scholarly journals Asymptotic Behavior of Odd-Even Partitions

10.37236/6874 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Min-Joo Jang
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Formula ◽  
Combinatorial Object ◽  
Similar Shape ◽  
Lost Notebook ◽  
Ramanujan's Lost Notebook

Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's "Lost" Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions. Nonetheless, Andrews found out that this function possesses combinatorial information, odd-even partition. In this paper, we provide the asymptotic formula for this combinatorial object. We also study its companion odd-even overpartitions.

The final problem: an identity from Ramanujan's lost notebook

2019 ◽  
Vol 100 (2) ◽  
pp. 568-591 ◽  
Author(s):  
Bruce C. Berndt ◽  
Junxian Li ◽  
Alexandru Zaharescu
Keyword(s):  
Lost Notebook ◽  
Ramanujan's Lost Notebook

FOUR IDENTITIES FOR THIRD ORDER MOCK THETA FUNCTIONS

2018 ◽  
Vol 239 ◽  
pp. 173-204 ◽  
Author(s):  
GEORGE E. ANDREWS ◽  
BRUCE C. BERNDT ◽  
SONG HENG CHAN ◽  
SUN KIM ◽  
AMITA MALIK
Keyword(s):  
Generating Functions ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Third Order ◽  
Lost Notebook ◽  
Ramanujan's Lost Notebook

In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3) 4 (1954), 84–106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. Math. 190 (2005), 278–299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan’s lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions are intimately connected with ranks of partitions. We prove new dissections for two rank generating functions, which are keys to our proof of the fourth, and the most difficult, of Ramanujan’s identities. In the last section of this paper, we establish new relations for ranks arising from our dissections of rank generating functions.

scholarly journals Ramanujan's “Lost” Notebook VI: The mock theta conjectures

1989 ◽  
Vol 73 (2) ◽  
pp. 242-255 ◽  
Author(s):  
George E Andrews ◽  
F.G Garvan
Keyword(s):  
Lost Notebook ◽  
Ramanujan's Lost Notebook
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