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In this paper, we state some q-analogs of the famous Ramanujan Master Theorem. As an application, we compute some values of Jackson’s q-integrals that involve q-special functions.

The famous French mathematician Pierre Fermat, like Ramanujan, made notes of his observations without proofs and communicated his findings in letters to contemporaries. In this article a comparison is made of certain mathematical contributions of Fermat and Ramanujan, and of their mathematical tastes. The famous Ramanujan taxi-cab equation is discussed as a Diophantine equation in four variables having solutions, but this equation becomes the cubic version of Fermat’s Last Theorem when one of the variables is set equal to zero, in which case there are no non-trivial solutions.

In this paper, the author proves some basic hypergeometric series which utilizes the same ideas that Margaret Jackson used to give a proof of Ramanujan’s [Formula: see text] summation formula.

Inspired by the recent work of Hirschhorn and Sellers, in this paper we present a unified proof of Ramanujan-type congruences modulo 5 for c ϕ 4 ( 5 n + 1 ) .

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono, without need for Hecke operators. In this paper we give a method for generalizing Lachterman, Schayer, and Younger's proof to include Ramanujan congruences for multipartition functions $p_k(n)$, and Ramanujan congruences for $p(n)$ modulo certain prime powers., Comment: 16 pages

In this paper, we give an elementary proof of Ramanujan's Eisenstein series of level 7. In the process, we also prove four Eisenstein series of level 14 due to S. Cooper and D. Ye [4].

In 1742, Leonhard Euler invented the generating function for P(n). Godfrey Harold Hardy said Srinivasa Ramanujan was the first, and up to now the only, Mathematician to discover any such properties of P(n). In 1916, Ramanujan defined the generating functions for X(n),Y(n) . In 2014, Sabuj developed the generating functions for . In 2005, George E. Andrews found the generating functions for In 1916, Ramanujan showed the generating functions for , , and . This article shows how to prove the Theorems with the help of various auxiliary functions collected from Ramanujan’s Lost Notebook. In 1967, George E. Andrews defined the generating functions for P1r (n) and P2r (n). In this article these generating functions are discussed elaborately. This article shows how to prove the theorem P2r (n) = P3r (n) with a numerical example when n = 9 and r = 2. In 1995, Fokkink, Fokkink and Wang defined the in terms of , where is the smallest part of partition . In 2013, Andrews, Garvan and Liang extended the FFW-function and defined the generating function for FFW (z, n) in differnt way.

Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function $\overline{C}_{k, i}(n)$ which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. He also proved that $\overline{C}_{3, 1}(9n + 3) \equiv \overline{C}_{3, 1}(9n + 6) \equiv 0 ({\rm mod} 3)$. Chen, Hirschhorn and Sellers then found infinite families of congruences modulo 3 and modulo powers of 2 for $\overline{C}_{3, 1}(n)$, $\overline{C}_{6, 1}(n)$ and $\overline{C}_{6, 2}(n)$. In this paper, we find new congruences for $\overline{C}_{3, 1}(n)$ modulo 4, 18 and 36, infinite families of congruences modulo 2 and 4 for $\overline{C}_{8, 2}(n)$, congruences modulo 2 and 3 for $\overline{C}_{12, 2}(n)$, $\overline{C}_{12, 4}(n)$, and congruences modulo 2 for $\overline{C}_{24, 8}(n)$ and $\overline{C}_{48, 16}(n)$. We use simple p-d...

Ramanujan sums have been studied and generalized by several authors. For example, Nowak studied these sums over quadratic number fields, and Grytczuk defined that on semigroups. In this note, we deduce some properties on sums of generalized Ramanujan sums and give examples on number fields. In particular, we have a relational expression between Ramanujan sums and residues of Dedekind zeta functions., Comment: 10 pages

We give direct proofs of some of Ramanujan’s P-Q modular equations based on simply proved elementary identities from Chapter 16 of his Second Notebook. Наведено прямі доведення деяких P-Q модульних РІВНЯНЬ Рамануджана на підставі елементарних тотожностей з глави 16 його Другого зошита, що просто доводяться.

Article embodies significant contributions of great mathematician Ramanujan in different branches of Mathematics. The article also provides the knowledge about latest researches related to Ramanujan’s Summation formula.

