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In this paper, we state some q-analogs of the famous Ramanujan Master Theorem. As an application, we compute some values of Jackson’s q-integrals that involve q-special functions.

Hafner and Stopple proved a conjecture of Zagier on the asymptotic expansion of a Lambert series involving Ramanujan’s tau function with the main term involving the nontrivial zeros of the Riemann zeta function. Recently, Chakraborty et. al. have extended this result to any cusp form over the full modular group and also over any congruence subgroup. The aim here is to study the asymptotic behaviour of a similar Lambert series involving the coefficients of Maass cusp forms over the full modular group.

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono, without need for Hecke operators. In this paper we give a method for generalizing Lachterman, Schayer, and Younger's proof to include Ramanujan congruences for multipartition functions $p_k(n)$, and Ramanujan congruences for $p(n)$ modulo certain prime powers., Comment: 16 pages

Hafner and Stopple proved a conjecture of Zagier relating to the asymptotic behaviour of the inverse Mellin transform of the symmetric square L-function associated with the Ramanujan tau function. In this paper, we prove a similar result for any cusp form over the full modular group.

We evaluate in closed form certain classes of infinite integrals containing trigonometric and hyperbolic trigonometric functions in their integrands. Although not apparent, the integrals are related to Jacobian elliptic functions, in particular, to theorems found in Ramanujan’s notebooks. One of our evaluations answers a question first posed by M. E. H. Ismail.

We establish a vast generalization of an observation made by Marvin Knopp half a century ago concerning the nonvanishing of Ramanujan’s tau-function.

The main purpose of this paper is to introduce new sums that are analogous to Dedekind sums. Using analysis and properties of Dirichlet \(L\)-functions, we study mean values for these new sums, and give a sharper mean value formula for it.

By using two known transformation formulas for basic hypergeometric series, we establish a direct extension of Bailey’s \(_6\psi _6\)-series identity. Subsequently, it and Milne’s identity are employed to drive multi-variable generalizations of Ramanujan’s reciprocity formula. Then we utilize also Milne’s identity to deduce a multi-variable generalization of the Askey–Wilson integral.

We show connection formulae between the origin and infinity for local solutions of the q-difference equation satisfied by the Ramanujan entire function. These solutions are given by the Ramanujan entire function, the q-Airy function, and the divergent basic hypergeometric series 2φ0(0,0;−;q,x). We use two different q-Borel–Laplace resummation methods to obtain our connection formulae.

We present a new proof of the monotonicity of the correction term θn in Ramanujan’s refinement of Stirling’s formula. Moreover we prove that θn is concave.

The average value of a certain normalization of Ramanujan sums is determined in terms of Bernoulli numbers and odd values of the Riemann zeta function. The distribution of values and limiting behavior of such a normalization are then studied along subsets of Beurling type integers with positive density and sequences of moduli with constraints on the number of distinct prime factors.

Pages 262–265 in Ramanujan’s Lost Notebook and Other Unpublished Papers are devoted to a draft of a manuscript on an unusual problem in diophantine approximation. Ramanujan finds the maximum value of a polynomial when the variable takes rational values, and he observes the maxima’s periodicity. Because definitions and theorems in the manuscript are imprecise, the authors have attempted to interpret Ramanujan’s ideas and provide rigorous proofs.

We prove a recent conjecture of B. Berndt and B. Kim regarding the positivity of the coefficients in the asymptotic expansion of a class of partial theta functions. This generalizes results found in Ramanujan’s second notebook and recent work of Galway and Stanley.

In this note, we give a generalization of Hirschhorn’s formulas on the 4-dissections of Ramanujan’s continued fraction R(q) and R−1(q) which were conjectured by Hirschhorn and proved by Lewis and Liu.

A central limit theorem is established for the absolute value of the modular Fourier-coefficient function defined by Ramanujan, and for that of the error term in the formula counting representations of integers as sums of twenty-four squares, in which the function appears.

We analyze a two-parameter q-series identity in Ramanujan’s Lost Notebook that generalizes the product part of the fundamental one-parameter Lebesgue identity. From reformulations of this two-parameter identity, we deduce new partition theorems including variants of the Gauss triangular number identity and Euler’s pentagonal number theorem. We discuss connections with a partial theta identity of Ramanujan and with several classical results such as those of Sylvester and Gollnitz–Gordon.

In this paper, we give a completely elementary proof of Ramanujan’s circular summation formula of theta functions and its generalizations given by S.H. Chan and Z.-G. Liu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given.

