Theta Function Identities And Their Applications

In this paper,the products of theta functions,the general of the analog ofclassical theta function are discussed using the theory of theta function and el-liptic function.Moreover their applications to the theory of Ramanujan,numbertheory,modular and so on are studied.The author discovers many useful thetafunction identities and devote to their applications.The details are as follows:1.Schr(o|¨)ter formula,which is the product of two theta functions,is discussed.The author prove an important case of SchrSter formula using the properties ofcompleted residue system.From this identity,some well-known addition formulasfor theta functions,the Hirschhorn septuple product identity are rediscovered.This formula also allows us to derive two remarkable theta functions identitieswhich are extensions of two well-known Ramanujan's identities related to themodular equations of degree 5.Moreover,some series expansion formula for(q;q)_∞⁶,Jacobi quartic identity and one trigonometric identity are proved.2.The products of three theta functions are studied.Four general thetafunction identities for the products of three theta functions are derived fromthe theory of theta function,which generalize the Hirschhorn-Garvan-Borweincubic theta functions for two variables into the general cubic theta functions forthree variables.While establishing the general cubic theta functions,the authorstudy their properties,relations,the product forms and transformation formulas.From the four general theta function identities,a lot theta function identities arededuced,such as Ramanujan identity,Winquist's identity,the quintuple productidentity,Farkas-Kra identity and Macdonald identity and so on.And as a spacialcase,the results of Shen about the products of theta functions are contained.Moreover,the series expansions for (q;q)_∞²,(q;q)_∞⁴(q⁵;q⁵)_∞,(q;q)_∞⁴(q^1/5;q^1/5)_∞,(q;q)_∞⁸ and (q;q)_∞¹⁰ are obtained.3.Eisenstein series on the congruence subgroupΓ₀ (3) are mainly discussed.The Eisenstein series expansions for Borweins' cubic theta functions are obtained. Some identities connecting theta functions and sums of Eisenstein series areproved using the complex variable theory of elliptic functions,which containas a special case a famous identity connected with a²(q).