M2-Ranks of overpartitions modulo 4 and 8

• Published in: The Ramanujan journal Vol. 55; no. 1; pp. 369 - 392
• Main Authors: Gu, Nancy S. S., Su, Chen-Yang
• Format: Journal Article
• Language: English
• Published: New York Springer US 2021; Springer Nature B.V
• Subjects: Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory; 11P81; Mock theta function; Ramanujan’s theta function; 11B65; rank difference; Overpartition
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-019-00228-y

An overpartition is a partition in which the first occurrence of a number may be overlined. For an overpartition λ , let ℓ ( λ ) denote the largest part of λ , and let n ( λ ) denote its number of parts. Then the M 2 -rank of an overpartition is defined as M 2 -rank ( λ ) : = ℓ ( λ ) 2 - n ( λ ) + n ( λ 0 ) - χ ( λ ) , where χ ( λ ) = 1 if ℓ ( λ ) is odd and non-overlined and χ ( λ ) = 0 , otherwise. In this paper, we study the M 2 -rank differences of overpartitions modulo 4 and 8. Especially, we obtain some relations between the generating functions of the M 2 -rank differences modulo 4 and 8 and the second order mock theta functions. Furthermore, we deduce some inequalities on M 2 -ranks of overpartitions.