Search

We prove specific biases in the number of occurrences of parts belonging to two different residue classes $a$ and $b$, modulo a fixed non-negative integer $m$, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size $n$ that belong to these sets of partitions and have a symmetric residue bias (i.e., for $1\le a

In this paper, we examine the unimodality and strict unimodality of certain formal bivariate Laurent series with non-negative coefficients. We show that the sets of these formal bivariate Laurent series form commutative semirings under the operations of addition and multiplication of formal Laurent series. This result is used to establish the unimodality of sequences involving Gauss polynomials and certain refined color partitions. In particular, we solve an open problem posed by Andrews on the unimodality of generalized Gauss polynomials and establish an unimodal result for a statistic of plane partitions. We also establish many unimodal results for rank statistics in partition theory, including the rank statistics of concave and convex compositions studied by Andrews, as well as certain unimodal sequences studied by Kim-Lim-Lovejoy. Additionally, we establish the unimodality of the Betti numbers and Gromov-Witten invariants of certain Hilbert schemes of points., Comment: This version includes 22 pages, adds the study of strict unimodality, and improves upon some results

In this paper, we refine a result of Andrews and Merca on truncated pentagonal number series. Subsequently, we establish some positivity results involving Andrews--Gordon--Bressoud identities and $d$-regular partitions. In particular, we prove several conjectures of Merca and Krattenthaler--Merca--Radu on truncated pentagonal number series.

Motivated by recent work of George Andrews and Mircea Merca on the expansion of the quotient of the truncation of Euler's pentagonal number series by the complete series, we provide similar expansion results for averages involving truncations of selected, more general, basic hypergeometric series. In particular, our expansions include new results for averaged truncations of the series appearing in the Jacobi triple product identity, the $q$-Gau{\ss} summation, and the very-well-poised ${}_5\phi_5$ summation. We show how special cases of our expansions can be used to recover various existing results. In addition, we establish new inequalities, such as one for a refinement of the number of partitions into three different colors., Comment: 14 pp

Let $\beta>1$ be fixed. We consider the $(\frak{b, d})$ numeration system, where the base ${\frak b}=(b_k)_{k\geq 0}$ is a sequence of positive real numbers satisfying $\lim_{k\rightarrow \infty}b_{k+1}/b_k=\beta$, and the set of digits ${\frak d}\ni 0$ is a finite set of nonnegative real numbers with at least two elements. Let $r_{\frak{b, d}}(\lambda)$ denote the number of representations of a given $\lambda\in\mathbb{R}$ by sums $\sum_{k\ge 0}\delta_kb_k$ with $\delta_k$ in ${\frak d}$. We establish upper bounds and asymptotic formulas for $r_{\frak{b,d}}(\lambda)$ and its arbitrary moments, respectively. We prove that the associated zeta function $\zeta_{\frak{b, d}}(s):=\sum_{\lambda>0}r_{\frak{b, d}}(\lambda)\lambda^{-s}$ can be meromorphically continued to the entire complex plane when $b_k=\beta^{k}$, and to the half-plane $\Re(s)>\log_\beta |\frak{d}|-\gamma$ when $b_k=\beta^{k}+O(\beta^{(1-\gamma)k})$, with any fixed $\gamma\in(0,1]$, respectively. We also determine the possible poles, compute the residues at the poles, and locate the trivial zeros of $\zeta_{\frak{b, d}}(s)$ in the regions where it can be extended. As an application, we answer some problems posed by Chow and Slattery on partitions into distinct terms of certain integer sequences., Comment: 23+\epsilon$\;$pages, 2 figures

Let $N_k(m,n)$ denote the number of partitions of $n$ with Garvan $k$-rank $m$. It is well-known that Andrews-Garvan-Dyson's crank and Dyson's rank are the $k$-rank for $k=1$ and $k=2$, respectively. In this paper, we prove that the sequence $\{N_k(m,n)\}_{|m|\le n-k-71}$ is log-concave for all sufficiently large $n$ and each integer $k$. In particular, we partially solve the log-concavity conjecture for Andrews-Garvan-Dyson's crank and Dyson's rank, which was independently proposed by Bringmann-Jennings-Shaffer-Mahlburg and Ji-Zang., Comment: $27$ pages. Dedicated to the memory of Freeman John Dyson, 1923-2020. Improved presentation with typos fixed and extra details/references added. Comments are welcome!

