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Roberto Tauraso

Publications and source records attributed to Roberto Tauraso.

At least 19 recordsLinked to original sources
arXiv2026

Franklin's identity for $n$-color partitions and companion Beck-type identities

We show that some classical identities valid for ordinary partitions have precise analogues for $n$-color partitions, that is partitions in which a part of size $n\geq 1$ can occur in colors $1, 2, \ldots, n$. For $r \ge 2$ and $j \ge 0$, we write $\mathcal{O}_{j,r}(m)$ and $\mathcal{D}_{j,r}(m)$ for the sets of $n$-color partitions of $m$ with, respectively, exactly $j$ different parts whose size and color are divisible by $r$, and exactly $j$ different parts occurring at least $r$ times. We prove an $n$-color version of Franklin's theorem, $|\mathcal{O}_{j,r}(m)| = |\mathcal{D}_{j,r}(m)|$, along with two Beck-type identities. We give both analytic and combinatorial proofs for all theorems.

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arXiv2025

Arithmetic properties of MacMahon-type sums of divisors: the odd case

A century ago, P. A. MacMahon introduced two families of generating functions, $$ \sum_{1\leq n_1<n_2<\cdots<n_t}\prod_{k=1}^t\frac{q^{n_k}}{(1-q^{n_k})^2} \quad\text{ and } \sum_{\substack{1\leq n_1<n_2<\cdots<n_t\\ \text{$n_1,n_2,\dots,n_t$ odd}}}\prod_{k=1}^t\frac{q^{n_k}}{(1-q^{n_k})^2}, $$ which connect sum-of-divisors functions and integer partitions. These have recently drawn renewed attention. In particular, Amdeberhan, Andrews, and Tauraso extended the first family above by defining $$ U_t(a,q):=\sum_{1\leq n_1<n_2<\cdots<n_t}\prod_{k=1}^t\frac{q^{n_k}}{1+aq^{n_k}+q^{2n_k}} $$ for $a=0, \pm1, \pm2$ and investigated various properties, including some congruences satisfied by the coefficients of the power series representations for $U_t(a,q)$. These arithmetic aspects were subsequently expanded upon by the authors of the present work. Our goal here is to generalize the second family of generating functions, where the sums run over odd integers, and then apply similar techniques to show new infinite families of Ramanujan--like congruences for the associated power series coefficients.

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arXiv2025

Congruences for sums involving $\binom{rk}{k}$

We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ over the ranges $0<k<p$ and $0<k<p/r$, where $p$ is a prime larger than the positive integer $r$. Here $x$ is an indeterminate, thus allowing specialization to numerical congruences where $x$ takes certain algebraic numbers as values. We employ two different approaches that have complementary strengths. In particular, we obtain congruences modulo $p^2$ for the sum $\sum_{0<k<p}\binom{rk}{k}x^k$, expressed in terms of finite polylogarithms of certain quantities related to $x$.

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arXiv2024

Arithmetic properties of MacMahon-type sums of divisors

In this paper, we prove several new infinite families of Ramanujan--like congruences satisfied by the coefficients of the generating function $U_t(a,q)$ which is an extension of MacMahon's generalized sum-of-divisors function. As a by-product, we also show that, for all $n\geq 0$, $\overline{B}_3(15n+7)\equiv 0 \pmod{5}$ where $\overline{B}_3(n)$ is the number of almost $3$-regular overpartitions of $n$.

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arXiv2024

Further study on MacMahon-type sums of divisors

This paper is devoted to the study of $$ U_t(a,q):=\sum_{1\leq n_1<n_2<\cdots<n_t}\frac{q^{n_1+n_2+\cdots+n_t}}{(1+aq^{n_1}+q^{2n_1})(1+aq^{n_2}+q^{2n_2})\cdots(1+aq^{n_t}+q^{2n_t})} $$ when $a$ is one of $0, \pm 1, \pm2$. The idea builds on our previous treatment of the case $a=-2$. It is shown that all these functions lie in the ring of quasi-modular forms. Among the more surprising findings is $$U_2(1,q)=\sum_{n\geq1} \frac{q^{3n}}{(1-q^{3n})^2}.$$

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arXiv2024

Congruences for sums of MacMahon's $q$-Catalan polynomials

One variant of the $q$-Catalan polynomials is defined in terms of Gaussian polynomials by $\mathcal{C}_k(q)=\genfrac{[}{]}{0pt}{}{2k}{k}_q-q\genfrac{[}{]}{0pt}{}{2k}{k+1}_q$. Liu studied congruences of the form $\sum_{k=0}^{n-1} q^k\mathcal{C}_k$ modulo the cyclotomic polynomial $\Phi_n(q)^2$, provided that $n\equiv\pm 1\pmod3$. Apparently the case $n\equiv 0\pmod3$ has been missing from the literature. It is our primary purpose to fill this gap by the current work. In addition, we discuss certain fascinating link to Dirichlet character sum identities.

