| The theme of this thesis is to study a linear space(?)(P, Q) of a class of theta functions and its application in constructing of theta function identities, we are focused on two kinds of coefficients among Fi(x),Fj(x)∈(?)(P,Q), as well as two summation formulas of the cubic theta functions and how to use these results to find cubic theta identities.Chapter one, we recall some preliminary on the theory of theta functions, including necessary notations, definitions, and results, which will be used for later discussions.Chapter two is the main part of this paper. Therein, we define a linear space (?)(P, Q) formed by theta functions (given fixed base), the linear coefficient of arbitrary theta function, and the symmetric difference coefficients of arbitrary two theta func-tions. As specific results, all these coefficients for ten basic theta functions, including their applications to theta function identities, are examined in details.Chapter three is devoted to a general summation formula concerning cubic theta functions. Based on this formula, we find some new and elementary proofs for the well-known cubic theta identities orignidly due to Ramanujan, Farkas-Kra, and Berndt et al. Certain p-power analogous summations are also discussed.As a final part, Chapter four gives a two-variable summation formula of cubic theta functions. This formula is, certainly new, an extension of the four well-known cubic theta identities which were first proposed and discussed by Hirschhorn-Garvan-Borwein. |