In this paper we investigate the monotonicity properties related to the ratio of gamma functions, from which some related asymptotics and inequalities are established. Some special cases also confirm the conjectures of C.-P. Chen [Monotonicity properties, inequalities, and asymptotic expansions associated with the gamma function. Appl. Math. Comput. 283 (2016), 385--396.]., Comment: 10 pages, accepted for publication in Mediterranean Journal of Mathematics

Let $\kappa$ be a positive real number and $m\in\mathbb{N}\cup\{\infty\}$ be given. Let $p_{\kappa, m}(n)$ denote the number of partitions of $n$ into the parts from the Piatestki-Shapiro sequence $(\lfloor \ell^{\kappa}\rfloor)_{\ell\in \mathbb{N}}$ with at most $m$ times (repetition allowed). In this paper we establish asymptotic formulas of Hardy-Ramanujan type for $p_{\kappa, m}(n)$, by employing a framework of asymptotics of partitions established by Roth-Szekeres in 1953, as well as some results on equidistribution., Comment: 19 pages

In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $\prod_{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $\delta$. We use the Hardy--Ramanujan--Rademacher circle method to give an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $\delta$ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $p=3$ case., Comment: 24 pages; an appendix with several new conjectures added; to appear in the Ramanujan Journal; dedicated to the memory of Richard Allen Askey

Let $\alpha>1$ be an irrational number. We establish asymptotic formulas for the number of partitions of $n$ into summands and distinct summands, chosen from the Beatty sequence $(\lfloor\alpha m\rfloor)_{m\in\mathbb{N}}$. This improves some results of Erd\"{o}s and Richmond established in 1977., Comment: 13 pages, changed the title

Let $r, v, n$ be positive integers. This paper investigate the number of solutions $s_{r,v}(n)$ of the following infinite Diophantine equations $$ n=1^{r}\cdot |k_{1}|^{v}+2^{r}\cdot |k_{2}|^{v}+3^{r}\cdot |k_{3}|^{v}+\ldots, $$ for ${\bf k}=(k_1,k_2,k_3,\dots)\in\mathbb{Z}^{\infty}$. For each $(r,v)\in\mathbb{N}\times\{1,2\}$, a generating function and some asymptotic formulas of $s_{r,v}(n)$ are established., Comment: 10 pages, accepted for publication in Taiwanese Journal of Mathematics

Let $f: \mathbb{Z}_+\rightarrow \mathbb{Z}_+$ be a polynomial with the property that corresponding to every prime $p$ there exists an integer $\ell$ such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed results between the number of partitions of an integer $n$ whose parts are taken from the sequence $\{f(\ell)\}_{\ell=1}^{\infty}$ and the number of parts of those partitions which are in a certain arithmetic progression., Comment: 14 pages, accepted for publication in International Journal of Number Theory

Let $F({\bf x})\in\mathbb{Z}[x_1,x_2,\dots,x_n]$ be a quadratic polynomial in $n\geq 3$ variables with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum $$ \Sigma_{k,F}(X; {\mathcal{B}})=\sum_{{\bf x}\in X\mathcal{B}\cap\mathbb{Z}^{n}}\tau_{k}\left(F({\bf x})\right), $$ for $X$ tending to infinity, where $\mathcal{B}\subset\mathbb{R}^n$ is an $n$-dimensional box such that $\min\limits_{{\bf x}\in X\mathcal{B}}F({\bf x})\ge 0$ for all sufficiently large $X$, and $\tau_{k}(\cdot)$ is the $k$-th divisor function for any integer $k\ge 2$., Comment: this article supersedes arXiv:1703.10546

In this paper, we investigate $\pi(m,n)$, the number of partitions of the \emph{bipartite number} $(m,n)$ into \emph{steadily decreasing} parts, introduced by L.Carlitz ['A problem in partitions', Duke Math Journal 30 (1963), 203--213]. We give a relation between $\pi(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Using this relation, some uniform asymptotic formulas for $\pi(m,n)$ are established., Comment: 9 pages

In this paper, we use basic asymptotic analysis to establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions. In particular, we answer and improve some problems suggested and investigated by Bringmann, Manschot, and Dousse. As applications, we establish the asymptotic monotonicity properties for the rank and crank of the integer partitions introduced and investigated by Dyson, Andrews, and Garvan., Comment: 39 pages, accepted for publication in The Ramanujan Journal

In this paper we prove a conjectured modular equation of Farkas and Kra, which involving a half sum of certain modular form of weight $1$ for congruence subgroup $\Gamma_1(k)$ with any prime $k$. We prove that their conjectured identity holds for all odd integer $k\ge 2$. A new modular equation of Farkas and Kra type is also established.

