Related papers: Ramanujan's theta functions and linear combination…
Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $\omega(q)$ and $\nu(q)$. In this paper, we find several new exact generating functions for those partition…
Let ${p}_{3,3}(n)$ denote the number of $2$-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $3$. In this paper, we shall establish some interesting Ramanujan-type congruences for…
In Ramanujan's final letter to Hardy, he listed examples of a strange new class of infinite series he called "mock theta functions". It turns out all of these examples are essentially specializations of a so-called universal mock theta…
We extend some results recently obtained by Dan Romik about the Taylor coefficients of the theta function $\theta_{3}\left(1\right)$ to the case $\theta_{3}\left(q\right)$ of an arbitrary value of the elliptic modulus $k.$ These results are…
Multiranks and new rank/crank analogs for a variety of partitions are given, so as to imply combinatorially some arithmetic properties enjoyed by these types of partitions. Our methods are elementary relying entirely on the three classical…
Product identities in two variables $x, q$ expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi's triple product identity, Watson's quintuple identity, and Hirschhorn's…
We study Schroder paths drawn in a (m,n) rectangle, for any positive integers m and n. We get explicit enumeration formulas, closely linked to those for the corresponding (m,n)-Dyck paths. Moreover we study a Schroder version of…
We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-function with explicit bounds on the error term. We…
Recently, Andrews, Dixit and Yee defined two partition functions $p_{\omega}(n)$ and $p_{\nu}(n)$ that are related with Ramanujan's mock theta functions $\omega(q)$ and $\nu(q)$, respectively. In this paper, we present two variable…
Recently the author introduced two new integer partition quadruple functions, which satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$. Here we reprove the congruences modulo $3$, $5$, and $7$ by defining a rank-type…
We investigate here sums of triangular numbers $f(x):=\sum_i b_i T_{x_i}$ where $T_n$ is the $n$-th triangular number. We show that for a set of positive integers $S$ there is a finite subset $S_0$ such that $f$ represents $S$ if and only…
Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In…
In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and…
Liu established an addition formula for the Jacobian theta function by using the theory of elliptic functions. From this addition formula he obtained the Ramanujan cubic theta function identity, Winquist's identity, a theta function…
We determine explicit formulas for the number of representations of a positive integer $n$ by quaternary quadratic forms with coefficients $1$, $2$, $5$ or $10$. We use a modular forms approach.
We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $\pi=(\pi_1,\pi_2,\pi_3)$ of $n$ such that $\pi_1$ and $\pi_2$ comprise distinct odd parts and…
An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number into negative powers of the nth…
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…
In this paper, we first prove that given a nonnegative integer $m$ and an odd number $t$ not divisible by $3$, there exists a unique Collatz's Sequence \[ S_{c}(m,t)=\{n_{0}(m,t),n_{1}(m,t),n_{2}(m,t),\ldots,n_{m}(m,t),n_{m+1}(m,t)\} \]…
We investigate integer numbers which possess at the same time the properties to be triangulars and squares, that are, numbers $a$ for which do exist integers $m$ and $n$ such that $ a = n^2 = \frac{m \cdot (m+1)}{2} $. In particular, we are…