Ramanujan's Notebook Research Articles

On a higher-order version of a formula due to Ramanujan

A special case of an Entry in Part II of Ramanujan's Notebooks is such that 1 + 1 5 ( 1 2 ) 2 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 2 + ⋯ = Γ 4 ( 1 4 ) 16 π 2 . This formula leads us to consider the higher-order version of the above series given by replacing the squares of normalized central binomial coefficients with fourth powers. Using a Fourier–Legendre expansion introduced in a 2022 article by Cantarini, together with a multiple elliptic integral evaluation conjectured by Wan and proved by Zhou, we prove the very natural extension shown below of Ramanujan's formula: 1 + 1 5 ( 1 2 ) 4 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 4 + ⋯ = Γ 8 ( 1 4 ) 96 π 5 . Furthermore, and in a closely related way, we show how a main result in an article by Papanikolas et al. concerning a Calabi–Yau threefold is equivalent to the evaluation of a Clebsch–Gordan-type multiple elliptic integral related to the work of Zhou and Brychkov.

Some new identities for colored partitions with parts in multiples of $ 4 $

<p>To start, we provided the deformations of certain specific modular equations derived from Ramanujan's notebook, Part Ⅲ, by using the complete square formula. Subsequently, we established fifteen identities for partitions with distinct colors that arose from these deformed modular equations in the theory of modular equations. Throughout our investigation, we found that the fifteen identities have parts that were in multiples of $ 4 $.</p>

Untrodden pathways in the theory of the restricted partition function p(n,N)

We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for spt(n). The latter motivates us to extend the theory of the restricted partition function p(n,N), namely, the number of partitions of n with largest parts less than or equal to N, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for our finite analogue of the spt-function, namely spt(n,N), we prove an inequality between the finite second rank and crank moments. The other results obtained include finite analogues of a recent identity of Garvan, an identity relating d(n,N) and lpt(n,N), namely the finite analogues of the divisor and largest parts functions respectively, and a finite analogue of the Beck-Chern theorem.

On extensions of two results due to Ramanujan

The aim in this note is to provide a generalization of an interesting entry in Ramanujan's notebooks that relate sums involving the derivatives of a function $\varphi\left t\right $ evaluated at 0 and 1. The generalization obtained is derived with the help of expressions for the sum of terminating ${}_3F_2$ hypergeometric functions of argument equal to 2, recently obtained by Kim et. al. [Two results for the terminating ${}_3F_2$ 2 with applications, Bulletin of the Korean Mathematical Society 2012; 49: 621-633]. Several special cases are given. In addition we generalize a summation formula to include integral parameter differences.

Level 17 Ramanujan–Sato series

Two level 17 modular functions $$ r=q^{2}\prod_{n=1}^{\infty}(1-q^{n})^{\left(\frac{n}{17}\right)},\quad s=q^{2}\prod_{n=1}^{\infty}\frac{(1-q^{17n})^{3}}{(1-q^{n})^{3}}, $$ are used to construct a new class of Ramanujan-Sato series for $1/\pi$. The expansions are induced by modular identities similar to those level of 5 and 13 appearing in Ramanujan's Notebooks. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.

Ramanujan: A tale of two evaluations

In 1887, beneath a canopy of stars, Srinivasa Ramanujan commenced his brief existence on this planet. In a universe at the mercy of its entropy, against all odds, a genius was born. His destiny was mathematics, a subject born thousands of years earlier. The power of this discipline is not to be denied. After all, with our minds in the stars, we have placed footprints on the moon. The conquest of the moon was a triumph of applied mathematics; however, it was the landscape of pure mathematics that awaited Ramanujan. In time, he would explore it with passion, leaving footprints lasting for eternity. Professor Bruce C.~Berndt has done a remarkable job of editing the notebooks Ramanujan left behind. In particular, Berndt's Chapter 9 of \textit {Ramanujan's notebooks Part} I provides a magnificent in-depth look at raw mathematical talent in action. The primary purpose of this article is to present three Chapter~9 related results, a series evaluation and two new functional equations, that Ramanujan either missed or his work on them was lost. The secondary purpose is to present what Berndt calls a ``corrected version'' \cite [page 233]{Berndt} of an incorrect Chapter~9 formula of Ramanujan. This series evaluation represents one of the few serious mistakes to be found in Ramanujan's work.

Certain Integrals Arising from Ramanujan's Notebooks

In his third notebook, Ramanujan claims that $$ \int_0^\infty \frac{\cos(nx)}{x^2+1} \log x \,\mathrm{d} x + \frac{\pi}{2} \int_0^\infty \frac{\sin(nx)}{x^2+1} \mathrm{d} x = 0. $$ In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if $\log x$ were replaced by $\log^2x$ in the first integral and $\log x$ were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly relating large classes of infinite integrals and illustrate these by several examples.

