A universal identity for theta functions of degree eight and applications

An addition formula for the Jacobian theta function and its applications

A three-term theta function identity and its applications

The history of algebraic equations is very long. The necessity and the trial of solving algebraic equations existed already in the ancient civilizations. The Babylonians solved equations of degree 2 around 2000 B.C. as well as the Indians and the Chinese. In the 16th century, the Italians discovered the resolutions of the equations of degree 3 and 4 by radicals known as Cardano’s formula and Ferrari’s formula. However in 1826, Abel [1] (independently about the same epoch Galois [7]) proved the impossibility of solving general equations of degree ≥ 5 by radicals. This is one of the most remarkable event in the history of algebraic equations. Was there nothing to do in this branch of mathematics after the work of Abel and Galois? Yes, in 1858 Hermite [8] and Kronecker [15] proved that we can solve the algebraic equation of degree 5 by using an elliptic modular function. Since \( \sqrt[n]{a} = \exp \left( {\left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}} \right)\log a} \right) \) which is also written as exp((1/n) ∫ 1 a (1/x)dx), to allow only the extractions of radicals is to use only the exponential. Hence under this restriction, as we learn in the Galois theory, we can construct only compositions of cyclic extensions, namely solvable extentions. The idea of Hermite and Kronecker is as follows; if we use another transcendental function than the exponential, we can solve the algebraic equation of degree 5. In fact their result is analogous to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) . In the quintic equation they replace the exponential by an elliptic modular function and the integral ∫(1/x)dx by elliptic integrals. Kronecker [15] thought the resolution of the equation of degree 5 by an elliptic modular function would be a special case of a more general theorem which might exist. Kronecker’s idea was realized in few cases by Klein [11], [13]. Jordan [10] showed that we can solve any algebraic equation of higher degree by modular functions. Jordan’s idea is clarified by Thomae’s formula, 8 Chap, m (cf. Lindemann [16]). In this appendix, we show how we can deduce from Thomae’s formula the resolution of algebraic equations by a Siegel modular function which is explicitely expressed by theta constants (Theorem 2). Therefore Kronecker’s idea is completely realized. Our resolution of higher algebraic equations is also similar to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) In our resolution the exponential is replaced by tne Siegel modular function and the integral ∫(1/x)dx is replaced by hyperelliptic integrals. The existance of such resolution shows that the theta function is useful not only for non-linear differential equations but also for algebraic equations.

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan’s \({}_{1}\psi _{1}\) summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identity is proved.

In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for ∏ n = 1 ∞ ( 1 − q n ) 10 \prod _{n=1}^\infty (1-q^n)^{10} is given. The proofs are self-contained and elementary.

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan’s identities, Winquist’s identity and many other interesting identities.

Introduction. In spite of its wide applicability in various branches of the theory of functions, the elliptic modular function is often used with a certain hesitation. This is mainly due to the fact that its application presupposes familiarity with a comparatively intricate formalism, in particular when the determination of numerical constants is involved. In fact, the endeavor to avoid the elliptic modular function has given rise to an extensive mathematical literature aiming at proving certain theorems in an way, the word being used here as a synonym for without making use of the elliptic modular function. As an impressive example, Picard's theorem on integral functions might be quoted. The difficulties which beset the numerical treatment of the elliptic modular function go essentially back to the fact that, on the one hand, the formalism of this function can only be developed with the help of the Jacobian elliptic functions while, on the other hand, what is needed in the applications are the conformal mapping properties of the modular function, and the connection between these two different aspects of the modular function has to be established through the medium of the theory of Schwarz' differential parameter or by a very detailed study of the periodic properties of the Jacobian elliptic functions. The object of the first part of this paper is to show how those properties of the elliptic modular function which are required for the applications may be derived in a simple way by the exclusive use of elementary principles of the theory of conformal representation. It will be shown that once the modular surface' is defined, the functional equation

On page 241 of his Second Notebook, Ramanujan recorded one of his theta function identity, which involves the ratio of the fourth power of theta functions with respect to ψ(q). In this article, we give a new proof for this theta function identity. Also, we give a new proof of another identity with respect to ϕ(q) established by B. C. Berndt and we establish two new theta function identities analogous to Ramanujan’s theta function identities.

