Ramanujan-type congruences for overpartitions modulo 16

Let \(\bar{p}(n)\) denote the number of overpartitions of \(n\). Recently, Fortin–Jacob–Mathieu and Hirschhorn–Sellers independently obtained 2-, 3- and 4-dissections of the generating function for \(\bar{p}(n)\) and derived a number of congruences for \(\bar{p}(n)\) modulo 4, 8 and 64 including \(\bar{p}(8n+7)\equiv 0 \pmod {64}\) for \(n\ge 0\). In this paper, we give a 16-dissection of the generating function for \(\bar{p}(n)\) modulo 16 and show that \(\bar{p}(16n+14)\equiv 0\pmod {16}\) for \(n\ge 0\). Moreover, using the \(2\)-adic expansion of the generating function for \(\bar{p}(n)\) according to Mahlburg, we obtain that \(\bar{p}(\ell ^2n+r\ell )\equiv 0\pmod {16}\), where \(n\ge 0\), \(\ell \equiv -1\pmod {8}\) is an odd prime and \(r\) is a positive integer with \(\ell \not \mid r\). In particular, for \(\ell =7\) and \(n\ge 0\), we get \(\bar{p}(49n+7)\equiv 0\pmod {16}\) and \(\bar{p}(49n+14)\equiv 0\pmod {16}\). We also find four congruence relations: \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n) \pmod {16}\) for \(n\ge 0\), \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {32}\) where \(n\) is not a square of an odd positive integer, \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {64}\) for \(n\not \equiv 1,2,5\pmod {8}\) and \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {128}\) for \(n\equiv 0\pmod {4}\).     

Ramanujan-type congruences modulo 8 for 7- and 23-core partition functions

Recently, Radu and Sellers proved numerous congruences modulo powers of 2 for \( (2k+1)\)-core partition functions by employing the theory of modular forms. In this paper, employing Ramanujan’s theta function identities, we prove many infinite families of congruences modulo 8 for 7-core partition function. Our results generalize the congruences modulo 8 for 7-core partition function discovered by Radu and Sellers. Furthermore, we present new proofs of congruences modulo 8 for 23-core partition function. These congruences were first proved by Radu and Sellers.     

Ramanujan-type congruences for broken 2-diamond partitions modulo 3

The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).     

Ramanujan-type congruences modulo powers of 5 and 7

Let b _? (n) denote the number of ?-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of ?. In particular, they showed that for α, n ≥ 0, b ₂₅ (3^2α+3 n+2 · 3^2α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where c _N (n) counts the number of bipartitions (λ₁,λ₂) of n such that each part of λ₂ is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, \({c_{25}}\left( {{5^{2j}}n + \frac{{11 \cdot {5^{2j}} + 13}}{{12}}} \right) \equiv 0\) (mod 5 ^j+1), \({c_{49}}\left( {{7^{2j}}n + \frac{{11 \cdot {7^{_{2j}}} + 25}}{{12}}} \right) \equiv 0\) (mod 7 ^j+1) and b ₂₅ (3^2α+3 · n+2 · 3^2α+2-1) ≡ 0 (mod 3 · 5^2j-1).     

Transformation of certain bilinear generating functions

Summary Erdélyi's generalizations of the Hardy-Hille formula are extended to series involving arbitrary coefficients; see(1.6), …,(1.9) below. These identities may be thought of as identities in formal power series. The proofs are simple: each identity in shown to be equivalent to an algebraic identity that is easily verified. Supported in part by NSF grant GP-7855. Entrata in Redazione il 18 novembre 1969.     

On certain generating functions in positive characteristic

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit connection with certain deformations of the Carlitz logarithm introduced by M. Papanikolas and involve the Anderson-Thakur function and the Carlitz exponential function. They collect certain functional identities in families for a new class of L-functions introduced by the first author. This paper also deals with specializations at roots of unity of these generating functions, producing a link with Gauss-Thakur sums.     

New families of generating functions for certain class of three-variable polynomials

Recently, Srivastava, Özarslan and Kaanoglu have introduced certain families of three and two variable polynomials, which include Lagrange and Lagrange-Hermite polynomials, and obtained families of two-sided linear generating functions between these families [H.M. Srivastava, M.A. Özarslan, C. Kaanoglu, Some families of generating functions for a certain class of three-variable polynomials, Integr. Transform. Spec. Funct. iFirst (2010) 1-12]. The main object of this investigation is to obtain new two-sided linear generating functions between these families by applying certain hypergeometric transformations. Furthermore, more general families of bilinear, bilateral, multilateral finite series relationships and generating functions are presented for them.     

On the distribution of the roots of certain congruences and a problem for additive arithmetic functions

The asymptotic distribution of the roots of the congruence ax ≡ b (mod D), 1 ≤ x ≤ D, as D varies, is investigated. Quantitative estimates are obtained by means of exponential sums combined with sieve methods. As an application of the results it is shown that if an additive arithmetic function satisfies f(an + b) ? f(cn + d) = O(1) for all positive integers n, ad ≠ bc, then f(n) = O((log n)³) must hold. This result is apparently the first bound of any kind in such a situation.     

Some congruences for modulus 13 related to partition generating function

In this study, we obtain some congruences modulo 13 related to the partition generating function with the help of a congruence between q-series. We use an elementary method which appears in the work of Kolberg.     

Exceptional congruences for the coefficients of certain eta-product newforms

Let F(z)=∑_n=1^∞a(n)qⁿ denote the unique weight 16 normalized cuspidal eigenform on . In the early 1970s, Serre and Swinnerton-Dyer conjectured that

     

On some congruences of certain binomial sums

For any prime \(p>3,\) we prove that

$$\begin{aligned} \sum _{k=0}^{p-1}\frac{3k+1}{(-8)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-1}{p}\right) +p^3E_{p-3}\pmod {p^4}, \end{aligned}$$

where \(E_{0},E_{1},E_{2},\ldots \) are Euler numbers and \(\left( \frac{\cdot }{p}\right) \) is the Legendre symbol. This result confirms a conjecture of Z.-W. Sun. We also re-prove that for any odd prime \(p,\)

$$\begin{aligned} \sum _{k=0}^{\frac{p-1}{2}}\frac{6k+1}{(-512)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-2}{p}\right) \pmod {p^2} \end{aligned}$$

using WZ method.

     

Ramanujan-type supercongruences

We present several supercongruences that may be viewed as p-adic analogues of Ramanujan-type series for 1/π and 1/π², and prove three of these examples.