# A Series Considered By Ramanujan

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### Special values of trigonometric Dirichlet series and Eichler

series of the form P n>1 f(ˇn˝) s, where f is an arbitrary product of the elementary trigonometric functions, ˝a real quadratic irrationality and s an integer of the appropriate parity. This uni es a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt.

### COMBINATORICS OF GENERALIZED -EULER NUMBERS

The column on the right contains the numbers considered by Prodinger for which the series (1.4) is an associated generating function [22, Theorem 2.2]. For each value of A,B,Cand D, the quotient (1.4) induces a corresponding recursion relation for the function f 2n+1(q). From these formulas, we obtain the following related polynomials.

### Automorphic Forms and Automorphic Representations

About ninety years ago, Ramanujan considered the following power series of q ∆(q) = q Y n≥1 (1 − qn)24. Expanding this out formally, we have: ∆(q) = X n>0 τ(n)qn = q − 24q2 + Ramanujan made a number of conjectures about the coeﬃcients τ(n). These conjectures have turned out to be very inﬂuential. They say: 2

### The Partition Function Revisited

Ramanujan considered the 24th power of the η-function: ( z):= η(z)24 = ∞ n=1 τ(n)qn, q = e2πiz, and showed that the coefﬁcients τ(n) are of sufﬁcient arithmetic interest. This moti-vated his celebrated conjectures regarding the τ-function and these conjectures had a pivotal role in the development of 20th century number theory.

### CONTRIBUTIONS OF MATHEMATICS GENIUS SRINIVASA RAMANUJAN IN

mathematicians. Considered to be a mathematical genius, Srinivasa Ramanujan, was regarded at par with the likes of Leonhard Euler and Carl Jacobi. In spite of having almost no formal training in mathematics, Ramanujan s knowledge of mathematics was astonishing. Even though he had no

### Infinite Sums and Reaching Infinity 006241-0012 Mr. Cavazos

Ramanujan was involved in. It revolved heavily around number theory and even made an equation for infinite series called the Ramanujan Summation. From this I also saw the calculations he made during this research in order to prove many infinite series, two of which are going to be presented in this paper.

### COMBINATORICS OF RAMANUJAN-SLATER TYPE IDENTITIES James

36, the other members having been found by Ramanujan and Slater. We examine combina-torial implications of the identities in this family, and of some of the identities we considered in Identities of the Ramanujan-Slater type related to the moduli 18 and 24, [preprint, 2007]. 1. Introduction The Rogers-Ramanujan identities, X1 j=0 qj2 (q;q)j

### COMBINATORIAL PROOFS OF IDENTITIES IN RAMANUJAN S LOST

Abstract. Ramanujan s lost notebook contains several identities arising from the Rogers Fine iden-tity and/or Rogers false theta functions. Combinatorial proofs for many of these identities are given. 1. Introduction Ramanujan s lost notebook [8] contains several hundred q-series identities. At least

### Third Ordered Transformations of KEYWORDS : Oblique fissure

Main Series Let M= 108 and Thus this transform can be considered as an third order series. Now a series of Is transformed into Where is as defined earlier Here the following values converge when transformed accord-ingly. Conclusion We conclude that Ramanujan type formulas can be still be ex-

### www.cambridge.org

Nagoya Math. J., 239 (2020), 232{293 DOI 10.1017/nmj.2018.38 GENERALIZED LAMBERT SERIES, RAABE S COSINE TRANSFORM AND A GENERALIZATION OF RAMANUJAN S FORMULA FOR (2m+1) ATUL

### Ramanujan sums as supercharacters

Nevertheless, Ramanujan was the ﬁrst to appreciate the importance of the sum and to use it systematically, according to G.H. Hardy [19, p. 159]. Ramanujan s interest in the sums (1.1) originated in his desire to obtain expres-sions for a variety of well-known arithmetical functions of n in the form of a series s ascs(n). This

### GENERALIZED LAMBERT SERIES AND ARITHMETIC NATURE OF ODD ZETA

Notebook [31] is that at the end of this page Ramanujan starts writing the exact same series considered by Kanemitsu, Tanigawa and Yoshimoto in their above result, that is, the series on the left side of (1.6), but with more general conditions on the associated parameters subsuming the ones given by Kanemitsu et al.

### WEIGHTED DIVISOR SUMS AND BESSEL FUNCTION SERIES, V

Each of these identities has three different interpretations. First, the double series in (1.3) and (1.4) can be interpreted as iterated series in, of course, two ways. Second, there is some evidence that Ramanujan considered the double sums in a third way, namely as series in which the product of the indices tends to inﬁnity.

