Infinite Families of Congruences for 3-Regular Partitions with Distinct Odd Parts
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Infinite Families of Congruences for 3-Regular Partitions with Distinct Odd Parts Nipen Saikia1 Received: 29 January 2018 / Revised: 1 October 2018 / Accepted: 19 March 2019 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Let pod3 (n) denote the number of 3-regular partitions with distinct odd parts (and even parts are unrestricted) of a non-negative integer n. In this paper, we present infinite families of Ramanujan-type congruences modulo 2 and 3 for pod3 (n). Keywords 3-Regular partition · Distinct odd parts · Congruence Mathematics Subject Classification 11P83 · 05A15 · 05A17
1 Introduction A partition of a positive integer n is a non-increasing sequence of positive integers, called parts, whose sum equals n. An -regular partition of n is a partition of n with no part divisible by . For example, n = 5 has seven partitions, namely 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1, and the number of 3-regular partitions of 5 is five, namely 5, 4 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1. The number of partitions of n = 5 with distinct odd parts (and even parts are unrestricted) is four, namely 5, 4 + 1, 3 + 2, 2 + 2 + 1.
B 1
Nipen Saikia [email protected] Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India
123
N. Saikia
In this paper, we consider 3-regular partitions of n with distinct odd parts (and even parts are unrestricted). Let pod3 (n) denote the number of 3-regular partitions with distinct odd parts (and even parts are unrestricted) of n. For example, number of 3-regular partitions of n = 5 with distinct odd parts is 3, that is, pod3 (5) = 3, namely 5, 4 + 1, 2 + 2 + 1. The generating function of pod3 (n) [3] is given by ∞
pod3 (n)q n =
n=0
ψ(−q 3 ) , ψ(−q)
(1.1)
f 22 f1
(1.2)
where ψ(q) =
∞
q n(n+1)/2 =
n=0
and for any positive integer k, f k :=
∞
(1 − q kn ), |q| < 1.
(1.3)
n=1
From [1, p. 39, Entry 24], we also note that ψ(−q) =
f1 f4 . f2
(1.4)
In [3], authors found the following internal congruences for pod3 (n): pod3 (9n + 2) ≡ pod3 (n)
(mod 9),
pod3 (27n + 20) ≡ pod3 (3n + 2)
(mod 27)
and pod3 (243n + 182) ≡ pod3 (27n + 20)
(mod 81).
The purpose of this paper is to establish some Ramanujan-type congruences for pod3 (n). In particular, we prove infinite families of congruences modulo 2 and 3 for pod3 (n). To prove our results, we use some q-identities. We list them in Sect. 2. In Sect. 3, we prove congruences modulo 2, and in Sect. 4, we present infinite families of congruences modulo 3 for pod3 (n).
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Infinite Families of Congruences for 3-Regular Partitions…
2 Some q-Identities This section is devoted to record some q-identities which will be used to prove congruences in later sections. Lemma 2.1 [2, Theorem 2.2] For any prime p ≥ 5, we have p−1
f1 =
2
k (3k 2 +k)/2
(−1) q
3 p2 +(6k+1) p 3 p2 −(6k+1) p 2 2 f −q , −q
k=− p−1 2 k= ± p−1 6
+ (−1)
± p−1 6
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