His most famous work was on the number p(n) of partitions of an integer n into summands. me in my work. The Rogers-Ramanujan functions and continued fractions. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. T1 - Solving Ramanujan's differential equations for Eisenstein series via a first order Riccati equation. The resulting generalized hypergeometric function is written. 2012. Part 6: General Forms for Ramanujan's Pi Formulas by Tito Piezas III Abstract: A simple overview of Ramanujan's four types of formulas for 1/pi will be given in terms of general forms using only the hypergeometric function and Dedekind eta function.. These series were first studied systematically by Heine, but many . Based on numerical computations, Van Hamme recently conjectured p-adic analogues to such formulae. Rao agreed to give him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately convinced him of Ramanujan's brilliance. The most important special case is when j = k + 1, when it becomes The topics include partitions, hypergeometric series, Ramanujan's \(\tau\)-function and round numbers. If a series converges, the individual terms of the series must approach zero. In particular, two new evaluations of ${}_7 F_6 $'s with four parameters are stated. In this paper, Ramanujan's 1ˆ1 summation and the q-Gauss summation are established combinatorially. * It is a series of essays rather than . He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. Perhaps his most famous work was on the number of partitions p(n) of an integer. Some new Rogers- Ramanujan identities are given. Ramanujan made significant contributions to the fields of elliptic functions, continued fractions, infinite series, hypergeometric series and the analytical theory of numbers. Nathan Jacob Fine. A connection between the work of Rogers and Andrews, and q-Lagrange inversion is stated. Abstract. His papers were published in English and European journals. Ramanujan type identities. Abstract. Ramanujan's widow, Smt. In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. The primary reference for q-series is the book of An Introduction to q-analysis by Warren P. Johnson, and the primary reference for theta functions is the book Ramanujan's Theta Functions . 4.4 and 4.11 in chap. Srinivasa Ramanujan FRS (/ ˈ s r iː n ɪ v ɑː s ə r ɑː ˈ m ɑː n ʊ dʒ ən /; born Srinivasa Ramanujan Aiyangar, IPA: [sriːniʋaːsa ɾaːmaːnud͡ʑan ajːaŋgar]; 22 December 1887 - 26 April 1920) was an Indian mathematician who lived during the British Rule in India. We were drawn towards calculating the number of rational points through a particular characterization in terms of a particular Gaussian hypergeometric series. These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi. Ramanujan's Eisenstein series and new hypergeometric-like series for 1 / π 2. Introduction Dirichlet L-values formulas for π hypergeometric series lattice sums Ramanujan MSC classification Secondary: 33C20: Generalized hypergeometric series, $_pF_q$ 11F11: Holomorphic modular forms of integral weight 11F03: Modular and automorphic functions 11Y60: Evaluation of constants 33C75: Elliptic integrals as hypergeometric functions 33E05 . Srinivasa Ramanujan Aiyangar. Srinivasa Ramanujan Biography: Srinivasa Ramanujan was an Indian greatest mathematician who made pioneering contributions to number theory, functions, and infinite series. : 80 When Rao asked him what he wanted, Ramanujan replied that he needed work and financial support. Gauss, hypergeometric series and Ramanujan. ramanujan saw the gauss summation theorem in carr's synopsis and it remains a mystery till date as to how in one sweep of intuitive imagination he was able to arrive at the most general summation theorem with only a hint of the gauss summation theorem (eqs. They were especially helpful in my study of Ramanujan's "Lost" Notebook which overlaps the present book in significant ways. In [5;p. 1-2, 09.2009, p. 135-153. These functions are in general solutions to certain second order ordinary linear differential equations which occur in a number of problems in the mathematical sciences. Sander Zwegers showed that Ramanujan's mock theta functions are q-hypergeometric series, whose q-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. In this paper we manage to prove three of the supercongruences by means of the Wilf-Zeilberger algorithmic technique. XII, (XII, 43, Ex. In chapter 9, we present in brief the work on the hypergeometric series by Gauss, its further developments in Europe, and the genius of the Indian mathematician Srinivasa Ramanujan, which enabled him to discover the results for himself with only a hint of the Gauss summation theorem. 