Zeros of entire functions and a problem of Ramanujan

We derive representations for certain entire q-functions and apply our technique to the Ramanujan entire function (or q-Airy function) and q-Bessel functions. This is used to show that the asymptotic series of the large zeros of the Ramanujan entire function and similar functions are also convergent series. The idea is to show that the zeros of the functions under consideration satisfy a nonlinear integral equation.     

Plancherel–Rotach Asymptotics for q-Orthogonal Polynomials

We establish the Plancherel–Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases, our results include this type of asymptotics for q ^?1-Hermite polynomials of Askey, Ismail, and Masson; q-Laguerre polynomials; and the Stieltjes–Wigert polynomials. We also introduce a one-parameter family of solutions to the q-difference equation of the Ramanujan function.     

The Moment Problem Associated with the q -Laguerre Polynomials

Abstract. We consider the indeterminate Stieltjes moment problem associated with the q -Laguerre polynomials. A transformation of the set of solutions, which has all the classical solutions as fixed points, is established and we present a method to construct, for instance, continuous singular solutions. The connection with the moment problem associated with the Stieltjes—Wigert polynomials is studied; we show how to come from q -Laguerre solutions to Stieltjes—Wigert solutions by letting the parameter α —> ∞ , and we explain how to lift a Stieltjes—Wigert solution to a q -Laguerre solution at the level of Pick functions. Based on two generating functions, expressions for the four entire functions from the Nevanlinna parametrization are obtained.     

A new proof of Ramanujan’s modular equation relating R(q) with R(q 5)

We give a new proof of Ramanujan’s modular identity relating R(q) with R(q ⁵), 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.     

A note on generalized q-difference equations for q-beta and Andrews–Askey integral

Two q-difference equations with solutions expressed by q-exponential operator identities are investigated. As applications, two extensions of Ramanujan?s formulas for q-beta integral are given, two generalizations of Andrews–Askey integral are obtained. In addition, generating functions for generalized Al-Salam–Carlitz polynomials are deduced. At last, a generalized transformation identity is gained.     

Liouville type results for a <Emphasis Type="Italic">p</Emphasis>-Laplace equation with negative exponent

Positive entire solutions of the equation \(\Delta _p u = u^{ - q} in \mathbb{R}^N (N \geqslant 2)\) where 1 < p ≤ N, q > 0, are classified via their Morse indices. It is seen that there is a critical power q = q _c such that this equation has no positive radial entire solution that has finite Morse index when q > q _c but it admits a family of stable positive radial entire solutions when 0 < q ≤ q _c. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q ≤ q _c relies on Caffarelli–Kohn–Nirenberg’s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p ≤ N and q > q _c. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.     

Identities and congruences for Ramanujan’s ω(q)

Recently, the authors constructed generalized Borcherds products where modular forms are given as infinite products arising from weight 1/2 harmonic Maass forms. Here we illustrate the utility of these results in the special case of Ramanujan’s mock theta function ω(q). We obtain identities and congruences modulo 512 involving the coefficients of ω(q).     

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 Γq(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.     

Identities and congruences for the general partition and Ramanujan’s tau functions

We present some identities and congruences for the general partition function p _r (n). In particular, we deduce some known identities for Ramanujan’s tau function and find simple proofs of Ramanujan’s famous partition congruences for modulo 5 and 7. Our emphasis throughout this paper is to exhibit the use of Ramanujan’s theta functions to generate identities and congruences for general partition function.     

An overpartition analogue of the <Emphasis Type="Italic">q</Emphasis>-binomial coefficients

We define an overpartition analogue of Gaussian polynomials (also known as q-binomial coefficients) as a generating function for the number of overpartitions fitting inside the \(M \times N\) rectangle. We call these new polynomials over Gaussian polynomials or over q-binomial coefficients. We investigate basic properties and applications of over q-binomial coefficients. In particular, via the recurrences and combinatorial interpretations of over q-binomial coefficients, we prove a Rogers–Ramanujan type partition theorem.     

