in [27].In the final chapter, we propose three unsolved problems which are worth to befurther considered.