Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence
$$p^{°f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then
$$\frac{1}{[n/p^{a-1}]!} \sum_{k=r(mod p^a)} \binom{pn+s}{pk+t}(-1)^{pk}((k-r)/p^{a-1})^l =\frac {1}{[n/p^{a-1}]!} \sum_{k=r(mod p^a)}\binom{n}{k}\binom{s}{t}(-1)^k((k-r)/p^{a-1})^l (mod p).$$
We also present an application of the first congruence to Bernoulli polynomials, and apply the second congruence to show that a p-adic order bound given by the authors in a previous paper can be attained when p=2.
We present two new Ramanujan-type congruences modulo 5 for overpartition. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.
It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for rank parity function is f(q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.
We prove congruence relations modulo cyclotomic polynomials for multisums of $q$-factorial ratios, therefore generalizing many well-known $p$-Lucas congruences. Such congruences connect various classical generating series to their $q$-analogs. Using this, we prove a propagation phenomenon: when these generating series are algebraically independent, this is also the case for their $q$-analogs.
Motivated by a question of Lovejoy \cite{lovejoy}, we show that three-colored Frobenius partition function $\c3$ and related arithmetic fuction $\cc3$ vanish modulo some powers of 5 in certain arithmetic progressions.
Let $b_\ell(n)$ be the number of $\ell$-regular partitions of $n$. Recently, Hou et al established several infinite families of congruences for $b_\ell(n)$ modulo $m$, where $(\ell,m)=(3,3),(6,3),(5,5),(10,5)$ and $(7,7)$. In this paper, by the vanishing property given by Hou et al, we show an infinite family of congruence for $b_{11}(n)$ modulo $11$. Moreover, for $\ell= 3, 13$ and $25$, we obtain three infinite families of congruences for $b_{\ell}(n)$ modulo $3, 5$ and $13$ by the theory of Hecke eigenforms.
The Fishburn numbers $ξ(n)$ are defined by the formal power series \[ \sum_{n \geq 0} ξ(n) q^n = \sum_{n \geq 0} \prod_{j = 1}^n (1 - (1 - q)^j). \] Recently, G. Andrews and J. Sellers discovered congruences of the form $ξ(p m + j) \equiv 0$ modulo $p$, valid for all $m \geq 0$. These congruences have then been complemented and generalized to the case of $r$-Fishburn numbers by F. Garvan. In this note, we answer a question of Andrews and Sellers regarding an extension of these congruences to the case of prime powers. We show that, under a certain condition, all these congruences indeed extend …
Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence: $$\sum_{0<i_1<...<i_n<p}(i_1/3)(-1)^{i_1}/(i_1...i_n)=0 (mod p).$$
Using Watson's terminating $_8ϕ_7$ transformation formula, we prove a family of $q$-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636--646]. As an application, we deduce two supercongruences modulo $p^4$ ($p$ is an odd prime) and their $q$-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.
We compute the congruence class modulo 16 of the number of unique path partitions of $n$ (as defined by Olsson), thus generalising previous results by Bessenrodt, Olsson and Sellers [Ann. Combin. 13 (2013), 591-602].
Let $p_r(n)$ denote the number of $r$-component multipartitions of $n$, and let $S_{γ,λ}$ be the space spanned by $η(24z)^γϕ(24z)$, where $η(z)$ is the Dedekind's eta function and $ϕ(z)$ is a holomorphic modular form in $M_λ({\rm SL}_2(\mathbb{Z}))$. In this paper, we show that the generating function of $p_r(\frac{m^k n +r}{24})$ with respect to $n$ is congruent to a function in the space $S_{γ,λ}$ modulo $m^k$. As special cases, this relation leads to many well known congruences including the Ramanujan congruences of $p(n)$ modulo $5,7,11$ and Gandhi's congruences of $p_2(n)$ modulo 5 and $p…
The notion of broken $k$-diamond partitions was introduced by Andrews and Paule. Let $Δ_k(n)$ denote the number of broken k-diamond partitions of $n$. They also posed three conjectures on the congruences of $Δ_2(n)$ modulo 2, 5 and 25. Hirschhorn and sellers proved the conjectures for modulo 2, and Chan proved cases of modulo 5. For the case of modulo 3, Radu and Sellers obtained an infinite family of congruences for $Δ_2(n)$. In this paper, we obtain two infinite families of congruences for $Δ_2(n)$ modulo 3 based on a formula of Radu and Sellers, the 3-dissection formula of the generating fu…
In this paper we prove some transformation formulae for congruences modulo a prime and deduce some congruences for Domb numbers and Almkvist-Zudilin numbers. We also pose some conjectures on congruences modulo prime powers.
Let $cϕ_{k}(n)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $cϕ_{3}(n)$ and $cϕ_{9}(n)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\ge 3$ and $n\ge 0$, we prove that \[cϕ_{3}\Big(3^{2k}n+\frac{7\cdot 3^{2k}+1}{8}\Big) \equiv 0 \pmod{3^{4k+5}}.\] We give two different proofs to the congruences satisfied by $cϕ_{9}(n)$. One of the proofs uses an relation between $cϕ_{9}(n)$ and $cϕ_{3}(n)$ due to Kolitsch, for which we provide a new proof in this paper.