1. KW - Hypergeometric functions PY - 2007. Introduction "Divergent" Ramanujan-type series for 1/π and 1/π² provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. The generalized hypergeometric function is given by a Hypergeometric Series, i.e., a series for which the ratio of successive terms can be written. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series. The q-analogs of the so-called "strange evaluations" are also corollaries. American Mathematical Soc., 1988 - Mathematics - 124 pages. Many evaluations of terminating hypergeometric series at arguments other than 1 are given. Born: December 22, 1887 Died: April 26, 1920 Achievements: Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. His most famous work was on the number p(n) of partitions of an integer n into summands. "Divergent" Ramanujan-type series for 1/π and 1/π² provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. The development of our ideas Ramanujan's series representations for 1/afii9843 depend upon Clausen's product formulas for hypergeometric series and Ramanujan's Eisenstein series P (q):= 1 − 24 ∞ summationdisplay k=1 kq k 1 − q k , |q| < 1. Hypergeometric series, elliptic curves and supercongruences. 8. Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. III. Other notable contributions by Ramanujan include hypergeometric series, the Riemann series, the elliptic integrals, the theory of divergent series, and the functional equations of the zeta function. I. (The factor of in the Denominator is present for historical reasons of notation.) Some identities arising in basic hypergeometric series can be interpreted in the theory of partitions using F-partitions. Basic Hypergeometric Series and Applications The theory of partitions, founded by Euler, has led in a natural way to the idea of basic hypergeometric series, also known as Eulerian series. the elliptic integrals, hypergeometric series, and the functional equations of the zeta function. Introduction. Although many of his discoveries were in the previous mathematical literature. This is one of the simplest and famous series given by Ramanujan and it's value is 2 / π. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, . On a Claim of Ramanujan about Certain Hypergeometric Series @inproceedings{Bradley1994OnAC, title={On a Claim of Ramanujan about Certain Hypergeometric Series}, author={David M. Bradley}, year={1994} } Keywords: Unfortunately Ramanujan's technique requires a reasonable amount of effort to understand. gamma functions $\Gamma$) involved, unlike his other similar infinite series. Basic Hypergeometric Series and Applications. N. H. Abel, letter to Holmboe, January 1826, reprinted in volume 2 of his collected papers. International Journal of Modern Physics: Conference Series Vol. 1 Introduction and Preliminaries A natural generalization of Gauss function 2F1 is the general hypergeometric series pFq [3, p.42, 160, No. Perhaps from this infinite series' very peculiar form, some kind of hypergeometric series could be involved (and Ramanujan was very highly fond of that area of study). We adopt the following notation and terminology in [5]. He was to discover later that he had been studying elliptic functions. But, unless disguised, there does not seem to be any "advanced" functions (e.g. As we shall see, the second follows from a theorem of J. Dougall [2]. The Rogers-Ramanujan identity is an equality between a certain "q-series" (given as an infinite sum) and a certain modular form (given as an infinite product). discuss some conjectural identities for certain q-hypergeometric series that came out of a reduction of other conjectures of Kanade and Russell. II. Y1 - 2007. His several other works include the Riemann series, the elliptic integrals, hypergeometric series, the functional equations of the zeta function, and his own theory of divergent series. In 1914, S. Ramanujan gave 17 series for 1/ π which are of the following form: \sum_ {n=0}^ {\infty} z^n \frac { ( \frac {1} {2} )_n (s)_n (1-s)_n} { (1)_n^3} (a+bn)= \frac {1} {\pi}, \qquad (x)_k=\frac {\varGamma (x+k)} {\varGamma (x)}, (1) where s ∈ {1/2,1/4,1/3,1/6} and the parameters z, a, b are algebraic numbers. 