Finally, we describe the relationship between k run overpartitions and certain sequences of random events, and use probabilistic estimates in order to determine the asymptotic growth behavior of the number of k. We give relations between partial mock theta functions of andrews and partal mock theta functons of brngmann et al in the limit we have the 0, mock theta functions expressed as a series in partial mock theta function s. Ramanujan began investigating mock theta functions in 1919, for which he left behind no published work. This book provides a systematic development of ramanujans results and extends them to a general theory. Ramanujan did not rigorously define mock theta functions and their orders. For example, ramanujans claims about their analytic properties remain open. Sander zwegers 2001, 2002 discovered that adding certain. This book contains the essential features of the theory of harmonic maass forms and mock modular forms, together with a wide variety of applications to algebraic number theory, combinatorics.
In particular, the jacobi triple product takes on a particularly elegant form when written in terms of. In relation to the results in the present paper, we recall two areas of investigation in the subject of mock theta functions. Watsons 1936 presidential address \ramanujans discovery of the mock theta functions makes it obvious that his skill and ingenuity did not desert him at the oncoming of his untimely end. False, partial, and mock jacobi theta functions as qhypergeometric series kathrin bringmann, amanda folsom, and robert c. It should be emphasized that the material on mock theta functions is perhaps ramanujans deepest work. Pdf we consider the secondorder mock theta function. Radial limits of mock theta functions connecting repositories. We find that the coefficients of the three mock theta functions of order 7 are surprisingly related. Of course, ramanujan would not have used this terminology.
Trigonometric limits california state university, northridge. In chapter 2 we consider indefinite theta functions of type r1,1. Mock theta functions and quantum modular forms 5 qhypergeometric series u. A report on complete mock theta functions of order eight pathak, maheshwar and srivastava, pankaj, tbilisi mathematical journal, 20. It is the purpose of this paper to prove the third and fourth of ramanujans tenth order mock theta function identities. The delta functions in ud give the derivative of the square wave. The basic bilateral hypergeometric series and the mock theta. Not to be confused with the mock theta functions discovered by ramanujan. We show how they can be interpreted in the theory of realanalytic modular forms. Since trigonometric functions are manyone over their domains, we restrict their domains and codomains in order to make them oneone and onto and then find their inverse. The lost notebook contains considerable material on mock theta functions and so undoubtedly emanates from the last year of ramanujans life. To complete the proof it suffices to prove that is not a mock theta function according to the modern definition. The modern view of mock theta functions is based on the work of zwegers 29,30, who showed that the mock theta functions are holomorphic parts of certain harmonic weak maass forms.
On basic hypergeometric series, mock theta functions, and partitions i, the quarterly journal of mathematics, volume 17, issue 1, 1. Introduction in his last letter to hardy, ramanujan gave a list of seventeen functions fq, where qis a complex number and jqj mock theta functions. Ramanujan sent to hardy january 12 1920 defined 17 jacobilike functions for complex called mock theta functions since then. The f q mock theta function conjecture and partition ranks.
Welcome to the homepage of 100 years of mock theta functions. Trigonometry formulas for functions, ratios and identities. The theory of modular forms has important applications to many areas of mathematics, e. A variety of applications appear in the fields of hypergeometric functions, number theory, mordell integrals, probability theory, and mathematical physics, where they are used to determine critical dimensions in some string theories. Pdf mock theta functions, weak maass forms, and applications. The six trigonometric functions are sine, cosine, secant, cosecant, tangent and cotangent. Gispen, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op woensdag 30 oktober 2002 des ochtends te. Mock theta function identities deriving from bilateral. New directions in partitions, modular forms, and mock. The student will recall that if f is a onetoone function with domain a and. Chapter 3 deals with fourier coefficients of meromorphic jacobi forms. See 1 for a wideranging survey and references on such problems i. Relations among partial mock theta functions of andrews and partial mock theta functions of bringmann et al. Introduction on pages 2 and 17 in his lost notebook 23, ramanujan recorded four identities involving the rank generating function.
Ramanujan introduced the mock theta functions in his last letter to g. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Inverse trigonometric functions trigonometric equations. In chapter 4 we use the results from chapter 2 to give explicit results for 8 of the 10 fifth order mock theta functions and all 3 seventh order functions, that were originally defined by ramanujan. Bilateral series and ramanujan radial limits of mock false. Pdf a mock theta function of second order researchgate. This is the motivation for the study of the modularity of fourier coe. Two new mock theta functions of the sixth order are defined. Hecke modular form expansions for eighth order mock theta functions srivastava, bhaskar, tokyo journal of mathematics, 2005. Mock theta functions ranks and maass forms springerlink. Download pdf theta functions and knots ebook full get.
