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Combinatorial proofs of Ramanujan's 1ψ1 summation and the q -Gauss summation

Theorems in the theory of partitions are closely related to basic hypergeometric series. Some identities arising in basic hypergeometric series can be interpreted in the theory of partitions using F-partitions. In this paper, Ramanujans 1 ψ1 summation and the q-Gauss summation are established combinatorially.

Formulas of Ramanujan for the power series coefficients of certain quotients of Eisenstein series

In their last published paper [9], [16, pp. 310–321], G. H. Hardy and S. Ramanujan derived infinite series representations for the coefficients of certain modular forms of negative weight which are not analytic in the upper half-plane. In particular, they examined in detail the coefficients of the reciprocal of the Eisenstein series E6(τ). While confined to the sanitarium, Matlock House, in 1918, Ramanujan wrote several letters to Hardy about the coefficients in the power series expansions of certain quotients of Eisenstein series. These letters are photocopied in [18, pp. 97–126], and printed versions with commentary can be found in [6, pp. 175–191]. In these letters, Ramanujan recorded formulas for the coefficients of several quotients of Eisenstein series not examined by Hardy and him in [9]. These claims fall into two related classes. In the first class are formulas for coefficients that arise from the main theorem of Hardy and Ramanujan, or a slight modification of it, and these results have been proved in a paper by Berndt and Bialek [5]. Those in the second class, which we prove in this paper, are much harder to prove. To establish the first main result, we need an extension of Hardy and Ramanujan’s theorem due to H. Petersson [11]. To prove the second primary result, we need to first extend work of H. Poincare [14], Petersson [11], [12], [13], and J. Lehner [10] to cover double poles. In all cases, the formulas have a completely different shape from those arising from modular forms analytic in the upper half-plane, such as the famous infinite series for the partition function p(n) arising from the reciprocal of the Dedekind eta-function. As we shall see in the sequel, the series examined in this paper are very rapidly convergent, even more so than those arising from modular forms analytic in the upper half-plane, so that truncating a series, even with a small number of terms, provides a remarkable approximation. Using Mathematica, we calculated several coefficients and series approximations for the two primary functions 1/B(q) and 1/B(q) (defined below) examined by Ramanujan. As will be seen from the first table, the coefficient of q in 1/B(q), for example, has 17 digits, while just two terms of Ramanujan’s infinite series representation calculate this coefficient with

Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions

Combinatorial proofs are given for certain entries in Ramanujans lost notebook. Bijections of Sylvester, Franklin, and Wright, and applications of Algorithm Z of Zeilberger are employed. A new bijection, involving the new concept of the parity sequence of a partition, is used to prove one of Ramanujans fascinating identities for a partial theta function.

ON THE PARITY OF PARTITION FUNCTIONS

Let S denote a subset of the positive integers, and let pS(n) be the associated partition function, that is, pS(n) denotes the number of partitions of the positive integer n into parts taken from S. Thus, if S is the set of positive integers, then pS(n) is the ordinary partition function p(n). In this paper, working in the ring of formal power series in one variable over the field of two elements Z/2Z, we develop new methods for deriving lower bounds for both the number of even values and the number of odd values taken by pS(n), for n ≤ N. New very general theorems are obtained, and applications are made to several partition functions.

A page on Eisenstein series in Ramanujan's lost notebook

Page 188 in Ramanujans lost notebook is devoted to a certain class of infinite series connected with Eulers pentagonal number theorem. These series are represented in terms of Ramanujans famous Eisenstein series

A Note on Partitions into Distinct Parts and Odd Parts

P, Q

Overpartitions and Generating Functions for Generalized Frobenius Partitions

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Integrals of Eisenstein series and derivatives of L-functions

R

On the Combinatorics of Lecture Hall Partitions

. The purpose of this paper is to prove all the formulas on page 188 and to show that one of them leads to an interesting, new recurrence formula for

A truncated Jacobi triple product theorem

\sigma (n)