Some theta function identities related to the Rogers-Ramanujan continued fraction

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction . In order to prove some of these claims, G. N. Watson established two important theorems about that he found in Ramanujan's notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.

     

A theta function identity and its implications

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 is given. The proofs are self-contained and elementary.

     

A three-term theta function identity and its applications

In this paper, we establish a three-term theta function identity using the complex variable theory of elliptic functions. This simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities. From which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived. We also give a new proof of the addition theorem for the Weierstrass elliptic function ℘. An identity involving the products of four theta functions is given and from which one theta function identity by McCullough and Shen is derived. The quintuple product identity is used to derive some Eisenstein series identities found in Ramanujan's lost notebook and our approach is different from that of Berndt and Yee. The proofs are self contained and elementary.     

On a theta function identity

Liu [An extension of the quintuple product identity and its applications. Pacific J Math. 2010;246:345–390] established a theta function identity. In this paper, we will give an equivalent form of Liu's identity, from which some non-trivial identities on circular summation of theta functions are deduced.     

The t-coefficient method II: A new series expansion formula of theta function products and its implications

By means of Jacobi?s triple product identity and the t  -coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewell?s and Chen–Chen–Huang?s octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system _A2$A_2$ as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented.     

The products of three theta functions and the general cubic theta functions

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.     

Szegö quadrature formulas for certain Jacobi-type weight functions

In this paper we are concerned with the estimation of integrals on the unit circle of the form by means of the so-called Szegö quadrature formulas, i.e., formulas of the type with distinct nodes on the unit circle, exactly integrating Laurent polynomials in subspaces of dimension as high as possible. When considering certain weight functions related to the Jacobi functions for the interval nodes and weights in Szegö quadrature formulas are explicitly deduced. Illustrative numerical examples are also given.

     

Generalizations of Ramanujan's reciprocity theorem and their applications

First, we briefly survey Ramanujan's reciprocity theorem fora certain q-series related to partial theta functions and givea new proof of the theorem. Next, we derive generalizationsof the reciprocity theorem that are also generalizations ofthe ₁₁ summation formula and Jacobi triple product identityand show that these reciprocity theorems lead to generalizationsof the quintuple product identity, as well. Last, we presentsome applications of the generalized reciprocity theorems andproduct identities, including new representations for generatingfunctions for sums of six squares and those for overpartitions.     

New weighted Rogers-Ramanujan partition theorems and their implications

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts (mod 6), and a set of three theorems involving partitions into parts (mod 7), .

     

Several identities for certain products of theta functions

By means of a technique used by Carlitz and Subbarao to prove the quintuple product identity (Proc. Am. Math. Soc. 32(1):42–44, 1972), we recover a general identity (Chu and Yan, Electron. J. Comb. 14:#N7, 2007) for expanding the product of two Jacobi triple products. For applications, we briefly explore identities for certain products of theta functions φ(q), ψ(q) and modular relations for the Göllnitz-Gordon functions.     

Factorization of functions in generalized Nevanlinna classes

For functions in the classical Nevanlinna class analytic projection of produces where is the outer part of i.e., this projection factors out the inner part of . We show that if is area integrable with respect to certain measures on the disc, then the appropriate analytic projections of factor out zeros by dividing by a natural product which is a disc analogue of the classical Weierstrass product. This result is actually a corollary of a more general theorem of M. Andersson. Our contribution is to give a simple one complex variable proof which accentuates the connection with the Weierstrass product and other canonical objects of complex analysis.

     

Vector-valued Maass-Poincaré series

Shortly before his death, Ramanujan wrote about his discovery of mock theta functions, functions with interesting analytic properties. Recently, Zweger showed that mock theta functions could be ``completed' to satisfy the transformation properties of a weight real analytic vector-valued modular form. Using Maass-Poincaré series, Bringmann and Ono proved the Andrews-Dragonette conjecture, establishing an exact formula for the coefficients of Ramanujan's mock theta function . In this paper we study vector-valued Maass-Poincaré series of all weights, and give their Fourier expansions.

     

Primes of the form in short intervals

In this note, we prove that for every and , the short interval contains at least one prime number of the form with . This improves a similar result due to Huxley and Iwaniec, which requires .

     

The class number one problem for some non-abelian normal CM-fields of degree 48

We prove that there is precisely one normal CM-field of degree 48 with class number one which has a normal CM-subfield of degree 16: the narrow Hilbert class field of with .

     

A simple proof of the modular identity for theta functions

The modular identity arises in the theory of theta functions in one complex variable. It states a relation between theta functions for parameters and situated in the complex upper half-plane. A standard proof uses Poisson summation and hence builds on results from Fourier theory. This paper presents a simple proof using only a uniqueness property and the heat equation.

     

On the characteristic polynomial of the almost Mathieu operator

Let be the rotation C*-algebra for angle . For with and relatively prime, is the sub-C*-algebra of generated by a pair of unitaries and satisfying . Let

be the almost Mathieu operator. By proving an identity of rational functions we show that for even, the constant term in the characteristic polynomial of is .

     

Local behaviour of polynomials

In this paper we study the local behaviour of a trigonometric polynomial around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in . The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form are also proved.

     

Heegner zeros of theta functions

Heegner divisors play an important role in number theory. However, little is known on whether a modular form has Heegner zeros. In this paper, we start to study this question for a family of classical theta functions, and prove a quantitative result, which roughly says that many of these theta functions have a Heegner zero of discriminant . This leads to some interesting questions on the arithmetic of certain elliptic curves, which we also address here.

     

Cusp forms for congruence subgroups of and theta functions

In this paper we use theta functions with rational characteristic to construct cusp forms for congruence subgroups of .The action of the quotient group on these forms is conjugate to the linear action of on . We show that these forms are higher-dimensional analogues of the Fricke functions.

     

Local automorphisms and derivations on

Let be an algebra. A mapping is called a -local automorphism if for every there is an automorphism , depending on and , such that and (no linearity, surjectivity or continuity of is assumed). Let be an infinite-dimensional separable Hilbert space, and let be the algebra of all linear bounded operators on . Then every -local automorphism is an automorphism. An analogous result is obtained for derivations.