A strongly concave composition of $n$ is an integer partition with strictly decreasing and increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews, Rhoades and Zwegers ['Modularity of the concave composition generating function', Algebra \& Number Theory 7 (2013), no. 9, 2103--2139].

Let $k$ be a positive integer and $m$ be an integer. Garvan's $k$-rank $N_k(m,n)$ is the number of partitions of $n$ into at least $(k-1)$ successive Durfee squares with $k$-rank equal to $m$. In this paper give some asymptotics for $N_k(m,n)$ with $|m|\ge \sqrt{n}$ as $n\rightarrow \infty$. As a corollary, we give a more complete answer for the Dyson's crank distribution conjecture. We also establish some asymptotic formulas for finite differences of $N_k(m,n)$ with respect to $m$ with $m\gg \sqrt{n}\log n$., Comment: typos corrected

Let $p_n$ denote the $n$-th prime number, and let $d_n=p_{n+1}-p_{n}$. Under the Hardy--Littlewood prime-pair conjecture, we prove \begin{align*} \sum_{n\le X}\frac{\log^{\alpha}d_n}{d_n} \sim\begin{cases} \frac{X\log\log\log X}{\log X}~\qquad\quad~ &\alpha=-1,\\ \frac{X}{\log X}\frac{(\log\log X)^{1+\alpha}}{1+\alpha}\qquad &\alpha>-1, \end{cases} \end{align*} and establish asymptotic properties for some series of $d_n$ without the Hardy--Littlewood prime-pair conjecture., Comment: 5 pages

Let $(a;q)_{\infty}$ be the $q$-Pochhammer symbol and $\mathrm{li}_2(x)$ be the dilogarithm function. Let $\prod_{\alpha,\beta,\gamma}$ be a finite product with every triple $(\alpha,\beta,\gamma)\in(\mathbb{R}_{>0})^3$ and $S_{\alpha\beta\gamma}\in\mathbb{R}$. Also let the triple $(A,B,v)\in\left(\mathbb{R}_{>0}\times\mathbb{R}^2\right)\cup\left(\{0\}^2\times\mathbb{R}_{>0}\right)\cup\left(\{0\}\times\mathbb{R}_{<0}\times\mathbb{R}\right)$. In this work, we let $z=e^v$, denote by $H_{-1}(u)=vu-Au^2+\sum_{\alpha}\mathrm{li}_2(e^{-\alpha u})\sum_{\beta,\gamma} \beta^{-1}S_{\alpha\beta\gamma}$ and consider the Eulerien series \[\mathcal{H}(z;q)=\sum_{m=0}^{\infty}\frac{q^{Am^2+Bm}z^{m}}{\prod\limits_{\alpha,\beta,\gamma}(q^{\alpha m+\gamma};q^{\beta})_{\infty}^{S_{\alpha\beta\gamma}}}.\] We prove that if there exist an $\varepsilon>0$ such that $H_{-1}(u)$ is an increasing function on $[0,\varepsilon)$, then as $q\rightarrow 1^-$, \[\mathcal{H}(z;q)=\left(1+o\left(|\log q|^p\right)\right)\int\limits_{0}^{\infty}\frac{q^{Ax^2+Bx}z^{x}}{\prod\limits_{\alpha,\beta,\gamma}(q^{\alpha x+\gamma};q^{\beta})_{\infty}^{S_{\alpha\beta\gamma}}}\,dx\] holds for each $p\ge 0$. We also obtain full asymptotic expansions for $\mathcal{H}(z;q)$ which satisfy above condition as $q\rightarrow 1^{-}$. The complete asymptotic expansions for related basic hypergeometric series could be derived as special cases., Comment: 3^3 pages