Some general theorems on the explicit evaluations of Ramanujan''s cubic continued fraction

In 2001, Jinhee Yi found many explicit values of the famous Rogers–Ramanujan continued fraction by using modular equations and transformation formulas for theta-functions. In this paper, we use her method to find some general theorems for the explicit evaluations of Ramanujan's cubic continued fraction.     

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.     

Triple product identity, Quintuple product identity and Ramanujan's differential equations for the classical Eisenstein series

In this article, we use the triple product identity and the quintuple product identity to derive Ramanujan's famous differential equations for the Eisenstein series.

     

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 multi-variable theta product

A multi-variable theta product is examined. It is shown that, under very general choices of the parameters, the quotient of two such general theta products is a root of unity. Special cases are explicitly determined. The second main theorem yields an explicit evaluation of a sum of series of cosines; which greatly generalizes one of Ramanujan's theorems on certain sums of hyperbolic cosines.     

On the Expansion of Ramanujan's Continued Fraction

If Ramanujan's continued fraction (or its reciprocal) is expanded as a power series, the sign of the coefficients is (eventually) periodic with period 5. We give combinatorial interpretations for the coefficients from which the result is immediate. We make use of the quintuple product identity, which we prove.     

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.     

三次分拆数模5的幂

三次分拆由Hei-Chi Chan引入,并由Byungchan Kim命名,因为它和Ramanujan的三次连分数联系在一起.Hei-Chi Chan证明了三次分拆函数具有模3的幂的Ramanujan型同余.在最近的一篇文章中,William Y.C.Chen和Bernard L.S.Lin研究了三次分拆函数模5的同余...     

Analogues of Jacobi's inversion formula for the incomplete elliptic integral of the first kind

In this article, we revisit Ramanujan's cubic analogue of Jacobi's inversion formula for the classical elliptic integral of the first kind. Our work is motivated by the recent work of Milne (Ramanujan J. 6(1) (2002) 7-149), Chan and Chua (Ramanujan J., to appear) on the representations of integers as sums of even squares.     

A 1Ψ1 Summation Theorem for Macdonald Polynomials

In this note we extend the Ramanujan's 11 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8].     

The Rogers-Ramanujan continued fraction and a new Eisenstein series identity

With two elementary trigonometric sums and the Jacobi theta function θ₁, we provide a new proof of two Ramanujan's identities for the Rogers-Ramanujan continued fraction in his lost notebook. We further derive a new Eisenstein series identity associated with the Rogers-Ramanujan continued fraction.     

A combinatorial correspondence related to Göllnitz' (big) partition theorem and applications

In recent work, Alladi, Andrews and Gordon discovered a key identity which captures several fundamental theorems in partition theory. In this paper we construct a combinatorial bijection which explains this key identity. This immediately leads to a better understanding of a deep theorem of Göllnitz, as well as Jacobi's triple product identity and Schur's partition theorem.

     

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 Göllnitz–Gordon.     

Modular Equations for Ramanujan's Cubic Continued Fraction

In this paper we present two new identities providing relations between Ramanujan's cubic continued fraction G(q) and the two continued fractions G(q⁵) and G(q⁷).     

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.     

On the Davenport–Heilbronn theorems and second order terms

We give simple proofs of the Davenport–Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic fields having bounded discriminant. We also establish second main terms for these theorems, thus proving a conjecture of Roberts. Our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings.     

Weighted forms of Euler's theorem

In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's “lost” notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujan's identities and Euler's theorem. Our combinatorial formulations of Ramanujan's identities rely on the notion of rooted partitions. Pak's iterated Dyson's map and Sylvester's fish-hook bijection are the main ingredients in the weighted forms of Euler's theorem.     

Tenth order mock theta functions in Ramanujan's lost notebook (IV)

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.

     

Summation theorems for multidimensional basic hypergeometric series by determinant evaluations

We derive summation formulas for a specific kind of multidimensional basic hypergeometric series associated to root systems of classical type. We proceed by combining the classical (one-dimensional) summation formulas with certain determinant evaluations. Our theorems include A_r extensions of Ramanujan's bilateral ₁ψ₁ sum, C_r extensions of Bailey's very-well-poised ₆ψ₆ summation, and a C_r extension of Jackson's very-well-poised ₈φ₇ summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A_r, B_r, C_r, and D_r, respectively, of Chu's bilateral transformation formula for basic hypergeometric series of Gasper–Karlsson–Minton type. Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation theorems of the classical theory of basic hypergeometric series.