Let $p$ be an odd prime, and let $m$ be an integer with $p\nmid m$. In this paper show that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak\binom{-1-a}k}{m^k} \equiv 0\pmod p \quad\hbox{implies}\quad\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak \binom{-1-a}k}{m^k}\equiv 0\pmod {p^2}.$$
We provide several new $q$-congruences for truncated basic hypergeometric series with the base being an even power of $q$. Our results mainly concern congruences modulo the square or the cube of a cyclotomic polynomial and complement corresponding ones of an earlier paper containing $q$-congruences for truncated basic hypergeometric series with the base being an odd power of $q$. We also give a number of related conjectures including $q$-congruences modulo the fifth power of a cyclotomic polynomial and a congruence for a truncated ordinary hypergeometric series modulo the seventh power of a pr…
Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo p for all primes p>3 which were conjectured earlier by the author. We extend Ono's method to handle the powers of 5, 7 and 13 congruences. We need the theory of weak Maass forms as well as certain classical modular equations for the Dedekind eta-function.
Recently, Lin introduced two new partition functions PD$_t(n)$ and PDO$_t(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$ with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for PD$_t(n)$ and PDO$_t(n)$, and conjectured some congruences modulo 8. Very recently, Adansie, Chern, and Xia found two new infinite families of congruences modulo 9 for PD$_t(n)$. In this paper, we prove the congruences modulo 8 conjectured by Lin and also…
In this work, we study the function $B_{s,t}(n)$, which counts the number of $(s,t)$-regular bipartitions of $n$. Recently, many authors proved infinite families of congruences modulo $11$ for $B_{3,11}(n)$, modulo $3$ for $B_{3,s}(n)$ and modulo $5$ for $B_{5,s}(n)$. Very recently, Kathiravan proved several infinite families of congruences modulo $11$, $13$ and $17$ for $B_{5,11}(n)$, $B_{5,13}(n)$ and $B_{81,17}(n)$. In this paper, we will prove infinite families of congruences modulo $5$ for $B_{2,15}(n)$, modulo $11$ for $B_{7,11}(n)$, modulo $11$ for $B_{27,11}(n)$ and modulo $17$ for $B_…
Let $B_{l,m}(n)$ denote the number of $(l,m)$-regular bipartitions of $n$. Recently, many authors proved several infinite families of congruences modulo $3$, $5$ and $11$ for $B_{l,m}(n)$. In this paper, using theta function identities to prove infinite families of congruences modulo $m$ for $(l,m)$-regular bipartitions, where $m\in\{7,3,11,13,17\}$.
Recently, Andrews, Hirschhorn and Sellers have proven congruences modulo 3 for four types of partitions using elementary series manipulations. In this paper, we generalize their congruences using arithmetic properties of certain quadratic forms.
We find and prove a class of congruences modulo 4 for Andrews' partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{EO}}(n)$ and further prove that $\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for almost all $n$. This study was inspired by similar congruences modulo 4 in the work by the second author and Garvan.
We study the properties of the odd Catalan numbers, C_n, modulo 2^k for k >= 2. We show that there exist only k - 1 different congruences of the odd Catalan numbers modulo 2^k. Moreover, these congruences can be obtained by C_{2^m - 1} (mod 2^k) for m = 1, 2, ..., k - 1.
Let $p_k(n)$ denote the number of $2$-color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$. We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo $5$ for $p_k(n)$. Moreover, we will present some new congruences modulo $7$ for $p_4(n)$.
This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.
Recently, Nayaka and Naika (2022) proved several congruences modulo $16$ and $32$ for $t$-colored overpartitions with $t=5,7,11$ and $13$. We extend their list using an algorithmic technique.
We introduce two new integer partition functions, both of which are the number of partition quadruples of $n$ with certain size restrictions. We prove both functions satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$ by use of generalized Lambert series identities and $q$-series techniques.
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special cases.
By finding the congruent relations between the generating function of the 5 dots bracelet partitions and that of the 5-regular partitions, we get some new congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for a given prime $p$, we study the arithmetic properties modulo $p$ of the $k$ dots bracelet partitions.
Inspired by recent congruences by Andersen with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author for the function spt(n). We show that a normalized form of the generating function of spt(n) is an eigenform modulo 32 for the Hecke operators T(l^2) for primes l >= 5 with l congruent to 1, 11, 17, 19 (mod 24), and an eigenform modulo 16 for l congruent to? 13, 23 (mod 24).
We establish the existence of congruences between a fixed endoscopic cuspidal automorphic representation $Π$ of $\mathrm{GSp}(4)$ of square-free conductor and stable cuspidal automorphic representations of the same level and weight modulo certain prime factors of the value at $1$ of the adjoint $L$-function of $Π$ normalized by a suitable period.