In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial result of Ramanujan’s congruence modulo 11. In 1988, Andrews and Garvan stated such functions and described the celebrated result that the crank simultaneously explains the three Ramanujan congruences modulo 5, 7 and 11. Dyson wrote the article, titled Some Guesses in the theory of partitions, for Eureka, the undergraduate mathematics journal of Cambridge. He discovered the many conjectures in this article by attempting to find a combinatorial explanation of Ramanujan’s famous congruences for P (n), the number of partitions of n indeed, Ramanujan’s formulas lay unread until 1976 when Dyson found In the Trainty College Library of Cambridge University among papers from the estate of the late G.N.Watson. In 1986, F.Garvan wrote his Pennsylvania state Ph.D. Thesis Precisely on the formulas of Ramanujan relative to the crank. In view of this theoretical description, the story of the crank is a long romantic tale and the crank functions are intimately connected to all partitions congruences. In 2005, Mahlburg stated that the crank functions themselves obey Ramanujan type congruences.

We prove two identities associated with Ramanujan's continued fraction of order 12. We further establish several Eisenstein series identities associated with Ramanujan's continued fraction of order 12.

In the case of Siegel modular forms of degree $n$, we prove that, for almost all prime ideals $\frak{p}$ in any ring of algebraic integers, mod $\frak{p}^m$ cusp forms are congruent to true cusp forms of the same weight. As an application of this property, we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan's congruence. We will conclude by giving numerical examples.

We present a short and simple proof of Ramanujan's partition congruence p ( 11 n + 6 ) ≡ 0 ( mod 11 ) , and dedicate it to him on the occasion of his 125th birthday.

The aim of this research paper is to establish an extension of a theorem due to Ramanujan. The result is obtained with the help of two terminating results for the series $_3F_{2}$ very recently obtained by Rakha et al. A few interesting special cases are also given.

We present some identities and congruences for the general partition function p r (n). In particular, we deduce some known identities for Ramanujan’s tau function and find simple proofs of Ramanujan’s famous partition congruences for modulo 5 and 7. Our emphasis throughout this paper is to exhibit the use of Ramanujan’s theta functions to generate identities and congruences for general partition function.

We present a new proof of the monotonicity of the correction term θn in Ramanujan’s refinement of Stirling’s formula. Moreover we prove that θn is concave.

In his famous paper, "On certain arithmetical functions", Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan's tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for the Euler products appearing in his lost notebook, we provide or sketch proofs using elementary methods, binary quadratic forms, and modular forms.

In this paper, we give a pedagogical introduction to several beautiful formulas discovered by Ramanujan. Using these results, we evaluate a Ramanujan-type integral formula. The result can be expressed in terms of the Golden Ratio.

Let p ( n ) p(n) denote the number of partitions of n n . Let A , B ∈ N A,B\in \mathbb {N} with A > B A>B and ℓ ≥ 5 \ell \geq 5 a prime, such that \[ p ( A n + B ) ≡ 0 ( mod ℓ ) , n ∈ N . p(An+B)\equiv 0\pmod {\ell }, \quad n\in \mathbb {N}. \] Then we will prove that ℓ | A \ell |A and ( 24 B − 1 ℓ ) ≠ ( − 1 ℓ ) \left (\frac {24B-1}{\ell }\right ) \neq \left (\frac {-1}{\ell }\right ) . This settles an open problem by Scott Ahlgren and Ken Ono. Our proof is based on results by Deligne and Rapoport.

Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form ∑ k = 0 ∞ ( 1 2 ) k ( 1 d ) k ( d - 1 d ) k k ! 3 ( a k + 1 ) ( λ d ) k = δ π for d=2,3,4,6, where łd are singular values that correspond to elliptic curves with complex multiplication, and a,δ are explicit algebraic numbers. In this paper we prove a p-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.

We conduct a detailed analysis of Ramanujan’s function k = k(q) defined by $$k = q\prod _{j=1}^{\infty }\frac{(1 - q^{10j-9})(1 - q^{10j-8})(1 - q^{10j-2})(1 - q^{10j-1})} {(1 - q^{10j-7})(1 - q^{10j-6})(1 - q^{10j-4})(1 - q^{10j-3})}$$ and obtain results that are analogues of theorems in Chapters 5–9

We present a variety of techniques for analyzing some remarkable series for 1∕π, such as $$\sum _{n=0}^{\infty }{2n\choose n}^{3}\left (n + \frac{5} {42}\right )\left ( \frac{1} {4096}\right )^{n} = \frac{8} {21} \times \frac{1} {\pi },$$ that were first studied by Ramanujan. It is shown how to classify the series by level and degree. We also obtain iterative processes that converge rapidly to 1∕π.

A simple proof is given of what is often described as “Ramanujan’s most beautiful identity”.

Earlier results are re-presented from Ramanujan’s point of view. Two proofs of Ramanujan’s celebrated 5-dissection of Euler’s product are given, including one by factorisation.

