Theta Identities Research Articles

On the identities of Ramanujan – a q-series approach

The idea of this paper is to prove two general theorems using q-series and deduce some partial theta identities and false theta identities. Some of the identities of Ramanujan come as special cases.

Rank partition functions and truncated theta identities

In 1944, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G. E. Andrews and F. G. Garvan. In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function p(n). The number of Garden of Eden partitions are also considered in this context in order to provide other infinite families of linear inequalities for p(n).

On a generalized Weierstrass theta identity

The classical Weierstrass theta identity plays a central role in the study of modular, elliptic and theta hypergeometric series. In this paper, we establish a multi-parameter Weierstrass theta identity with the use of the Cauchy determinant and an anti-symmetric difference of a product of two theta functions. Some applications of this generalized Weierstrass theta identity to basic and elliptic hypergeometric series are also discussed.

A universal identity for theta functions of degree eight and applications

Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

On three identities of Ramanujan

In this paper, we represent the generating function of the rank function as a summation of four parts—a constant, two Lambert series and a product. Applying it to three of Ramanujan’s identities we get new mock theta identities. Furthermore, through this representation, new relations between ranks modulo certain integers are provided.

Elliptic classes, McKay correspondence and theta identities

We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to the equivariant local situation. We study theta function identities having a geometric origin. In the case of quotient singularities {mathbb {C}}^n/G, where G is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity A_n leads to a remarkable formula.

A Truncated Theta Identity of Gauss and Overpartitions into Odd Parts

We examine two truncated series derived from a classical theta identity of Gauss. As a consequence, we obtain two infinite families of inequalities for the overpartition function $$\overline{p_o}(n)$$ counting the number of overpartitions into odd parts. We provide partition-theoretic interpretations of these results.

Bisected theta series, least r-gaps in partitions, and polygonal numbers

The least r-gap, $$g_r(\lambda )$$, of a partition $$\lambda $$ is the smallest positive integer that does not appear at least r times as a part of $$\lambda $$. In this article, we introduce two new partition functions involving least r-gaps. We consider a bisection of a classical theta identity and prove new identities relating Euler’s partition function p(n), polygonal numbers, and the new partition functions. To prove the results, we use an interplay of combinatorial and q-series methods. We also give a combinatorial interpretation for $$\begin{aligned} \sum _{n=0}^\infty (\pm 1)^{k(k+1)/2} p(n-r\cdot k(k+1)/2). \end{aligned}$$

Formulas for the Riemann zeta-function and certain Dirichlet series

ABSTRACTUsing known theta identities and formulas of S. Ramanujan and G. Hardy among others we prove several formulas for the Riemann zeta-function and two Dirichlet series.

Ramanujan-type partial theta identities and conjugate Bailey pairs, II. Multisums

In the first paper of this series, we described how to find conjugate Bailey pairs from residual identities of Ramanujan-type partial theta identities. Here we carry this out for four multisum residual identities of Warnaar and two more due to the authors. Applying known Bailey pairs gives expressions in the algebra of modular forms and indefinite theta functions.

PARTIAL INDEFINITE THETA IDENTITIES

Using a result of Warnaar, we prove a number of single- and multi-sum identities in the spirit of Ramanujan’s partial theta identities, but with partial indefinite binary theta functions in the role of partial theta functions. We also calculate the corresponding residual identities and use a result of Ji and Zhao to recast our identities in terms of indefinite ternary theta functions.

Ramanujan-Type Partial Theta Identities and Rank Differences for Special Unimodal Sequences

We consider three types of unimodal sequences related to Ramanujan-type partial theta identities. In each case we compute generating functions related to the rank and use the partial theta identity to give formulas for certain rank differences.