S. Ramanujan introduced a technique, known as Ramanujan’s Master Theorem, which provides an explicit expression for the Mellin transform of a function in terms of the analytic continuation of its Taylor coefficients. The history and proof of this result are reviewed, and a variety of applications is presented. Finally, a multi-dimensional extension of Ramanujan’s Master Theorem is discussed.

We give a natural derivation of a formula of Ramanujan, described by B.C. Berndt as “enigmatic”, for the harmonic series.

An integer a is said to be regular (modr) if there exists an integer x such that a 2 x≡a (mod r). In this paper we introduce an analogue of Ramanujan’s sum with respect to regular integers (modr) and show that this analogue possesses properties similar to those of the usual Ramanujan’s sum.

The aim of this paper is to improve some approximation formulas of Ramanujan type discussed by E.A. Karatsuba [J. Comput. Appl. Math. 135 (2001), 225–240].

The aim of this paper is to extend and refine an approximation formula of the gamma function by Ramanujan.

In this paper, we give an easy and short proof of the well-known Askey–Wilson integral by means of the five-variable Ramanujan’s reciprocity theorem.

A partial theta series identity from Ramanujan’s lost notebook has a connection with some parity problems in partitions studied by Andrews in Ramanujan J., to appear where 15 open problems are listed. In this paper, the partial theta series identity of Ramanujan is revisited and answers to Questions 9 and 10 of Andrews are provided.

We use the Algorithm Z on partitions due to Zeilberger, in a variant form, to give a combinatorial proof of Ramanujan’s 1ψ1 summation formula.

This paper considers a variety of parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordon. We begin by restricting the partitions in the Rogers-Ramanujan-Gordon identities to those wherein even parts appear an even number of times. We then take up questions involving sequences of alternating parity in the parts of partitions. This latter study leads to: (1) a bi-basic q-binomial theorem and q-binomial series, (2) a new interpretation of the Rogers-Ramanujan identities, and (3) a new natural interpretation of the fifth-order mock theta functions f0(q) along with a new proof of the Hecke-type series representation.

Recently, Bruinier and Ono investigated the arithmetic of the coefficients of Ramanujan’s mock theta function ω(q). In Ramanujan J. (submitted) they obtained congruences with respect to the modulus 512. Here we show that ω(q) modulo 5 is dictated by an elliptic curve.

We study the divisibility properties of the coefficients c(n) defined by $$\prod_{n=1}^\infty\frac{1}{(1-q^n)^2(1-q^{3n})^2}=\sum _{n=0}^\infty c(n)q^n.$$ An analogue of Ramanujan’s partition congruences is obtained for certain coefficients c(n) modulo powers of 2. Furthermore, an analogue of the identity that Hardy regarded as Ramanujan’s most beautiful is proved.

In this paper, we give a new proof of two identities involving Ramanujan’s cubic continued fraction. These identities are the key ingredients to an analog of Ramanujan’s “Most Beautiful Identity” discovered recently.

We will interpret a partial theta identity in Ramanujan’s Lost Notebook as a weighted partition theorem involving partitions into distinct parts with smallest part odd. A special case of this yields a new result on the parity of the number of parts in such partitions, comparable to Euler’s pentagonal numbers theorem. We will provide simple and novel proofs of the weighted partition theorem and the special case. Our proof leads to a companion to Ramanujan’s partial theta identity which we will explain combinatorially.

In both his second and lost notebooks, Ramanujan introduced a function, related to the Rogers–Ramanujan continued fraction and its quadratic transformation, and listed several of its properties. We extend these results and develop a systematic theory.

A survey of Ramanujan’s work on cranks in his lost notebook is given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujan worked was cranks—not mock theta functions.

We give a new proof of Ramanujan’s modular identity relating R(q) with R(q5), where R(q) is the famous Rogers–Ramanujan continued fraction. Our formulation is stronger than those of preceding authors; in particular, we give for the first time identities for the expressions appearing in the numerator and the denominator of Ramanujan’s identity. A related identity for R(q) that has partition-theoretic connections is also proved.

Two modular identities of Gordon, McIntosh, and Robins are shown to be connected to the Rogers–Ramanujan continued fraction R(q), and in particular to Ramanujan’s parameter k:=R(q)R2(q2). Using this connection, we give new modular relations for R(q), and offer new and uniform proofs of several results of Ramanujan. In particular, we give a new proof of a famous and fundamental modular identity satisfied by the Rogers–Ramanujan continued fraction. We furthermore show that many analogous results hold for Ramanujan’s parameters μ:=R(q)R(q4) and ν:=R2(q1/2)R(q)/R(q2). New proofs are offered for modular relations connecting R(q) to R(−q), R(q2), and R(q4), and new relations connecting R(q) at these arguments are offered. Eleven identities for the Rogers–Ramanujan functions are proved, including four new identities.