We discuss progress towards the Ramanujan conjecture for the group G L n \mathrm {GL}_n and its relation to various other topics in analytic number theory.

At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli. In this paper, we offer some general formulas for the explicit evaluations of Ramanujan’s singular moduli by parameterizations of Ramanujan’s theta-functions and give examples.

The polynomial Ramanujan sum was first introduced by Carlitz [7], and a generalized version by Cohen [10]. In this paper, we study the arithmetical and analytic properties of these sums, derive various fundamental identities, such as H older formula, reciprocity formula, orthogonality relation and Davenport{Hasse type formula. In particular, we show that the special Dirichlet series involving the polynomial Ramanujan sums are, indeed, the entire functions on the whole complex plane, we also give a square mean values estimation. The main results of this paper are new appearance to us, which indicate the particularity of the polynomial Ramanujan sums., 29 pages

Pages 262–265 in Ramanujan’s Lost Notebook and Other Unpublished Papers are devoted to a draft of a manuscript on an unusual problem in diophantine approximation. Ramanujan finds the maximum value of a polynomial when the variable takes rational values, and he observes the maxima’s periodicity. Because definitions and theorems in the manuscript are imprecise, the authors have attempted to interpret Ramanujan’s ideas and provide rigorous proofs.

Among the many astonishing formulae stated by Ramanujan we find in his Notebooks the following sequence (see [1], p 96, Entry 43):which hold for any triple of complex numbers (a, b, c) such that a + b + c = 0.Ramanujan concluded this list by ‘And so on’, which suggests that he had some kind of method or algorithm allowing him to extend this list to any even power of the quadratic form p(a, b, c) = ab + bc + ca, when a + b + c = 0. To explain this, Bruce Brendt details a theorem by S. Bhargava [2], which can be used to produce identities of the kind above, and infers that Ramanujan is likely to have used the same proof to establish his identities (see [1], pp. 97-100)

Ramanujan's famous congruences for the partition function modulo powers of 5, 7, and 11 have inspired much further research. For example, in 2002 Lovejoy and Ono found subprogressions of 5jn + β5(j) for which Ramanujan's congruence mod 5j could be strengthened to a statement modulo 5j+1. Here we provide the analogous results modulo powers of 7 and 11. We require the arithmetic properties of two special elliptic curves.

Cataloged from PDF version of article. In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. With one exception all of Ramanujan's identities were proved. In this paper, we provide a proof for the remaining identity together with new elementary proofs for two identities of Ramanujan which were previously proved using the theory of modular forms. Ramanujan stated that each of his formula was the simplest of a large class. Our proofs are constructive and permit us to obtain several analogous identities which could have been stated by Ramanujan and may very well belong to his large class of identities. © 2011 Elsevier Inc.

Srinivasa Ramanujan was a self-taught mathematical genius, born in 1887 in India. In the short span of 32 years, he attained great distinction. Ramanujan's theorems, questions, and solutions and the famous notebooks in which he recorded his findings have inspired many scholars and researchers since his death in 1920. A citation analysis of Ramanujan's articles shows the impact of his work on the research and academic community well into the twenty-first century.

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.

Two congruences are proved for an infinite family of Appell–Lerch sums. As corollaries, special cases imply congruences for some of the mock theta functions of order two, six and eight.

We give a natural derivation of a formula of Ramanujan, described by B.C. Berndt as “enigmatic”, for the harmonic series.

This paper summarizes the development of Ramanujan expansions of arithmetic functions since Ramanujan's paper in 1918, following Carmichael's mean-value-based concept from 1932 up to 1994. A new technique, based on the concept of related arithmetic functions, is introduced that leads to considerable extensions of preceding results on Ramanujan expansions. In particular, very short proofs of theorems for additive and multiplicative functions going far beyond previous borders are presented, and Ramanujan expansions that formerly have been considered mysterious are explained.

In 2007, George E. Andrews and Peter Paule (Acta Arithmetica 126:281–294, 2007) introduced a new class of combinatorial objects called broken k-diamonds. Their generating functions connect to modular forms and give rise to a variety of partition congruences. In 2008, Song Heng Chan proved the first infinite family of congruences when k=2. In this note, we present two non-standard infinite families of broken 2-diamond congruences derived from work of Oliver Atkin and Morris Newman. In addition, four conjectures related to k=3 and k=5 are stated.

In this paper, we study the divisibility of the function a(n) defined by $\sum_{n\geq 0} a(n) q^n := (q;q)^{-1}_\infty (q^2; q^2)^{-1}_\infty $. In particular, we prove certain "Ramanujan type cong...