We determine the ring structure of Siegel modular forms of degree g modulo a prime p, extending Nagaoka's result in the case of degree g=2. We characterize U(p) congruences of Jacobi forms and Siegel modular forms, and surprisingly find different behaviors of Siegel modular forms of even and odd degrees.
In 2009, Corteel, Savelief and Vuletić generalized the concept of overpartitions to a new object called plane overpartitions. In recent work, the author considered a restricted form of plane overpartitions called $k$-rowed plane overpartions and proved a method to obtain congruences for these and other types of combinatorial generating functions. In this paper, we prove several restricted and unrestricted plane overpartition congruences modulo $4$ and $8$ using other techniques.
For any non-negative integer $n$ and non-zero integer $r$, let $p_r(n)$ denote Ramanujan's general partition function. By employing $q$-identities, we prove some new Ramanujan-type congruences modulo 5 for $p_r(n)$ for $r=-(5λ+1), -(5λ+3), -(5λ+4), -(25λ+1)$, $-(25λ+2)$, and any integer $λ$.
A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using theta function identities due to Ramanujan, we establish some new infinite families of congruences for $b_l(n)$ modulo $l$, where $l=13,17,23$.
We prove a two-parameter family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews' multiseries extension of the Watson transformation, and a Karlsson--Minton type summation for very-well-poised basic hypergeometric series due to George Gasper. The new family of $q$-congruences is then used to prove two conjectures posed earlier by the authors.
We observe that a sequence satisfies Lucas congruences modulo $p$ if and only if its values modulo $p$ can be described by a linear $p$-scheme, as introduced by Rowland and Zeilberger, with a single state. This simple observation suggests natural generalizations of the notion of Lucas congruences. To illustrate this point, we prove explicit generalized Lucas congruences for integer sequences that can be represented as the constant terms of $P(x,y)^n Q(x,y)$ where $P$ and $Q$ are certain Laurent polynomials.
We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p. In particular, we show that there exists a positive integer $n\ll p^{1/2+ε}, such that n! is a primitive root modulo p. We also show that every nonzero congruence class a \not \equiv 0 \pmod p can be represented as a product of 7 factorials, a \equiv n_1! ... n_7! \pmod p, where $\max \{n_i | i=1,... 7\}=O(p^{11/12+ε}), and we find the asymptotic formula for …
In 2017, Andrews, Dixit, Schultz and Yee introduced the function $\overline{\textrm{spt}}_ω(n)$, which denotes the number of smallest parts in the overpartitions of $n$ in which the smallest part is always overlined and all odd parts are less than twice the smallest part. Recently, Baruah and Begum established several internal congruences and congruences modulo small powers of $5$ for $\overline{\textrm{spt}}_ω(n)$. Moreover, they conjectured a family of internal congruences modulo any powers of $5$ and two families of congruences modulo any even powers of $5$. In this paper, we confirm three …
We give bounds on the number of solutions to the Diophantine equation (X+1/x)(Y+1/y) = n as n tends to infinity. These bounds are related to the number of solutions to congruences of the form ax+by = 1 modulo xy.
For any positive integers $n$ and $r$, let $p_r(n)$ denotes the number of partitions of $n$ where each part has $r$ distinct colours. Many authors studied the partition function $p_r(n)$ for particular values of $r$. In this paper, we prove some general congruences modulo $5$ and $7$ for the colour partition function $p_r(n)$ by considering some general values of $r$. To prove the congruences we employ some $q$-series identities which is also in the spirit of Ramanujan.
Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several congruences satisfied by the corresponding coefficients. In this paper, we find some new families of congruences modulo powers of primes. We also find analogous results for the coefficients of the fractional powers of the generating function for the 2-color partition function.
Let $NT(m, k, n)$ denote the total number of parts in the partitions of n with rank congruent to m modulo k. Andrews proved Beck's conjecture on congruences for $NT(m, k, n)$ modulo 5 and 7. Generalizing Andrews'results, Chern obtain congruences for $NT(m, k, n)$ modulo 11 and 13. More recently, the second author use the theory of Hecke operators to establish congruences for such partition statistics modulo powers of primes $\ell \ge 7$. In this paper, we obtain Andrews-Beck type congruences modulo powers of 5.
The partition function $ p_{[1^c11^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{11}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{11 n})^d}.\] In this paper, we prove infinite families of congruences for the partition function $ p_{[1^c11^d]}(n)$ modulo powers of $11$ for any integers $c$ and $d$, which generalizes Atkin and Gordon's congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions of the congruence subgroup $Γ_0(11)$.
We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),
Let $b_{\ell, k}(n), b_{\ell, k, r}(n)$ count the number of $(\ell, k)$, $(\ell, k, r)$-regular partitions respectively. In this paper we shall derive infinite families of congruences for $b_{\ell, k}(n)$ modulo $2$ when $ (\ell, k) = (3,8), (4, 7)$, for $b_{\ell, k}(n)$ modulo $8$, modulo $9$ and modulo $12$ when $(\ell, k) = (4, 9)$ and $b_{\ell, k, r}(n)$ modulo $2$ when $(\ell, k, r) = (3, 5, 8)$.
We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews' multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the…