Bruggeman, we study the mock theta functions, that were introduced by ramanujan. On the mathieu mock theta function miezaki, tsuyoshi, proceedings of the japan academy. Mar 20, 2014 not only does our formula prove classical hecke. Mock theta function identities deriving from bilateral basic. People with an interest in theta functions or knot theory can learn how the two are related. Watson presidential address delivered at the meeting of 14 november, 1935. This is provided by the universal mock theta function g2 of gordon and mcintosh. Threemanifold quantum invariants and mock theta functions. Mock theta functions wolfram demonstrations project. Finally, we describe the relationship between krun overpartitions and certain sequences of random events, and use probabilistic estimates in order to determine the asymptotic growth behavior of the number of krun.
That sawtooth ramp rr is the integral of the square wave. From the eulerian forms of mock theta functions, it is difficult to observe any significant transformation properties. In mathematics, a mock modular form is the holomorphic part of a harmonic weak maass form, and a mock theta function is essentially a mock modular form of weight 12. On basic hypergeometric series, mock theta functions, and partitions i, the quarterly journal of mathematics, volume 17, issue 1, 1 january 19. In his letter and lost notebook, he presented mock theta functions and their identities. For each mock theta function, we correspondingly establish some parameterized identities.
Ramanujan discovered functions he called mock theta functions which over eighty years later were recognized as pieces of harmonic maass forms. The mock theta functions were introduced to the world by ramanujan in his last letter to g. In the special case of 1run overpartitions we further relate the generating function to one of ramanujans mock theta functions. May 07, 20 this grouptheoretic structure was discovered by zwegers. They linked ramanujans functions to partitions of a given integerthe ways of writing an integer as a sum of smaller integers. Further identities will be proved in subsequent papers. Each of these identities gives us a representation for the mock theta function and produces new interesting identities. Ramanujan did not explain precisely what he meant by a mock theta function, and ramanujans james mc laughlin. Andrews, atul dixit, daniel schultz, and ae ja yee abstract.
Dec 17, 2019 mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by ramanujan nearly a century ago. The mock theta functions 2 watson 1937 proceedings of. Since then, mock theta functions have attracted the attention of many mathematicians. A mock theta function of second order article pdf available in international journal of mathematics and mathematical sciences 20092 january. Ramanujan sent to hardy january 12, 1920 defined 17 jacobilike functions fq for complex q lt 1, called mock theta functions since then.
There are basically 6 laws used for finding the elements in trigonometry. Theta functions were studied extensively by ramanujan. Two identities on the mock theta function v0q sciencedirect. More mock theta functions were discovered afterward, including some of the 10th order 2, 3. A survey of classical mock theta functions springerlink. False, partial, and mock jacobi theta functions as q. Combinatorial identities for tenth order mock theta functions. On the mathieu mock theta function miezaki, tsuyoshi, proceedings of the japan academy, series a, mathematical sciences, 2012. Mock theta functions and appelllerch sums journal of. Previously, the author proved six of the eight tenth order mock theta function identities. Request pdf four identities for third order mock theta functions in 2005, using a famous lemma of atkin and swinnertondyer some properties of partitions, proc. The first examples of mock theta functions were described by srinivasa ramanujan in his last 1920 letter to g.
We first recall the definition it is wellknown that g2 is a mock jacobi form, which implies that if. Building on the work by andrews and garvan, dean hickerson proved that five identities in each of the two families are equivalent, at the same time confirming that these mock theta functions are truly mock theta. The authors treatment of the subject is comprehensive, providing a detailed study of theta functions. In this paper, we provide the combinatorial interpretations of many mock theta functions and some generalizations using frobenius partitions with attached weights. New zealand journal of mathematics volume 36 2007, 287294. Ramanujan emphasized that a mock theta function should not have the same singularities. As we said then, the extent to which we can express modular forms in terms of. Jan 01, 20 pdf ramanujans last letter to hardy concerns the asymptotic properties of modular forms and his mock theta functions. New fifth and seventh order mock theta function identities. As a result of his thesis, it is known that all of ramanujans examples are essentially the holomorphic part of weight weak harmonic maass forms whose nonholomorphic parts are period integrals of weight unary theta functions. Pdf ramanujans last letter to hardy concerns the asymptotic properties of modular forms and his mock theta functions.
