On some new arithmetic properties of certain restricted color partition functions

Abstract

AbstractVery 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 ) .