Ramanujan Identities Research Articles

Partial Theta Functions. I. Beyond the Lost Notebook

It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and a new generalization of the Jacobi triple product identity. By computing residues around the poles of our identities we find a surprising connection between partial theta function identities and Garrett–Ismail–Stanton-type extensions of multisum Rogers–Ramanujan identities. 2000 Mathematics Subject Classification 05A30, 33D15, 33D90.

Gordon's theorem for overpartitions

We give overpartition-theoretic analogues of certain combinatorial generalizations of the Rogers–Ramanujan identities.

On sums of certain trigonometric series

Let N be a fixed positive integer. In this paper, we apply a newly developed method of Eie and Lai to the evaluation of certain infinite trigonometric series. Indeed, we are able to express the sums of these series in terms of Bernoulli polynomials. That is, we obtain several new Bernoulli identities. A similar method leads to Ramanujan's identities.

Partition Identities and a Theorem of Zagier

In the study of partition theory and q-series, identities that relate series to infinite products are of great interest (such as the famous Rogers–Ramanujan identities). Using a recent result of Zagier, we obtain an infinite family of such identities that is indexed by the positive integers. For example, if m=1, then we obtain the classical Eisenstein series identity ∑ λ⩾1 odd (−1) (λ−1)/2q λ (1−q 2λ) =q ∏ n=1 ∞ (1−q 8n) 4 (1−q 4n) 2 . If m=2 and ( · 3 ) denotes the usual Legendre symbol modulo 3, then we obtain ∑ λ⩾1 ( λ 3 )q λ (1−q 2λ) =q ∏ n=1 ∞ (1−q n)(1−q 6n) 6 (1−q 2n) 2(1−q 3n) 3 . We describe some of the partition theoretic consequences of these identities. In particular, we find simple formulas that solve the well-known problem of counting the number of representations of an integer as a sum of an arbitrary number of triangular numbers.

Partial-Sum Analogues of the Rogers–Ramanujan Identities

A new polynomial analogue of the Rogers–Ramanujan identities is proven. Here the product-side of the Rogers–Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.

Identities for Ramanujan's Sixth-Order Mock Theta Functions

Identities for Ramanujan's Sixth-Order Mock Theta Functions

Finite Affine Groups: Cycle Indices, Hall–Littlewood Polynomials, and Probabilistic Algorithms

The study of asymptotic properties of the conjugacy class of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers. Four different probabilistic understandings of this measure are given—three using symmetric function theory and one using Markov chains. This leads to non-trivial enumerative results. Cycle index generating functions are derived and are used to compute the large dimension limiting probabilities that an element of the affine group is separable, cyclic, or semisimple and to study the convergence to these limits. The semisimple limit involves both Rogers–Ramanujan identities. This yields the first examples of such computations for a maximal parabolic subgroup of a finite classical group.

Vertex Algebras and Combinatorial Identities

In the 1980's, J. Lepowsky and R. Wilson gave a Lie-theoretic interpretation of Rogers–Ramanujan identities in terms of level 3 representations of affine Lie algebra sl(2,C)~. When applied to other representations and affine Lie algebras, Lepowsky and Wilson's approach yielded a series of other combinatorial identities of the Rogers–Ramanujan type. At about the same time, R. Baxter rediscovered Rogers–Ramanujan identities within the context of statistical mechanics. The work of R. Baxter initiated another line of research which yielded numerous combinatorial and analytic generalizations of Rogers–Ramanujan identities. In this note, we describe some ideas and results related to Lepowsky and Wilson's approach and indicate the connections with some results in combinatorics and statistical physics.

A BASIS OF THE BASIC $\mathfrak{sl} ({\bf 3}, {\mathbb C})^~$-MODULE

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers–Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard [Formula: see text]-modules. We obtained several infinite series of new combinatorial identities through the construction of all standard [Formula: see text]-modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper we extend our construction to the basic [Formula: see text]-module and, by using the principal specialization of the Weyl–Kac character formula, we obtain a Rogers–Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky–Wilson's approach for affine Lie algebras of higher ranks, say for [Formula: see text], n ≥ 2, in a way parallel to the next level of complexity seen when passing from the Rogers–Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli ≥7.

ON MULTI-COLOR PARTITIONS AND THE GENERALIZED ROGERS–RAMANUJAN IDENTITIES

Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions.

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.

A PROBABILISTIC PROOF OF THE ROGERS–RAMANUJAN IDENTITIES

The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given in this paper, leading to an elementary probabilistic proof of the Rogers–Ramanujan identities. This is compared with work on the uniform measure. The main case of Bailey's lemma is interpreted as finding eigenvectors of the transition matrix of a Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.

Residue Theorem and Theta Function Identities

In this paper we will use the residue theorem of elliptic functions to prove some theta function identities of Ramanujan. We also derive some new identities by this method.

On some Ramanujan's P-Q Identities

In this paper, we obtain some P-Q eta-function identities of Ramanujan on employing some modular equations in Ramanujan's alternative theory of elliptic functions of signature 4.

Some Eisenstein Series Identities

In this paper we use the theory of elliptic functions to provide different proofs of some Eisenstein series identities of Ramanujan from those given in a recent paper by B. C. Berndt, S. Bhargava, and F. G. Garvan (1995, Trans. Amer. Math. Soc.347, 4136–4244). From one of these identities we derive the inversion formula for the Borweins cubic theta functions via Venkatachanliengar's method. We also derive some striking Eisenstein series identities associated with the Borweins' cubic theta functions.

On the Number of Homomorphisms from a Finite Group to a General Linear Group

We study the number of homomorphisms from a finite group to a general linear group over a finite field. In particular, we give a generating function of such numbers. Then the Rogers–Ramanujan identities are applicable.

On Certain Identities Of Ramanujan

In this paper we will start with one identity of Ramanujan about Lambert series related to modular relations of degree 7 to give some completely new proofs of several important theta functions identities proved by Berndt and Zhang [1994, J. Number Theory48, 224–242] via the theory of modular forms. This study also yields a simple proof of one important identity of Ramanujan connected with partitions modulo 7. We also derive some new theta-function identities.

Umbral Calculus, Bailey Chains, and Pentagonal Number Theorems

We revisit the umbral methods used by L. J. Rogers in his second proof of the Rogers–Ramanujan identities. We shall study how subsequent methods such as the Bailey chains and their variants arise naturally from Rogers' insights. We conclude with the introduction of multi-dimensional Bailey chains and apply them to prove some new Pentagonal Number Theorems.

An Infinite Family of Engel Expansions of Rogers–Ramanujan Type

The extended Engel expansion is an algorithm that leads to unique series expansions of q-series. Various examples related to classical partition theorems, including the Rogers–Ramanujan identities, have been given recently. The object of this paper is to show that the new and elegant Rogers–Ramanujan generalization found by Garrett, Ismail, and Stanton also fits into this framework. This not only reveals the existence of an infinite, parameterized family of extended Engel expansions, but also provides an alternative proof of the Garrett, Ismail, and Stanton result. A finite version of it, which finds an elementary proof, is derived as a by-product of the Engel approach.

Engel Expansions and the Rogers–Ramanujan Identities

We study the previously developed extension of the Engel expansion to the field of Formal Laurent series. We examine three separate aspects. First we consider the algorithm in relation to the work of Ramanujan. Second we show how the algorithm can be used to prove expansions such as those found by Euler, Rogers, and Ramanujan. Finally we remark briefly on its use in acceleration of convergence.