1. KW - Hypergeometric differential equation. Download Citation | Ramanujan's Radial Limits and Mixed Mock Modular Bilateral q -Hypergeometric Series | Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof . 7884 SHAUN COOPER AND DONGXI YE By(1.2)and[7,(3.1.6)-(3.1.8),(4.2.1)-(4.2.3)andTheorem5.7(a)(i)],theidentity . G. Ólafsson, A. Pasquale. Srinivasa Ramanujan was a mathematician par excellence. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. As usual for any complex number a, we write (a)n:= 1 . Ramanujan's mock theta functions and some recent developments. Hardy was the first to recognize the brilliance of Ramanujan's ideas. Ramanujan's Master theorem for the hypergeometric Fourier transform on root systems. Ramanujan was shown how to solve cubic equations in 1902 andhe went on to find his own method to solve the quartic. PROF. G. H. HARDY who, some thirteen years ago, supervised the editing of Ramanujan's collected papers, has now produced a new volume dealing with Ramanujan. Keywords and phrases. One example is In fact, we prove a more general identity (which may have been the identity that Ramanujan actually had, and from which he recorded the most The first gives a formula for the derivative of a quotient of two certain bilateral hypergeometric series. Some of his most important contributions include the Riemann series, Divergent Series Theory, elliptic integrals, and Hypergeometric series. Bilateral basic hypergeometric series, g-series, multiple basic hypergeometric series associated to the root system Ani U(n + 1) series, q-binomial theorem, Ramanujan's itpi summation, Macdonald identities, Bailey's 2^2 transformations, 2^2 summation. During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan-type series for 1/π. Srinivasa Ramanujan was . 0 Reviews. In this paper we manage to prove three of the supercongruences by means of the Wilf-Zeilberger algorithmic technique. Answer (1 of 4): Srinivasa Ramanujan(22 December 1887 - 26 April 1920) was an Indian mathematician who lived during the British Rule in India. Perhaps from this infinite series' very peculiar form, some kind of hypergeometric series could be involved (and Ramanujan was very highly fond of that area of study). Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical . The unilateral basic hypergeometric series is defined as where and is the q -shifted factorial . 1.Introduction During year 1903-1914 Ramanujan recorded most of his mathematical discoveries without proof in his notebooks. 4 π = 1 + 7 4 ( 1 2) 3 + 13 4 2 ( 1 ⋅ 3 2 ⋅ 4) 3 + …. Their q-analogues lead to many of the continued c 2015 American Mathematical Society 7883. There are now major interactions with Lie algebras, combinatorics, special functions, and number theory. An incomplete survey of hypergeometric series is given, and some of Ramanujan's work on hypergeometric and basic hypergeometric series is put into the general framework as we understand it now. The book concludes with a chapter by chapter . Research on g-hypergeometric series is significantly more active now than when Fine began his researches. Introduction Srinivasa Ramanujan Iyengar, one of the India's greatest mathematical geniuses, was born on AU - Hill, James M. AU - Berndt, Bruce C. AU - Huber, Tim. On page 200 of his lost notebook, in the pagination of [4], Ramanujan offers two results on certain bilateral hypergeometric series. 1906. Basic hypergeometric series; Lambert series; elliptic functions; mock theta functions. is ``the'' Hypergeometric Function, and is the . Ramanujan's results on continued fractions are simple consequences of three-term relations between hypergeometric series. Janaki Ammal, moved to Bombay; in 1931 she returned to Madras and settled in Triplicane, where she supported herself on a pension from . Ramanujan's Eisenstein series and new hypergeometric-like series for 1/ 2 Nayandeep Deka Baruaha,1, Bruce C. Berndtb ∗ 2 aDepartment of Mathematical Sciences, Tezpur Universtiy, Napaam 784028, Sonitpur, Assam, India bDepartment of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA Mathematics. While this sounds quite simple, somehow a full proof has eluded us. Ramanujan has enriched world mathematics with more than 3,500 equations and formulae. Eisenstein series and the Borweins' general formula for p = 1. Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant D. Borwein, J.M. I have presented the proof for this series and it's friend. Theorems in the theory of partitions are closely related to basic hypergeometric series. 7. series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas. Glasser zand J.G Wanx May 28, 2011 Abstract We undertake a thorough investigation of the moments of Ramanujan's alterna-tive elliptic integrals and of related hypergeometric functions. Ramanujan theta function is used to determine the critical dimensions in Bosonic string theory, superstring theory, and M-theory. Special cases include quadratic and cubic transformations for basic hypergeometric series. Ramanujan in England had made advances mainly in the partition of numbers. But, unless disguised, there does not seem to be any "advanced" functions (e.g. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. ditions) of Srinivasa Ramanujan, using Laplace transform, hypergeometric summation theorem for 4F3( 1), properties of Pochhammer's symbol and Gamma function. Ramanujan Number Theory The two subject areas of Ramanujan number theory are q-series and theta functions, both of which are introduced in the book Number Theory in the Spirit of Ramanujan by Bruce C. Berndt. What is so special about the special functions? It has fascinated many mathematicians,. / Baruah, Nayandeep Deka; Berndt, Bruce C. In: Journal of Approximation Theory, Vol. gamma functions $\Gamma$) involved, unlike his other similar infinite series. Introduction Ramanujan's sum is a useful extension of Jacobi's triple product formula, and has recently become important in the treatment of certain orthogonal polynomials defined by basic hypergeometric series. Duration July 25 - August 6, 2016 series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas. 7), in the rst Notebook of Ramanujan, using well-known transformation and summation theorems of hypergeometric series. Borweiny, M.L. Keywords-q-hypergeometric series, Ramanujan's Theta function, q-bionomial theorem . In any case, Ramanujan rediscovered not only all that was known in Europe on hypergeometric series at that time, but he also discovered several new theorems, and, in particular, theorems on products of hypergeometric series [12], as well as several types of asymptotic expansions. George Andrews, Henri Cohen, Freeman Dyson, and Dean Hickerson found a pair of q-hypergeometric series each of which contains half of the Fourier . The first author was partially supported by National Security Agency grant ditions) of Srinivasa Ramanujan, using Laplace transform, hypergeometric summation theorem for 4F3( 1), properties of Pochhammer's symbol and Gamma function. His most famous work was on the number p(n) of partitions of an integer n into summands. (3.1) In our papers [3,4], by combining two different relations between P (q 2 )andP (q 2n . Hardy, Highly composite numbers, Partition function. KW - Differential equations for Eisenstein series. In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit . In his book, 'The Discovery of Ramanujan's Lost Notebook', Georg Andrews wrote that he was already an advanced researcher in fields, such as mock theta functions and hypergeometric series.. Hypergeometric series and continued fractions K G RAMANATHAN A1 Sree Krishna Dham, 70 L B S Marg, Bombay 400080, India Al~raet. DOI: 10.1090/S0002-9939-1994-1189537-5 Corpus ID: 119621808. Some identities arising in basic hypergeometric series can be interpreted in the theory of partitions using F-partitions. 1. At the highschool, Ramanujan was to do well in all his school subjects and showed himself an able all round scholar. Research output: Contribution to journal › Article › peer-review 22 (2013) 679-685 World Scientific Publishing Company DOI: 10.1142/S2010194513010854 679 GENERALIZATION OF A RESULT INVOLVING PRODUCT OF GENERALIZED HYPERGEOMETRIC SERIES DUE TO RAMANUJAN NIDHI SHEKHAWAT Introduction In his short lifetime of 32 years, he came up with more than 3900 identities, equations and proofs, including completely novel discoveries like the Ramanujan prime . Srinivasa Ramanujan, born into a poor Brahmin family at Erode on Dec. 22, 1887, attended school in nearby Kumbakonam. 7), in the rst Notebook of Ramanujan, using well-known transformation and summation theorems of hypergeometric series. 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