### On a generalisation of a result of Ramanujan connected with

ON A GENERALISATIO OF A RESULN OTF RAMANUJAN CONNECTED WITH THE EXPONENTIAL SERIES by R. B. PARIS (Received 6th June 1980) 1. Introduction One of the man y interesting problems discusse by Ramanujad n is an approximatio n related to the exponential series for e , when n assumes large positive integer values If the numbe 6rn is defined by

### POWERS OF THE ETA-FUNCTION AND HECKE OPERATORS

Hecke action with respect to is de ned on its q-series by (1.2) X n˛1 a(n)qn T( ) := X n˛1 s=2+1˜(1= )a( n) + a(n= ) qn: Remark. The normalization in both de nitions is nonstandard, and is tailored so that the coe cient of the rst power of qin the q-series we considered here remains the same under

### MARKING AND SHIFTING A PART IN PARTITION THEOREMS

Ramanujan identities, the G ollnitz-Gordon identities, Euler s odd=distinct theorem, and the Andrews-Gordon identities. Generalizations of each of these theorems are given where a single part is marked or weighted. This allows a single part to be replaced by a new larger part, shift-

### ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 15, Number 2

He stated two results, a series transformation and the evaluation of a special series, but it is clear from the second of these results that he had more. Ramanujan considered this series and more general ones in Chapters 10 and 11, and found most of the series summations that are given in [7], and many of the transforma-

### ON A CONTINUED FRACTION OF RAMANUJAN

Ramanujan was a master of manipulatorics in the class of Euler himself. Thus it is appropriate that the continued fraction formulas of Ramanujan here are all derived by using the same approach as the one taken by Euler [8] for the trans-formation of the divergent series 1 mx + m(m + n)x2 m(m + n)(m +2n)x3 +

### Ramanujan-Fourier Series and Applications

We look at in nite series expansions for arithmetic functions, rst considered by Srinivasa Ramanujan in 1918. A basis for these expansions is investigated, for which several properties are proven. Examples of these in nite series are established using multiple techniques. Finally, we apply this theory to study the famous twin prime

### NONCOMMUTATIVE EXTENSIONS OF RAMANUJAN S SUMMATION

the base of the series. The bilateral 1 series in (2.2) reduces to a unilateral 1;2 134 series if one of the lower parameters, say < 1, equals (or more generally, an integral power of ). The basic hypergeometric 1;2K134 series terminates if one of the upper parameters, say 1, is of the form N 4ML, for a nonnegative integer If the basic

### RAMANUJAN S PLACE IN THE WORLD OF MATHEMATICS*

theory of partitions and q-hypergeometric series, the Ramanujan taxi cab equation, and sums involving the roots of unity. ===== 11. G. H. Hardy: Ramanujan s mentor Abstract: The British mathematician G. H. Hardy was the consummate Cambridge professor. Upon seeing Ramanujan s letter he concluded that Ramanujan was a true ge-nius and invited

### Lambert series and Ramanujan - ias

Lambert series on the development of the above areas. 2. Lambert series The series a~ (2.1) n=l 1 X n' was considered by Lambert [19-1 in connection with the convergence of power series. oo If the series ~a~ converges, then the Lambert series (2.1) converges for all values of 1 269

### University of Texas Rio Grande Valley ScholarWorks @ UTRGV

of x(˝) with [Q(x(˝)) : Q] 2 within the radius of convergence for zas a powers series in xand therefore provide a complete list of linear and quadratic Ramanujan-Sato series corresponding to x(˝). 2. Level 5 and 13 Series The product R(˝), de ned by (1.4), is the Rogers-Ramanujan continued fraction [19]. Together, R(˝) and S(˝), de ned by

### WEIGHTED DIVISOR SUMS AND BESSEL FUNCTION SERIES, IV

Possibly, Ramanujan had derived (1.2) with the intention of somehow using it with its extra parameter to attack the circle problem, but, of course, this is sheer speculation. Since double series appear on the right-hand sides of each identity, one can ask about different orders of summation. The authors [3] [6] have considered three different

### A Tiling Approach to Eight Identities of Rogers

numerous occasions; see, for example, Andrews [1] and Ramanujan s Collected Works [8]. As noted by Andrews [1], Rogers proved numerous series product identities in his three paper series, not just the two identities mentioned above. Many of these series have proven to be invaluable in the ﬁeld of q series.

