Ramanujan Congruences for a Class of Eta Quotients

Abstract

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes for which their coefficients c(n) obey congruences of the form c( n + a) ≡ 0 (mod ). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences c N (n) satisfy, where c N (n) is defined by(1−q n )(1−q N n ) for N > 1. This last result answers a question of H.-C. Chan.

Comment on “Some Congruence Properties of a Restricted Bipartition Function cN(n)”

“…Let ( ) denote the number of bipartitions ( , ) of such that part of is divisible by . The generating function for ( ) is given by [8] …”

Some congruences for 3-component multipartitions

“…A number of congruences satisfied by p r .n/ were discovered, see, for example Andrews [1], Eichhorn and Ono [3], Atkin [6], Baruah and Ojah [7], Boylan [8], Cheema and Haskell [9], Gordon [10], Kiming and Olsson [11], Newman [12], Sinick [13], Treneer [14], Xia [15] and Yao [16]. Recently, Baruah and Ojah [7] established a 3-dissection formula for the generating function for p 3 .n/ and proved that for n 0,…”

Congruences modulo powers of 2 for a certain partition function

“…For subsequent studies involving the partition function a(n), see the work of Byungchan Kim [8] and of Jonah Sinick [13].…”