On pages 338 and 339 in his first notebook, Ramanujan defined the remarkable product of theta-functions $a_{m, n}$. Also he recorded eighteen explicit values depending on two parameters, namely, $m$, and $n$, where these are odd integers. All these values were established by Berndt et al. In this paper, we initiate to study explicit evaluations of $a_{m, n}$ for even values of $m$. We establish a new general theorems for the explicit evaluations of $a_{m, n}$ involving class invariant of Ramanujan. For this purpose, we establish several new P-Q mixed modular equations. Using these theorems, we calculate several new explicit values of $a_{m, n}$ for $n = 3, 5, 7, 13$ and even values of $m$.
Eigenvectors of the discrete Fourier transform can be expressed using Ramanujan theta functions. New theta function identities, Ramanujan theta function identities, and generating functions for the quadratic numbers are a consequence.
Two classes of finite trigonometric sums, each involving only $\sin$'s, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and modular equations.
We prove some Lambert series which were stated by Gosper without proof or reference. As an application, we shall evaluate integrals involving Ramanujan theta function $ψ(q)$. Furthermore, motivated by Ramanujan's identities for $qψ^4(q^2)$ and $\fr{ψ^3(q)}{ψ(q^3)}$, we shall evaluate the squares of $qψ^4(q^2)$ and $\fr{ψ^3(q)}{ψ(q^3)}$ in terms of Lambert series.
Recently, Garthwaite-Penniston have shown that the coefficients of Ramanujan's mock theta function $ω$ satisfy infinitely many congruences of Ramanujan-type. In this work we give the first explicit examples of congruences for Ramanujan's mock theta function $ω$ and another mock theta function $\mathcal{C}$.
We present completions of mock theta functions to harmonic weak Maass forms of weight $1/2$ and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight $1/2$ that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift we compute finite algebraic formulas for the coefficients of the appearing mock theta functions in terms of traces of singular moduli.
In this paper, we initiate a generous amount of new-found general theorems for explicit evaluations of product of the theta functions $b_{m, n}$ using Kronecker's limit formula and other various novel explicit evaluations that were introduced thereupon. Also, we have obtained a few novel values of Ramanujan's class invariant $g_n$ using new-found values of $b_{m, n}$. Eventually, we have shown that $b_{m, n}$ is the unit in certain algebraic number fields.
In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.
The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function $ω(q)$ (resp. $ν(-q)$). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function $ϕ(q)$. Congruences for the smallest parts functions associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler's pentagonal theorem are also obtained.
We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\pm 1/2,\pm 1/3,\pm 1/4,...
We find two involutions on partitions that lead to partition identities for Ramanujan's third order mock theta functions $ϕ(-q)$ and $ψ(-q)$. We also give an involution for Fine's partition identity on the mock theta function f(q). The two classical identities of Ramanujan on third order mock theta functions are consequences of these partition identities. Our combinatorial constructions also apply to Andrews' generalizations of Ramanujan's identities.
Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$ approaches an even order $2k$ root of unity $ζ$, \[\lim_{q\to ζ} \big(f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 - \cdots)\big) = O(1),\] and hinted at the existence of similar statements for his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for the implied constant in this radial limit of $f(q)$. Here, by…
We employ spectral methods of automorphic forms to establish a holomorphic projection operator for tensor products of vector-valued harmonic weak Maass forms and vector-valued modular forms. We apply this operator to discover simple recursions for Fourier series coefficients of Ramanujan's mock theta functions.
The modular transformations of Ramanujan's tenth order mock theta functions are computed, beginning from Choi's Hecke-type identites and using Zwegers' results on indefinite theta series. Explicit completions and shadows are found as an intermediate step. Our result for the modular transformations confirms numerical work by Gordon and McIntosh, and a recent conjecture by Cheng, Duncan, and Harvey.
Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\ldots,a_k,n\in\Bbb N$ let $N(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by $a_1x_1^2+a_2x_2^2+\cdots+a_kx_k^2$, and let $t(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by $a_1\frac{x_1(x_1-1)}2+a_2\frac{x_2(x_2-1)}2+\cdots+a_k\frac{x_k(x_k-1)}2$ $(x_1,\ldots,x_k\in\Bbb Z$). In this paper, by using Ramanujan's theta functions $φ(q)$ and $ψ(q)$ we reveal many relations between $t(a_1,a_2,\ldots,a_k;n)$ and $N(a_1,a_2,\ldots,a_k;8n+a_1+\cdots+a_k)$ f…
Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom, Ono, and Rhoades for mock theta functions. Here we see that the author's previous work on the dual nature of Appell--Lerch sums and partial theta functions and on constructing bilateral $q$-series with mixed mock modular behaviour is well-suited for such radial limits. We present five more radial limit results which follow from mixed mock modular bilateral $q$-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function…
In his lost notebook, Ramanujan listed 5 identities related to the false theta function $$f(q)=\sum_{n=0}^\infty (-1)^nq^{n(n+1)/2}.$$ A new combinatorial interpretation and proof of one of these identities is given. The methods of the proof allow for new multivariate generalizations of this identity. Additionally, the same technique can be used to obtained a combinatorial interpretation of another one of the identities.
In this article, we define functions analogous to Ramanujan's function $f(n)$ defined in his famous paper "Modular equations and approximations to $π$". We then use these new functions to study Ramanujan's series for $1/π$ associated with the classical, cubic and quartic bases.