### SOME OBSERVATIONS ON LAMBERT SERIES, VANISHING COEFFICIENTS

Two identities that mat be considered as extensions of two Identities of Ramanujan are also derived. It is also shown how applying similar ideas to certain other Lambert series gives rise to some rather curious q-series identities, such as, for any positive integer m, q;q;a;q a;bq d;dq b;aq bd;bdq a;q 1 b;q b;d;q d; a b;bq a; d;dq a;q 1 = mX1 r

### Ramanujan 125 - American Mathematical Society

Ramanujan 125 International Conference to Commemorate the 125th Anniversary of Ramanujan s Birth Ramanujan 125 November 5 7, 2012 University of Florida, Gainesville, Florida Krishnaswami Alladi Frank Garvan Ae Ja Yee Editors American Mathematical Society Providence, Rhode Island

### Ramanujan summation of the Ln(n) series

x=0. This value is considered as the Ramanujan sum of the divergent series. First example. To obtain the value of the divergent series [8]: 1+1+1+1 (9) First, we represent the sum of the function (y axis) as a function of the number of the element we are summing (x axis). This is represented in grey (the ladder type function).

### ZEROS OF GENERALIZED ROGERS-RAMANUJAN SERIES: ASYMPTOTIC AND

liptic integral associated with the zeros of generalized Rogers-Ramanujan series. Our calculations provide an eﬃcient algorithm for the computation of series expansions for zeros of generalized Rogers-Ramanujan series. 1. Introduction Among the multitude of series appearing in Ramanujan s notebooks [20, 21], one sum of particular

### Hypergeometric series and continued fractions

Heine's q-series in the Lost' Notebook [13] discovered in 1976 by G E Andrews among the Ramanujan papers in the Trinity College, Cambridge. Our object in this note is to show that many of the results of Ramanujan on continued fractions can be obtained as limiting cases of results of Gauss, Euler and

### Srinivasa Ramanujan: Going Strong at 125, Part I

partitions and q-hypergeometric series with which I am most familiar and provide just a sample of some very important recent results. Research relating to Ramanujan s work in many other areas has been very signiﬁcant, and some of these ideas will be described by others in the following articles. Ramanujan s mock theta functions are mysteri-

### Summation of Glaisher- and Ap´ery-like Series

which are commonly referred to as Ap´ery-like series, have been considered by van der Poorten [11] [13], Leschiner [7], Lehmer [6], Zucker [14], J. Borwein and Bradley [Search- ing Symbolically for Apery-like formulae for values of the Riemann Zeta function] and J.

### REVIEW OF THE MOVIE ON THE MATHEMATICAL GENIUS RAMANUJAN

Hardy. The asymptotic series for the partition function they jointly obtained [12] is a crowning achievement of their collaboration. The lm is based on a book by Robert Kanigel [13] under the same title - a book which gives a complete and accurate account of Ramanujan s life. Starting brie y with Ramanujan s life and initial discoveries in

### Ramanujan and hypergeometric and basic hypergeometric series

series, and hypergeometric functions in different disguises occur in many places throughout the two Notebooks. The left hand pages in the First Notebook were used to record facts that Ramanujan had not fitted into a general framework yet. Hardy [55] considered Chapters 12 and 13 of the First Notebook in some

### n=1 (1 + q)(l + q2) (1 + qf) n S(n)qn - JSTOR

reveals that most of the q-series considered either have coefficients which tend to infinity in absolute value or are bounded. In studying Ramanujan's Lost Notebook [4], [5], I have come across some strikingly simple-looking q-series where the behavior appears to be quite different from that described above.

### WHAT IS A q-SERIES?

One could claim that the theory of q-series began with certain famous theorems of L. Euler [36, Chapter 16] and C.F. Gauss [40]. We begin non-chronologically with Gauss, who is generally considered to be the founder of the theory of theta functions. In Ramanujan s notation, perhaps the three most important special cases of (1.2) are deﬁned by

### ON A MODULARITY CONJECTURE OF ANDREWS, DIXIT, SCHULTZ, AND

yields a Hecke-type (inde nite) theta series representation by expanding the denominators using geometric series. In [3], Andrews, Dixit, Schultz, and Yee considered an overpartition analog of p!(n) and spt (n). In particular, they de ned p!(n) to be the number of overpartitions of nsuch that all odd parts are

### Sequences in Partitions, Double q-Series and the Mock Theta

We present generalization of the double series considered by them. Two particular in nite families are identi ed. Beyond the main theorems, ap-plications are made to Rogers-Ramanujan identities and mock theta func-tions. 1 Introduction This paper is devoted to the partition-theoretic aspects of (1.1) H r;s(k;a;x;q) = X n;j 0 ( 1)jxaj+nq(aj+n)2