Proofs of five conjectures on matching coefficients of Baruah, Das and Schlosser by an algorithmic approach. (May 2023)

• Record Type: Journal Article
• Authors: Du, Julia Q.D.; Tang, Dazhao
• Is Part Of: Journal of symbolic computation. Volume 116(2023)
• Page Start: 213
• Page End: 242
• Publication Date: 2023-05
• Subjects: 11F27 -- 30B10 -- 11F33; Matching coefficients -- Rogers–Ramanujan continued fraction -- Theta functions -- Vanishing coefficients -- Modular forms; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.09.004
• Languages: English
• ISSNs: 0747-7171

Abstract

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.