Parity in partition identities

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 f ₀(q) along with a new proof of the Hecke-type series representation.     

Some Theorems on the Rogers-Ramanujan Continued Fraction and Associated Theta Function Identities in Ramanujan's Lost Notebook

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.The parameter k = R(q) R₂(q₂) introduced by Ramanujan in his second notebook has not been recognized for its usefulness. In this work, we demonstrate how beautifully the parameter k works, as we prove several identities involving k stated by Ramanujan in the lost notebook.     

Critical exponents for the evolution p-Laplacian equation with a localized reaction

This paper deals with the large time behavior of nonnegative solutions to the equation $$u_t = div\left( {\left| {\nabla u} \right|^{p - 2} \nabla u} \right) + a\left( x \right)u^q ,\left( {x,t} \right) \in R^N \times (0,T),$$ where p > 2, q > 0, and the function a(x) ?? 0 has a compact support. We obtain the critical exponent for global existence q ₀ and the Fujita exponent q _c . In one-dimensional case N = 1, we have $q_0 = \frac{{2(p - 1)}} {p}$ and q _c = 2(p ? 1). Particularly, all solutions are global in time if 0 < q ?? q _o, but blow up if q ₀ < q ?? q _c ; while if q > q _c both blowing up solutions and global solutions exist. However, for the case N ?? p > 2, these two critical exponents are exactly the same. Namely, q ₀ = p ? 1 = q _c .     

Odd prime values of the Ramanujan tau function

We study the odd prime values of the Ramanujan tau function, which form a thin set of large primes. To this end, we define LR(p,n):=τ(p ^n?1) and we show that the odd prime values are of the form LR(p,q) where p,q are odd primes. Then we exhibit arithmetical properties and congruences of the LR numbers using more general results on Lucas sequences. Finally, we propose estimations and discuss numerical results on pairs (p,q) for which LR(p,q) is prime.     

On the 4-dissections of certain infinite products

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.     

Ramanujan-type formulae for 1/π: q-analogues

The hypergeometric formulae designed by Ramanujan more than a century ago for efficient approximation of π, Archimedes' constant, remain an attractive object of arithmetic study. In this note we discuss some q-analogues of Ramanujan-type evaluations and of related supercongruences.     

Continued fractions with three limit points

On page 45 in his lost notebook, Ramanujan asserts that a certain q-continued fraction has three limit points. More precisely, if A_n/B_n denotes its nth partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of A_n/B_n exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established.     

On the images of entire functions under the limit <Emphasis Type="Italic">q</Emphasis>-Bernstein operator

The limit q-Bernstein operator B _q comes out naturally as the limit for the sequence of q-Bernstein operators in the case 0 < q < 1: Alternatively, it can be viewed as a modification of the Szász-Mirakyan operator related to the Euler distribution. In this paper, a necessary and sufficient condition for a function g to be an image of an entire function under B _q is presented.     

Ramanujan-type congruences for certain generating functions

For nonnegative integers a, b, the function d _a,b (n) is defined in terms of the q-series $\sum_{n=0}^\infty d_{a,b}(n)q^n=\prod_{n=1}^\infty{(1-q^{ an})^b}/{(1-q^n)}$ . We establish some Ramanujan-type congruences for d _a,b (n) by the theory of modular forms with complex multiplication. As consequences, we generalize the famous Ramanujan congruences for the partition function p(n) modulo 5, 7, and 11.     

The q-Calkin-Wilf tree

We define a q-analogue of the Calkin-Wilf tree and the Calkin-Wilf sequence. We show that the nth term f(n;q) of the q-analogue of the Calkin-Wilf sequence is the generating function for the number of hyperbinary expansions of n according to the number of powers that are used exactly twice. We also present formulae for branches within the q-analogue of the Calkin-Wilf tree and predecessors and successors of terms in the q-analogue of the Calkin-Wilf sequence.