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arXiv2023

Extensions of MacMahon's sums of divisors

In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so we reveal a wealth of divisibility theorems and unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves rational function approximation to MacMahon-type generating functions. One such example involves multiple $q$-harmonic sums $$\sum_{k=1}^n\frac{(-1)^{k-1}\genfrac{[}{]}{0pt}{}{n}{k}_{q}(1+q^k)q^{\binom{k}{2}+tk}}{[k]_q^{2t} \genfrac{[}{]}{0pt}{}{n+k}{k}_{q}}=\sum_{1\leq k_1\leq\cdots\leq k_{2t}\leq n}\frac{q^{n+k_1+k_3\cdots+k_{2t-1}}+q^{k_2+k_4+\cdots+k_{2t}}}{[n+k_1]_q[k_2]_q\cdots[k_{2t}]_q}.$$

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arXiv2022

Congruences for partial sums of the generating series for $\binom{3k}{k}$

We produce congruences modulo a prime $p>3$ for sums $\sum_k\binom{3k}{k}x^k$ over ranges $0\le k<q$ and $0\le k<q/3$, where $q$ is a power of $p$. Here $x$ equals either $c^2/(1-c)^3$, or $4s^2/\bigl(27(s^2-1)\bigr)$, where $c$ and $s$ are indeterminates. In the former case we deal more generally with shifted binomial coefficients $\binom{3k+e}{k}$. Our method derives such congruences directly from closed forms for the corresponding series.

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arXiv2020

A local-global theorem for $p$-adic supercongruences

Let ${\mathbb Z}_p$ denote the ring of all $p$-adic integers and call $${\mathcal U}=\{(x_1,\ldots,x_n):\,a_1x_1+\ldots+a_nx_n+b=0\}$$ a hyperplane over ${\mathbb Z}_p^n$, where at least one of $a_1,\ldots,a_n$ is not divisible by $p$. We prove that if a sufficiently regular $n$-variable function is zero modulo $p^r$ over some suitable collection of $r$ hyperplanes, then it is zero modulo $p^r$ over the whole ${\mathbb Z}_p^n$. We provide various applications of this general criterion by establishing several $p$-adic analogues of hypergeometric identities.

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arXiv2020

Three pairs of congruences concerning sums of central binomial coefficients

Recently the first author proved a congruence proposed in 2006 by Adamchuk: $\sum_{k=1}^{\lfloor\frac{2p}{3}\rfloor}\binom{2k}{k}\equiv 0\pmod{p^2}$ for any prime $p=1 \pmod{3}$. In this paper, we provide more examples (with proofs) of congruences of the same kind $$\sum_{k=1}^{\lfloor\frac{ap}{r}\rfloor}\binom{2k}{k}x^k \pmod{p^2}$$ where $p$ is a prime such that $p\equiv 1 \pmod{r}$, $a/r$ is a fraction in $(1/2,1)$ and $x$ is a $p$-adic integer. The key ingredients are the $p$-adic Gamma functions $Γ_p$ and a special class of computer-discovered hypergeometric identities.

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arXiv2019

Two supercongruences related to multiple harmonic sums

Let $p$ be a prime and let $x$ be a $p$-adic integer. We provide two supercongruences for truncated series of the form $$\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}.$$

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arXiv2017

From generating series to polynomial congruences

Consider an ordinary generating function $\sum_{k=0}^{\infty}c_kx^k$, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form $C(x)$. Various instances are known where the corresponding truncated sum $\sum_{k=0}^{q-1}c_kx^k$, with $q$ a power of a prime $p$, also admits a closed form representation when viewed modulo $p$. Such a representation for the truncated sum modulo $p$ frequently bears a resemblance with the shape of $C(x)$, despite being typically proved through independent arguments. One of the simplest examples is the congruence $\sum_{k=0}^{q-1}\binom{2k}{k}x^k\equiv(1-4x)^{(q-1)/2}\pmod{p}$ being a finite match for the well-known generating function $\sum_{k=0}^\infty\binom{2k}{k}x^k= 1/\sqrt{1-4x}$. We develop a method which allows one to directly infer the closed-form representation of the truncated sum from the closed form of the series for a significant class of series involving central binomial coefficients. In particular, we collect various known such series whose closed-form representation involves polylogarithms ${\rm Li}_d(x)=\sum_{k=1}^{\infty}x^k/k^d$, and after supplementing them with some new ones we obtain closed-forms modulo $p$ for the corresponding truncated sums, in terms of finite polylogarithms $£_d(x)=\sum_{k=1}^{p-1}x^k/k^d$.

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arXiv2016

Revitalized automatic proofs: demonstrations

We consider three problems from the recent issues of the American Mathematical Monthly involving different versions of Catalan triangle. Our main results offer generalizations of these identities and demonstrate automated proofs with additional twists, and on occasion we furnish a combinatorial proof.

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