On 2- and 4-dissections for some infinite products

The 2- and 4-dissections of some infinite products are established in this paper. As corollaries of our results, we derive the 4-dissections of some continued fractions appearing in Ramanujan's notebooks and their reciprocals.

Entry 1.414… in Ramanujan's Notebooks: is irrational

Entry 1.414… in Ramanujan's Notebooks: is irrational

Generalization of two theorems due to Ramanujan

The aim in this paper is to provide generalizations of two interesting entries in Ramanujan's notebooks that relate sums involving the derivatives of a function φ(t) evaluated at 0 and 1. The generalizations obtained are derived with the help of expressions for the Gauss hypergeometric function 2 F 1(−n, a; 2a+j; 2) for non-negative integer n and j=0,±1, …,±5 given very recently by Kim et al. [Generalizations of Kummer's second theorem with applications, Comput. Math. Math. Phys. 50(3) (2010), pp. 387–402] and extension of Gauss’ summation theorem available in the literature. Several special cases that are closely related to Ramanujan's results are also given.

An alternate circular summation formula of theta functions and its applications

We prove a general alternate circular summation formula of theta functions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two formulaes for (q; q)2n?.

FINITE TRIGONOMETRIC CHARACTER SUMS VIA DISCRETE FOURIER ANALYSIS

We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were first systematically studied by Berndt and Zaharescu where their proofs involved complex contour integration. We show how to prove most of Berndt–Zaharescu's and some new identities by elementary methods of discrete Fourier analysis.

Some entries in Ramanujan's notebooks

Some of Ramanujan's original discoveries about hypergeometric functions and their relation to modular integrals, especially Eisenstein series ofnegative weight, are still not very well understood. These discoveries take the form of identities that he recorded, without proof, as entries in his notebooks. In the following sections I will introduce some of these entries, discuss their status, give new proofs of several of them and also provide new results of a similar nature.

New identities for 7-cores with prescribed BG-rank

Let π be a partition. BG-rank ( π ) is defined as an alternating sum of parities of parts of π [A. Berkovich, F.G. Garvan, On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo 5 and generalizations, Trans. Amer. Math. Soc. 358 (2006) 703–726. [1]]. Berkovich and Garvan [The BG-rank of a partition and its applications, Adv. in Appl. Math., to appear in 〈 http://arxiv.org/abs/math/0602362 〉 ] found theta series representations for the t-core generating functions ∑ n ⩾ 0 a t , j ( n ) q n , where a t , j ( n ) denotes the number of t-cores of n with BG-rank = j . In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme values of BG-rank. In this paper we discuss representations of this type for all 7-cores with prescribed BG-rank. We make an essential use of the Ramanujan modular equations of degree seven [B.C. Berndt, Ramanujan's Notebooks, Part III, Springer, New York, 1991] to prove a variety of new formulas for the 7-core generating function ∏ j ⩾ 1 ( 1 - q 7 j ) 7 ( 1 - q j ) . These formulas enable us to establish a number of striking inequalities for a 7 , j ( n ) with j = - 1 , 0 , 1 , 2 and a 7 ( n ) , such as a 7 ( 2 n + 2 ) ⩾ 2 a 7 ( n ) , a 7 ( 4 n + 6 ) ⩾ 10 a 7 ( n ) . Here a 7 ( n ) denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only.

Ramanujan: Essays and surveys

The fact that Ramanujan made path-breaking discoveries without much formal training and was a misfit under the regular curriculum (he failed in the FA examination), has been a source of mystery to many. This intriguing aspect has fired the imaginations of several authors who have penned his biography. Almost all of them have dealt more with his personal life than with his mathematical work. This volume can be said with some justification to be an attempt at a more mathematical biography. A facsimile edition of Ramanujan's famous notebooks had been published in 1957 by the Tata Ins t i tu te of Fundamenta l Research, Bombay and, as such, they were unedited.

Some Eisenstein series identities related to modular equations of the seventh order

In this paper we will use one well-known modular equation of seventh order, one theta function identity of S. McCullough and L.-C. Shen, 1994, and the complex variable theory of elliptic functions to prove some new septic identities for theta functions. Then we use these identities to provide new proofs of some Eisenstein series identities in Ramanujan's notebooks or lost notebook. We also derive a new identity for Eisenstein series and some curious trigonometric identities.

Eisenstein Series in Ramanujan's Lost Notebook

In his lost notebook, Ramanujan stated without proofs several beautifulidentities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.

A Ramanujan Sequence that Refines the Cayley Formula for Trees

A further combinatorial study of Entries 16 and 17 in chapter 3 of Ramanujan's second notebook [1] leads to a refinement of Cayley's formula for counting trees and generalizes the recent results derived by Shor [13], and Dumont and Ramamonjisoa [5]. Our approach also yields a new proof of Clarke's extension of Cayley's formula [4].

Conjectures in Ramanujan's Notebooks

Conjectures in Ramanujan's Notebooks

Conjectures in Ramanujan's Notebooks

Conjectures in Ramanujan's Notebooks