Apr 20, 2019 in this paper, the open problem posed by sareen and rana proc. These famous series play a prominent role in the study of integer partition congruences for example, see 5,8,16,23,33. We thus see the same points as above, but in a more complicated setting. Zwegers 2001 phd thesis was a breakthrough in the study of the mock theta functions. Developments in mathematics diophantine approximation. One of our exercises emphasizes their relation with eld theory and galois theory. New identities for ramanujans cubic continued fraction bairy, k. We establish the existence of infinitely many linear congruences of the form. We also study the specializations a n 1 3 q k, q and relate them to the fifth order mocktheta functions of. For a certain class of seifert threemanifolds, we describe a conjecture on the mock. Freeman dyson, 1987 in this lecture i will describe the solution to this challenge and give an indication of some of the open problems which have now been solved as a result.
Tenth order mock theta functions in ramanujans lost. It was some time before researchers understood the modularity of mock theta functions. We establish our results by providing the interpretations of unsigned versions of mock theta functions leading to the interpretations of the corresponding mock theta functions. Andrews in a visit to trinity college discovered some notebooks of ramanujans and called one of them the.
Theta functions and knots can be read in two perspectives. Those interested in chernocosimons theory find here an introduction using the simplest case, that of abelian chernocosimons theory. In mathematics, a mock modular form is the holomorphic part of a harmonic weak maass form, and a mock theta function is essentially a mock modular form of. Trigonometric functions laws for evaluating limits typeset by foiltex 2. It is the purpose of this paper to prove the fifth and sixth identities of ramanujans tenth order mock theta functions. The expression of each mock theta function relies on the hypergeometric series. The effective nature of our results is based on the. To properly understand the context of ramanujans last work, we must. Introduction the goal of this article is to provide an overview on mock theta functions and their connection to weak maass forms. On mock theta functions and weightattached frobenius. Mathematicians are probably several decades away from a complete understanding of those functions. The theory of modular forms has important applications to many.
In this proceeding we highlight a new area where mock modular forms start to play an important role, namely the study of threemanifold invariants. Representations of mock theta functions sciencedirect. This research underscores the fact that the theory of partitions has historically served as a testing ground for. By using a rightangled triangle as a reference, the trigonometric functions or identities are derived. Tenth order mock theta functions in ramanujans lost notebook ii. New directions in partitions, modular forms, and mock modular forms vanderbilt university this conference has been postponed and will be rescheduled. Here, we interpret two tenth order mock theta functions combinatorially in terms of lattice paths. Generating functions play a central role throughout number theory. The function v 0 q is an eighthorder mock theta function first studied by gordon and mcintosh in 14.
Motivated by the works of liu, we provide a unified approach to find appelllerch series and hecketype series representations for. Then we extend enumeration of one of these with benderknuth matrices. Demazure flags, chebyshev polynomials, partial and mock theta. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. This topic, which he began right at the end of his life, has been enigmatic all through the 20th century, andrews says. Mock theta functions and quantum modular forms cambridge core. This talk should be accessible to graduate students. Surprisingly, very little is known about mock theta functions in general. New zealand journal of mathematics volume 36 2007, 287294 mock theta functions and theta functions bhaskar srivastava received august 2004 1. The subject of mock theta functions has its roots in classical theta functions, which physicists use in their study. Ramanujans mock theta functions and their applications d. Curious qseries identities since rogers 32 introduced the false theta functions, they have played a curious role in the theory of partitions see for instance 1, 2, 12.
The properties of modular forms are used for the proofs of theta function identities. On the theorems of watson and dragonette for ramanujans mock theta functions, amer. In mathematics, particularly qanalog theory, the ramanujan theta function generalizes the form of the jacobi theta functions, while capturing their general properties. In chapter 1 we give results for lerch sums also called appell functions, or generalized lambert series. The proof of these and others for mock theta functions of seventh order were given by hickerson 21, 22.
Watson studied the relations between ramanujans fifth order mock theta functions and the function introduced by m. Ricci, bilateral series and ramanujans radial limits, proc. Ramanujans mock theta functions, dysons rank generating functions, borcherds products, and central values and. The first two of ramanujans tenth order mock theta function identities were proved by the author cy.
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