Two pairs of inverse relations for elliptic theta functions are established with the method of Fourier series expansion, which allow us to recover many classical results in theta functions. Many nontrivial new theta function identities are discovered. Some curious trigonometric identities are derived.

Previously, we proved an addition formula for the Jacobi theta function, which allows us to recover many important classical theta function identities. Here, we use this addition formula to derive a curious theta function identity, which includes Jacobi�s quartic identity and some other important theta function identities as special cases. We give new series expansions for ?2(t), ?6(t), ?8(t), and ?10(t), where ?(t) is Dedekind�s eta function. The series expansions for ?6(t) and ?10(t) lead to simple proofs of Ramanujan�s congruences p(7n + 5) = 0 (mod 7) and p(11n + 6) = 0 (mod 11), respectively.

The classical identities of the Jacobi theta functions are obtained from the multiplicities of the eigenvalues ik and the corresponding eigenvectors of the DFT Φ(n) expressed in terms of the theta functions. An extended version of the classical Watson addition formula and Riemann’s identity on theta functions is derived. Watson addition formula and Riemann’s identity are obtained as a particular case. An extensions of some classical identities corresponding to the theta functions θa,b(x, τ) with a,b ∈ \(\frac {1}{3}\mathbb {Z}\) are also derived.

A Combinatorial Proof of the Farkas-Kra Theta Function Identities and Their Generalizations

We know that the elliptic modular function j(z) gives the birational invariant of the elliptic curve with the analytic modulus z, and that the elliptic modular functions belonging to the congruence-subgroups are obtained from the values of elliptic functions at the points of finite order on elliptic curves. This is not only the origin of those functions, but one of the most essential points to which we may ascribe the significance of elliptic modular functions in number-theory. It is important to generalize these facts, namely, to investigate in a more general case the relation between automorphic functions with one or more variables and systems of algebraic varieties, especially, systems of abelian varieties. The object of this paper is to give some results on this subject. We shall deal with certain systems of polarized abelian varieties parametrized by holomorphic functions and show that there exist meromorphic functions whose values are considered as of the members of the systems (Theorem 1). The theory developed here will be chiefly concerned with systems of abelian varieties with non-trivial endomorphisms, since I have already given elsewhere a theory for Siegel's modular functions [11]. I am particularly interested, in the present paper, in the determination of the fields of definition for fields of automorphic functions. This is the first problem which confronts us when we proceed beyond a formal treatment in the arithmetic theory of automorphic functions. We obtain a criterion (Theorem 2) applicable even in the case of compact fundamental domain where one can not employ Fourier expansions. The last part of the paper is devoted to the theory of a certain type of automorphic functions of one variable known in the literature as functions belonging to indefinite ternary quadratic forms [8], [5]; they occur as moduli of abelian varieties of dimension 2 whose endomorphismrings are isomorphic to an order of an indefinite quaternion algebra. We get in this case the field of rational numbers as a field of definition for the function-field. We can also prove the congruence formulae for the modular correspondences, which give a generalization of what is obtained in [3], [10]. We shall treat these formulae in a subsequent paper together with the theory of the functions belonging to congruence-subgroups. Our method can be applied to other types of automorphic functions, for example, to Hilbert's modular functions, by considering a system of

We begin this chapter with an exposition of a theory of theta functions. Using the classical transformation properties of theta functions and the theta function identity (12.7.13), we show that the string functions are modular forms and find a transformation law for these forms. Furthermore, using the theory of modular forms, we prove the "very strange" formula (12.3.7), which in turn, is used to show that the string functions, multiplied by a "standard" cusp-form, are cusp-forms. All this is applied to find explicit formulas for the weight multiplicities and characters of integrable highest weight modules.

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.