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