Using results that were well-known to Ramanujan, we give proofs of some results for Eisenstein series in the lost notebook. Our proofs have the additional advantage that it is not necessary to know the results in advance; that is, the proofs are derivations as opposed to verifications.

On pages 338 and 339 in his first notebook (Notebooks (2 volumes), [1957]), Ramanujan records eighteen values for a certain product of theta-functions. All these have been proved by B.C. Berndt, H.H. Chan and L.-C. Zhang (Proc. Edinburg Math. Soc. 40:583–612, [1997]). The main purpose of this paper is to establish some new general theorems for the explicit evaluations of Ramanujan’s product of theta-functions and some new explicit evaluations there from.

In this paper, we find new proofs of modular relations for the Gollnitz-Gordon functions established earlier by S.-S. Huang and S.-L. Chen. We use Schroter’s formulas and some simple theta-function identities of Ramanujan to establish the relations. We also find some new modular relations of the same nature.

S. Ramanujan gave fourteen families of series in his Second Notebook in Chap. 17, Entries 13–17. In each case he gave only the first few examples, giving us the motivation to find and prove a general formula for each family of series. The aim of this paper is to develop a powerful tool (four versatile functions f 0,f 1,f 2, and f 3) to collect all of Ramanujan’s examples together.

The Dedekind–Holder theorem states that Ramanujan’s sum and the Von Sterneck function are identical. In this article, we extend the Dedekind–Holder theorem by generalizing both Ramanujan’s sum and the Von Sterneck function and showing that the two generalizations are identical.

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s 1ψ1 summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares.

This paper considers a generalization of an integral introduced by S. Ramanujan in his third notebook. Ramanujan’s integral is itself a version of the dilogarithm, $$Li_2(z) =-\int_{0}^{z}\frac{\log(1-x)}{x} \,\,dx.$$ We prove various functional equations and properties of the generalized integral.

We give short elementary proofs of a formula of Ramanujan as interpreted by Bradley, and a companion formula originally proved by W. Chu. We also give an elementary proof of a generalization of an identity originally proved using modular functions and used to study a generating function for the number of partitions with specified crank.

We show that the Ramanujan tau function τ(n) can be computed by a randomized algorithm that runs in time \(n^{\frac{1}{2}+\varepsilon}\) for every O(\(n^{\frac{3}{4}+\varepsilon}\)) assuming the Generalized Riemann Hypothesis. The same method also yields a deterministic algorithm that runs in time O(\(n^{\frac{3}{4}+\varepsilon}\)) (without any assumptions) for every e > 0 to compute τ(n). Previous algorithms to compute τ(n) require Ω(n) time.

Let f(a, b) denote Ramanujan's symmetric theta function. In his Lost Notebook, Ramanujan claimed that the “circular” summation of n-th powers of f satisfies a factorization of the form f(a, b)F(ab). He listed elegant identities for n = 2, 3, 4, 5 and 7. We present alternative proofs of his claims.

Let k be a positive number and t k(n) denote the number of representations of n as a sum of k triangular numbers. In this paper, we will calculate t 2k (n) in the spirit of Ramanujan. We first use the complex theory of elliptic functions to prove a theta function identity. Then from this identity we derive two Lambert series identities, one of them is a well-known identity of Ramanujan. Using a variant form of Ramanujan's identity, we study two classes of Lambert series and derive some theta function identities related to these Lambert series . We calculate t 12(n), t 16(n), t 20(n), t 24(n), and t 28(n) using these Lambert series identities. We also re-derive a recent result of H. H. Chan and K. S. Chua [6] about t 32(n). In addition, we derive some identities involving the Ramanujan function τ(n), the divisor function σ11(n), and t 24(n). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.

In this paper we give analogues of the Ramanujan functions and nonlinear differential equations for them. Investigating a modular structure of solutions for nonlinear differential systems, we deduce new identities between the Ramanujan and hypergeometric functions. Another result of this paper is a solution of transcendence problems concerning nonlinear systems.

In his lost notebook, Ramanujan stated without proofs several beautifulidentities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.

We prove three modular equations of Ramanujan using theta-function identities. Proofs via methods known to Ramanujan were not available hitherto. One had previously been proved by classical methods, and two had been proved using the theory of modular forms.

In his lost notebook, Ramanujan recorded several modular equations of degree 5 related to the Rogers-Ramanujan continued fraction R(q). We prove several of these identities and give factorizations of some of them in this paper.

We use the Ramanujan operator and some modular relations of degree 5 to give new proofs of two identities of Ramanujan.