Proofs of five conjectures on matching coefficients of Baruah, Das and Schlosser by an algorithmic approach. (May 2023) - Du, Julia Q.D.; Tang, Dazhao
Abstract: In this paper, we present an algorithm on vanishing coefficients with arithmetic progressions in the sum of two generalized eta-quotients by using the theory of modular forms, and utilize our algorithm to prove five conjectures on matching coefficients in series expansions of certain q -products and their reciprocals. One of which was provided by Baruah and Das, and the others were found by Schlosser. For instance, we prove that for any n ≥ 0, λ 12 ( 10 n + r ) = λ 12 ′ ( 10 n + r − 6 ), r ∈ 7, 9 , where the sequences λ 12 ( n ) n ≥ 0 and λ 12 ′ ( n ) n ≥ 0 are defined by ∑ n = 0 ∞ λ 12 ( n ) q n = 1 R ( q ) R ( q 2 ) R ( q 4 ) R ( q 8 ) = ( ∑ n = 0 ∞ λ 12 ′ ( n ) q n ) − 1, and where R ( q ) is the Rogers–Ramanujan continued fraction, which has the following celebrated product representation: R ( q ) = ∏ n = 0 ∞ ( 1 − q 5 n + 1 ) ( 1 − q 5 n + 4 ) ( 1 − q 5 n + 2 ) ( 1 − q 5 n + 3 ) . Finally, we find that this phenomenon also exists in other infinite q -series expansions.