Some identities for the partition function

Goswami, Ankush; Jha, Abhash Kumar; Singh, Anup Kumar

doi:10.1016/j.jmaa.2021.125864

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2024

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Cited by 2 publications

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“…The method of this paper is an extension of [6]. It seems that this extension can be applied for finding generating functions for the 2-color partition functions as well.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Witness identities for three Ramanujan congruences

Baruah¹,

Das²,

Sarma³

2022

Preprint

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“…The method of this paper is an extension of [6]. It seems that this extension can be applied for finding generating functions for the 2-color partition functions as well.…”

Section: Discussionmentioning

confidence: 99%

“…Such generating functions are called witness identities for Ramanujan congruences. A witness identity for (1.4) was first found by Lehner [8] and later by Atkin [1], Paule and Radu [9], and Goswami, Jha, and Singh [6]. Hirschhorn and Hunt [7] later found a witness identity for the infinite family (1.5) for arbitrary a and b = c = 0.…”

Section: Introductionmentioning

confidence: 97%

Witness identities for three Ramanujan congruences

Baruah¹,

Das²,

Sarma³

2022

Preprint

View full text Add to dashboard Cite

show abstract

On some new arithmetic properties of certain restricted color partition functions

Dasappa,

Channabasavayya,

Keerthana

2024

Arab. J. Math.

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Very recently, Pushpa and Vasuki (Arab. J. Math. 11, 355–378, 2022) have proved Eisenstein series identities of level 5 of weight 2 due to Ramanujan and some new Eisenstein identities for level 7 by the elementary way. In their paper, they introduced seven restricted color partition functions, namely $$P^{*}(n), M(n), T^{*}(n), L(n), K(n), A(n)$$ P ∗ ( n ) , M ( n ) , T ∗ ( n ) , L ( n ) , K ( n ) , A ( n ) , and B(n), and proved a few congruence properties of these functions. The main aim of this paper is to obtain several new infinite families of congruences modulo $$2^a\cdot 5^\ell $$ 2 a · 5 ℓ for $$P^{*}(n)$$ P ∗ ( n ) , modulo $$2^3$$ 2 3 for M(n) and $$T^*(n)$$ T ∗ ( n ) , where $$a=3, 4$$ a = 3 , 4 and $$\ell \ge 1$$ ℓ ≥ 1 . For instance, we prove that for $$n\ge 0$$ n ≥ 0 , $$\begin{aligned} P^{*}(5^\ell (4n+3)+5^\ell -1)&\equiv 0\pmod {2^3\cdot 5^{\ell }}. \end{aligned}$$ P ∗ ( 5 ℓ ( 4 n + 3 ) + 5 ℓ - 1 ) ≡ 0 ( mod 2 3 · 5 ℓ ) . In addition, we prove witness identities for the following congruences due to Pushpa and Vasuki: $$\begin{aligned} M(5n+4)\equiv 0\pmod {5},\quad T^{*}(5n+3)\equiv 0\pmod {5}. \end{aligned}$$ M ( 5 n + 4 ) ≡ 0 ( mod 5 ) , T ∗ ( 5 n + 3 ) ≡ 0 ( mod 5 ) .

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Some identities for the partition function

Cited by 2 publications

References 23 publications

Witness identities for three Ramanujan congruences

Witness identities for three Ramanujan congruences

On some new arithmetic properties of certain restricted color partition functions

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