Ramanujan's Function Research Articles

Plancherel–Rotach asymptotics for some q-orthogonal polynomials with complex scalings

In this work we study the Plancherel–Rotach type asymptotics for Stieltjes–Wigert, q-Laguerre and Ismail–Masson orthogonal polynomials with complex scalings. The main terms of the asymptotics for Stieltjes–Wigert and q-Laguerre polynomials (Ismail–Masson polynomials) contain Ramanujan function A q ( z ) for scaling parameters above the vertical line R ( s ) = 2 ( R ( w ) = 1 2 ); the main terms of the asymptotics involve theta function for scaling parameters in the vertical strip 0 < R ( s ) < 2 ( 0 < R ( w ) < 1 2 ). When scaling parameters in the vertical strips, the number theoretical properties of scaling parameters completely determine the orders of the error terms. These asymptotic formulas may provide some insights to new random matrix models and also add a new link between special functions and number theory.

Plancherel–Rotach asymptotics for certain basic hypergeometric series

In this work we study the chaotic and periodic asymptotics for the confluent basic hypergeometric series. For a fixed q ∈ ( 0 , 1 ) , the asymptotics for Euler's q-exponential, q-Gamma function Γ q ( x ) , q-Airy function of K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Ramanujan function ( q-Airy function), Jackson's q-Bessel function of second kind, Ismail–Masson orthogonal polynomials ( q −1 -Hermite polynomials), Stieltjes–Wigert polynomials, q-Laguerre polynomials could be derived as special cases.

ASYMPTOTICS OF ZEROS OF CERTAIN ENTIRE FUNCTIONS

We derive representations for some entire q-functions and use it to derive asymptotics and closed form expressions for large zeros of a class of entire functions including the Ramanujan function, and q-Bessel functions.

Remarks on Ramanujan function [formula omitted

In this short notes we will derive an inequality for scaled q −1 -Hermite orthogonal polynomials of Ismail and Masson, an inequality for scaled Stieltjes–Wigert, two inequalities for Ramanujan function and two definite integrals for Ramanujan function.

Modular relations for the nonic analogues of the Rogers–Ramanujan functions with applications to partitions

We define the nonic Rogers–Ramanujan-type functions D ( q ) , E ( q ) and F ( q ) and establish several modular relations involving these functions, which are analogous to Ramanujan's well known forty identities for the Rogers–Ramanujan functions. We also extract partition theoretic results from some of these relations.

COMMON SOURCE OF NUMEROUS THETA FUNCTION IDENTITIES

Abstract.Motivated by the recent work due to Warnaar (2005), two new and elementary proofs are presented for a very useful q-difference equation on eight shifted factorials of infinite order. As the common source of theta function identities, this q-difference equation is systematically explored to review old and establish new identities on Ramanujan's partition functions. Most of the identities obtained can be interpreted in terms of theorems on classical partitions.

P-adic Measures of Algebraic Independence for the Values of Ramanujan Functions and Klein Modular Functions

In this paper, we give the p–adic measures of algebraic independence for the values of Ramanujan functions and Klein modular functions at algebraic points.

Hecke operators on weighted Dedekind symbols

Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind symbols. We prove that these newly introduced operators are compatible with Hecke operators on the space of modular forms. As an application, we present formulae to give Fourier coefficients of Hecke eigenforms. In particular we give explicit formulae for generalized Ramanujan's tau functions.

Arithmetic properties of the Ramanujan function

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisorP (τ(n)) and the number of distinct prime divisors ω (τ (n)) of τ(n) for various sequences ofn. In particular, we show thatP(τ(n)) ≥ (logn)33/31+o(1) for infinitely many n, and $$P(\tau )(p)\tau (p^2 )\tau (p^3 )) > (1 + o(1))\frac{{\log \log p\log \log \log p}}{{\log \log \log \log p}}$$ for every primep with τ(ρ) ≠ 0.