The $$\mathrm {al}$$ al function of a cyclic trigonal curve of genus three

A cyclic trigonal curve of genus three is a \(\mathbb {Z}_3\) Galois cover of \(\mathbb {P}^1\), therefore can be written as a smooth plane curve with equation \(y^3 = f(x) =(x - b_1) (x - b_2) (x - b_3) (x - b_4)\). Following Weierstrass for the hyperelliptic case, we define an “\(\mathrm {al}\)” function for this curve and \(\mathrm {al}^{(c)}_r\), \(c=0,1,2\), for each one of three particular covers of the Jacobian of the curve, and \(r=1,2,3,4\) for a finite branchpoint \((b_r,0)\). This generalization of the Jacobi \(\mathrm {sn}\), \(\mathrm {cn}\), \(\mathrm {dn}\) functions satisfies the relation: $$\begin{aligned} \sum _{r=1}^4 \frac{\prod _{c=0}^2\mathrm {al}_r^{(c)}(u)}{f'(b_r)} = 1 \end{aligned}$$ which generalizes \(\mathrm {sn}^2u + \mathrm {cn}^2u = 1\). We also show that this can be viewed as a special case of the Frobenius theta identity.

A Ramanujan-type congruence modulo 5 for 4-colored generalized Frobenius partitions

In his 1984 AMS Memoir, Andrews introduced the $$k$$ -colored generalized Frobenius partition function $$c\phi _k(n)$$ which denotes the number of generalized Frobenius partitions of $$n$$ with $$k$$ colors. Recently, Baruah and Sarmah, Lin, and Sellers established several Ramanujan-type congruences for $$c\phi _4(n)$$ . In this paper, employing some theta identities due to Ramanujan, the $$(p, k)$$ -parametrization of theta functions given by Alaca, Alaca, and Williams, and some results of Baruah and Sarmah, we prove that $$c\phi _4(20n+11)\equiv 0\ (\mathrm{mod}\ 5)$$ .

On the equivalence of two fundamental theta identities

Two fundamental theta identities, a three-term identity due to Weierstrass and a five-term identity due to Jacobi, both with products of four theta functions as terms, are shown to be equivalent. One half of the equivalence was already proved by R. J. Chapman in 1996. The history and usage of the two identities, and some generalizations are also discussed.

The rank of a unimodal sequence and a partial theta identity of Ramanujan

We study the number of unimodal sequences of weight n and rank m using a partial theta identity discovered by Ramanujan. We obtain rank difference identities as well as a congruence for the second rank moment.

On two 10th-order mock theta identities

We give short proofs of conjectural identities due to Gordon and McIntosh involving two 10th-order mock theta functions.

A Franklin type involution for squares

We find an involution as a combinatorial proof of Ramanujanʼs partial theta identity. Based on this involution, we obtain a Franklin type involution on the set of partitions into distinct parts with the smallest part being odd. Compared with the involution of Bessenrodt and Pak, our involution possesses a weight-preserving property that leads to a combinatorial proof of a weighted partition theorem derived by Alladi from Ramanujanʼs partial theta identity. This gives an indirect answer to a question of Berndt, Kim and Yee. Moreover, we obtain a partition theorem based on Andrewsʼ identity and provide a combinatorial proof via certain weight assignment for our involution. A specialization of this partition theorem is related to an identity of Andrews concerning partitions into distinct nonnegative parts with the smallest part being even. Finally, we give an extension of our partition theorem which corresponds to a generalization of Andrewsʼ identity.

Ramanujan-type partial theta identities and conjugate Bailey pairs

Residual identities of Ramanujan-type partial theta identities are tailor-made for producing conjugate Bailey pairs. This is carried out for partial theta identities in Ramanujan’s lost notebook, a number of the partial theta identities of Warnaar, and for some new ones as well.

Analysis of a generalized Lebesgue identity in Ramanujan’s Lost Notebook

We analyze a two-parameter q-series identity in Ramanujan’s Lost Notebook that generalizes the product part of the fundamental one-parameter Lebesgue identity. From reformulations of this two-parameter identity, we deduce new partition theorems including variants of the Gauss triangular number identity and Euler’s pentagonal number theorem. We discuss connections with a partial theta identity of Ramanujan and with several classical results such as those of Sylvester and Gollnitz–Gordon.