This is a tutorial for using two new MAPLE packages, thetaids and ramarobinsids. The thetaids package is designed for proving generalized eta-product identities using the valence formula for modular functions. We show how this package can be used to find theta-function identities as well as prove them. As an application, we show how to find and prove Ramanujan's 40 identities for his so called Rogers-Ramanujan functions G(q) and H(q). In his thesis Robins found similar identities for higher level generalized eta-products. Our ramarobinsids package is for finding and proving identities for gene…
We provide new proofs to five of Ramanujan's intriguing identities on false theta functions without using the Rogers-Fine identity and Bailey transforms.
Inspired by the original definition of mock theta functions by Ramanujan, a number of authors have considered the question of explicitly determining their behavior at the cusps. Moreover, these examples have been connected to important objects such as quantum modular forms and ranks and cranks by Folsom, Ono, and Rhoades. Here we solve the general problem of understanding Ramanujan's definition explicitly for any weight $\frac12$ mock theta function, answering a question of Rhoades. Moreover, as a side product, our results give a large, explicit family of modular forms.
Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2$ $(x,y,z,w\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $φ(q)$ and $ψ(q)$ we present many formulas and conjectures on $t(a,b,c,d;n)$.
We demonstrate that quotients of septic theta functions appearing in S. Ramanujan's Notebooks and in F. Klein's work satisfy a new coupled system of nonlinear differential equations with interesting symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends a technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein's quartic relation induces new symmetric representations for low weight Eisenstein se…
This paper provides a survey of particular values of Ramanujan's theta function $φ(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-π\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to evaluate theta functions are given. Second, classical values are briefly discussed. Third, certain values due to Ramanujan and later authors are given. Fourth, the methods that are used to determine these values are described. Lastly, an incomplete evaluation found in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.
We present q-series proofs of four identities involving sixth order mock theta functions from Ramanujan's lost notebook. We also show how Ramanujan's identities can be used to give a quick proof of four sixth order identities of Berndt and Chan.
We show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan Delta-function $Δ(τ)=η^{24}(τ)$, $η^{12}(τ)$ and $η^{8}(τ)$ in terms of Cohen's numbers $H(3,N)$ and $H(5,N)$. We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the J…
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving his theta--functions $φ(q)$ and $ψ(-q)$ with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta--function and a continued fraction of order 12 are explicitly evaluated.
In this paper we consider the first four of the eight identities between the tenth order mock theta functions, found in Ramanujan's lost notebook. These were originally proved by Choi. Here we give an alternative (much shorter) proof.
We give simple proofs of Hecke-Rogers indefinite binary theta series identities for the two Ramanujan fifth order mock theta functions $χ_0(q)$ and $χ_1(q)$ and all three of Ramanujan's seventh order mock theta functions. We find that the coefficients of the three mock theta functions of order 7 are surprisingly related.
We evaluate regularized theta lifts for Lorentzian lattices in three different ways. In particular, we obtain formulas for their values at special points involving coefficients of mock theta functions. By comparing the different evaluations, we derive recurrences for the coefficients of mock theta functions, such as Hurwitz class numbers, Andrews' spt-function, and Ramanujan's mock theta functions.
We analyze the mock modular behavior of $\bar{P}_ω(q)$, a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta functions, one of which is $ω(q)$. We find that the modular completion of $\bar{P}_ω(q)$ is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions. We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies…
In his second notebook, Ramanujan recorded total of 23 P-Q modular equations involving theta-functions $f(-q)$, $φ(q)$ and $ψ(q)$. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving $f(-q)$. As a consequence, values of certain quotients of theta-function are evaluated.
Unary theta functions have played a significant role in the theory of holomorphic modular forms and modular $L$-functions. A partial theta functions is defined analogously, but the sum is over part of the integer lattice. Such sums fail to have modular properties. We give a construction which unifies these partial theta functions with the mock theta function introduced by Ramanujan. The modularity of Ramanujan's mock theta functions has only recently been understood by the work of Sander Zwegers.
In this Ph.D. thesis, written under the direction of D.B. Zagier and R.W. Bruggeman, we study the mock theta functions, that were introduced by Ramanujan. We show how they can be interpreted in the theory of (real-analytic) modular forms. In Chapter 1 we give results for Lerch sums (also called Appell functions, or generalized Lambert series). In Chapter 2 we consider indefinite theta functions of type (r-1,1). Chapter 3 deals with Fourier coefficients of meromorphic Jacobi forms. In Chapter 4 we use the results from Chapter 2 to give explicit results for 8 of the 10 fifth order mock theta fun…
Let $\Bbb Z$ be the set of integers. For positive integers $a,b,c$ and $n$ let $N(a,b,c;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2$, and let $t(a,b,c;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2$ $(x,y,z\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $φ(q)$ and $ψ(q)$ we reveal the relation between $t(2,3,3;n)$ and $N(1,3,3;n+1)$, and the relation between $t(1,1,6;n)$ and $N(1,1,3;n+1)$. We also obtain formulas for $t(a,3a,4b;n),$ $t(a,7a,4b;n),t(3a,5a,4b;n)$ and $t(a,15a,4b;n)$ under certain congruence conditions, where $a$…
The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's $_1ψ_1$ summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanuja…
A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $Ξ$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives as by-products, transformation formulas of the form $F(z,α)=F(iz,β)$, where $αβ=1$. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulas of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization o…
We study the divisibility properties of the partition function associated with the eighth order mock theta function $V_0(q)$, introduced by Gordon and McIntosh. We obtain congruences modulo powers of 2 for certain coefficients of the partition function, akin to Ramanujan's partition congruences. Further, we also present several infinite families of congruences molulo 13, 25 and 27.
Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $ω(q)$ and $ν(q)$. In this paper, we find several new exact generating functions for those partition functions as well as the associated smallest parts functions and deduce several new congruences modulo powers of 5.