New identities for the Rogers–Ramanujan functions

where (a; q)∞ := lim n→∞ (a; q)n, |q| < 1. At the end of his brief communication [9], [10, p. 231] announcing his proofs of the Rogers–Ramanujan identities (1.2), Ramanujan remarks, “I have now found an algebraic relation between G(q) and H(q), viz.: H(q) {G(q)} − qG(q) {H(q)} = 1 + 11q {G(q)H(q)} . (1.3) Another noteworthy formula is H(q)G(q)− qG(q)H(q) = 1. (1.4) Each of these formulae is the simplest of a large class.” Ramanujan did not indicate how he had proved the identities. They are two identities from a set of forty identities for G(q) and H(q) that he never published. In the years since then, most of the 40 identities were established in a series of papers by L. J. Rogers [13] in 1919, G. N. Watson [14] in 1933, D. Bressoud [6], [7] in 1977, and A. J. F. Biagioli [3] in 1989. It should be remarked that the complete list of identities was not brought to the mathematical public until 1975 when B. J. Birch [4] found them in the Oxford University Library. Although all the identities can be proved using the theory of modular forms, the method employed by Biagioli, it is more instructive to

The Hypergeometric Equation and Ramanujan Functions

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.

An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers

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.

Septic analogues of the Rogers–Ramanujan functions

Septic analogues of the Rogers–Ramanujan functions

On Ramanujan's cubic transformation formula for 2 F 1(1/3, 2/3; 1; z)

The main aim of this paper is to provide two new proofs of Ramanujan's cubic transformation formula for 2F1(1/3, 2/3; 1; z) (see (1·8) below). For our first proof, we have to develop Ramanujan's elliptic functions in the theory of signature 3 using a different approach from that given in a recent paper by Berndt, Bhargava and Garvan. For our second proof, we use two of Goursat's formulas.

On Modular Relations for the Göllnitz–Gordon Functions with Applications to Partitions

In a manuscript of Ramanujan, published with his Lost Notebook [21, pp. 236–237], there are forty identities involving the Rogers–Ramanujan functions. According to G. N. Watson, the beauty of these identities are comparable to that of the Rogers– Ramanujan identities. In the paper, we establish modular relations involving the Göllnitz–Gordon functions which are analogous to Ramanujan's forty identities. Furthermore, we extract interesting partition results from some of the modular relations.

Ramanujan's q-hypergeometric functions in lattice statistics problems

Mathematical techniques leading to exact q-series type generating functions for several lattice and stacking models are presented. The structure of the generating function singularities is related to universality. These results have been obtained in collaboration with V. Privman and were extensively reviewed in Ref. 3.

Dirichlet and Poincaré series

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerningwhere T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet seriesdefines an entire function of s and satisfies the functional equationThe most penetrating statements that have been made on T(n) and LΔ(s)are:Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε &gt; 1/2, it was established in [2] for every ε&gt;1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

Deviations from critical density in the generalised hard hexagon model

Ramanujan's elliptic function identities and others are used to obtain infinite product forms for the deviations rho - rho c from the critical density rho c=1/2(1-5-12/) in the four regimes of the generalised hard hexagon model. Product forms are also obtained for the singular part rho sing of the density confirming the simple scaling forms recently conjectured by Huse, (1983).

Transformation formulae for Dirichlet polynomials

Formulae of Voronoi-Atkinson type are proved for Dirichlet polynomials related to the Dirichlet series ζ 2( s) = Σ d( n) n − s or ϕ( s) = Σ a( n) n − s , where the a( n) are the Fourier coefficients of a cusp form, a typical example being a( n) = τ( n), the Ramanujan function. Applications are given to a formula of Atkinson ( Acta Math. 81 (1949), 353–376) for the mean square of |ζ( 1 2 + it)| and to the differences between consecutive zeros of ϕ( s) on the critical line in the case when all the a( n) are real.

Exponential sums connected with Ramanujan's function τ( n )

Exponential sums connected with Ramanujan's function τ( <i>n</i> )