Related papers: Generalizing Ramanujan's J Functions

In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\leq j\leq m-1, {n_1+n_{m}\brack…

In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\to1$. These $q$-series generalize Ramanujan function $A_{q}(z)$; Jackson's $q$-Bessel function $J_{\nu}^{(2)}$(z;q), Ismail-Masson orthogonal…

We generalize some widely used mother wavelets by means of the q-exponential function $e_q^x \equiv [1+(1-q)x]^{1/(1-q)}$ ($q \in {\mathbb R}$, $e_1^x=e^x$) that emerges from nonextensive statistical mechanics. Particularly, we define…

Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. George Andrews, Henri Cohen, Freeman Dyson,…

In this paper, by applying a range of classic summation and transformation formulas for basic hypergeometric series, we obtain a three-term identity for partial theta functions. It extends the Andrews-Warnaar partial theta function…

We state and prove a number of unilateral and bilateral $q$-series identities and explore some of their consequences. Those include certain generalizations of the $q$-binomial sum which also generalize the $q$-Airy function introduced by…

In this paper, for a positive integer $n\ge 1$, we look at the size and prime factors of the iterates of the Ramanujan $\tau$ function applied to $n$.

The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…

Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan's tau-function. Recently, Xiong showed that these prime…

For a function $f\colon \mathbb{N}\to\mathbb{N}$, let $$ N^+_f(x)=\{n\leq x: n=k+f(k) \mbox{ for some } k\}. $$ Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and…

We will tackle a conjecture of S. Seo and A. J. Yee, which says that the series expansion of $1/(q,-q^3;q^4)_\infty$ has nonnegative coefficients. Our approach relies on an approximation of the generally nonmodular infinite product…

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the…

We study frequency moments of partition statistics arising from Euler products $A(q)=\prod_{r\ge1}(1-q^r)^{-c(r)}$ via a transform that expresses the moment generating functions as $B(q)$ times explicit divisor--sum series determined by…

The Lie-Trotter formula $e^{\hat{A}+\hat{B}} = \lim_{N\to \infty} (e^{\hat{A}/N} e^{\hat{B}/N})^N$ is of great utility in a variety of quantum problems ranging from the theory of path integrals and Monte Carlo methods in theoretical…

All arithmetical functions $F$ satisfying Ramanujan Conjecture, i.e., $F(n)\ll_{\varepsilon}n^{\varepsilon}$, and with $Q-$smooth divisors, i.e., with Eratosthenes transform $F':=F\ast \mu$ supported in $Q-$smooth numbers, have a kind of…

Ramanujan's $q$-continued fractions are a central part of Ramanujan's development of basic hypergeometric series. They appear in Chapter 16 of Part III and Chapter 32 of Part V of {\em Ramanujan's Notebooks} edited by Berndt, and in Volume…

In this work we shall apply the Bochner's theorem to prove certain combinations of Euler's q-exponentials are positive definite functions. Then we apply this positivity to prove curious inequalities for the Jacobi theta function…

We study in detail the Ramanujan smooth expansions, for arithmetic functions; we start with the most general ones, for which we supply the "$P-$local expansions", for arguments with all prime-factors $p\le P$ (namely, $P-$smooth arguments),…

We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of…

Two level 17 modular functions $$ r=q^{2}\prod_{n=1}^{\infty}(1-q^{n})^{\left(\frac{n}{17}\right)},\quad s=q^{2}\prod_{n=1}^{\infty}\frac{(1-q^{17n})^{3}}{(1-q^{n})^{3}}, $$ are used to construct a new class of Ramanujan-Sato series for…