Odd-balanced unimodal sequences and related functions: parity, mock modularity and quantum modularity

2016 ◽  
Vol 144 (9) ◽  
pp. 3687-3700 ◽  
Author(s):  
Byungchan Kim ◽  
Subong Lim ◽  
Jeremy Lovejoy
Keyword(s):  
Unimodal Sequences

scholarly journals Generating Functions for Certain Weighted Cranks

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Shreejit Bandyopadhyay ◽  
Ae Yee
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Colored Partitions ◽  
Unimodal Sequences

Recently, George Beck posed many interesting partition problems considering the number of ones in partitions. In this paper, we first consider the crank generating function weighted by the number of ones and obtain analytic formulas for this weighted crank function under conditions of the crank being less than or equal to some specific integer. We connect these cumulative and point crank functions to the generating functions of partitions with certain sizes of Durfee rectangles. We then consider a generalization of the crank for $k$-colored partitions, which was first introduced by Fu and Tang, and investigate the corresponding generating function for this crank weighted by the number of parts in the first subpartition of a $k$-colored partition. We show that the cumulative generating functions are the same as the generating functions for certain unimodal sequences.

A novel discrete grey Riccati model and its application

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Liang Zeng
Keyword(s):  
Difference Equation ◽  
Special Kind ◽  
Grey Model ◽  
Aviation Industry ◽  
Data Series ◽  
Grey Prediction ◽  
Development Cost ◽  
Content Type ◽  
Riccati Difference Equation ◽  
Unimodal Sequences

PurposeTo develop the theory and application of the grey prediction model, this investigation constructs a novel discrete grey Riccati model termed DGRM(1,1).Design/methodology/approachBy examining a special kind of Riccati difference equation and the structure of the conventional discrete grey model (DGM), we advance a novel DGRM, and the model's prediction effect is evaluated by two numerical examples and an application case and compared with that of other conventional grey models.FindingsThe average relative simulation error of DGRM(1,1) does not change if the model is built after the original sequence has been transformed by a multiplier, and the new model is suitable to predict monotonically increasing, monotonically decreasing and unimodal sequences.Practical implicationsDGRM(1,1) is utilized to forecast the development cost of a small plane owned by the Aviation Industry Corporation of China (AVIC) with an original data sequence from 2006 to 2013. The outcomes indicate that DGRM(1,1) exhibits high precision and potential in development cost prediction.Originality/valueCombining the Riccati difference equation with the conventional DGM, the author advances a new grey model that is suitable to predict three kinds of data series with different changing trends.

scholarly journals Quantum Jacobi forms and balanced unimodal sequences

2018 ◽  
Vol 186 ◽  
pp. 16-34 ◽  
Author(s):  
Michael Barnett ◽  
Amanda Folsom ◽  
Obinna Ukogu ◽  
William J. Wesley ◽  
Hui Xu
Keyword(s):  
Jacobi Forms ◽  
Unimodal Sequences

scholarly journals Asymptotic distribution of odd balanced unimodal sequences with rank congruent to a modulo c

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Taylor Garnowski
Keyword(s):  
Partition Function ◽  
Modular Form ◽  
Asymptotic Distribution ◽  
Generating Function ◽  
Modular Forms ◽  
Asymptotic Estimate ◽  
Asymptotic Properties ◽  
Unimodal Sequences ◽  
The World ◽  
Mock Modular Form

AbstractKim et al. (Proc Am Math Soc 144:687–3700, 2016) introduced the notion of odd-balance unimodal sequences in 2016. Like was shown by Bryson et al. (Proc Natl Acad Sci USA 109:16063–16067, 2012) for the generating function of strongly unimodal sequences, the generating function for odd-balanced unimodal sequences also has quantum modular behavior. Odd-balanced unimodal sequences thus appear to be a fundamental piece in the world of modular forms and combinatorics, and understanding their asymptotic properties is important for understanding their place in this puzzle. In light of this, we compute an asymptotic estimate for odd balanced unimodal sequences for ranks congruent to $$a \pmod {c}$$ a ( mod c ) for $$c\ne 2$$ c ≠ 2 or a multiple of 4. We find the interesting result that the odd balanced unimodal sequences are asymptotically related to the overpartition function. This is in contrast to strongly unimodal sequences which, are asymptotically related to the partition function. Our proofs of the main theorems rely on the representation of the generating function in question as a mixed mock modular form.

Unimodal sequences of Gorenstein ideals of codimension 4

1998 ◽  
Vol 14 (4) ◽  
pp. 563-568
Author(s):  
Hyoung J. Ko ◽  
Yong Su Shin
Keyword(s):  
Unimodal Sequences

scholarly journals Jacobi’s triple product, mock theta functions, unimodal sequences and the q-bracket

2018 ◽  
Vol 14 (07) ◽  
pp. 1961-1981
Author(s):  
Robert Schneider
Keyword(s):  
Unit Circle ◽  
Theta Function ◽  
Modular Forms ◽  
Statistical Physics ◽  
Theta Functions ◽  
Triple Product ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Quantum Modular Forms ◽  
Unimodal Sequences

In Ramanujan’s final letter to Hardy, he listed examples of a strange new class of infinite series he called “mock theta functions”. It turns out all of these examples are essentially specializations of a so-called universal mock theta function [Formula: see text] of Gordon–McIntosh. Here we show that [Formula: see text] arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms—under the action of an operator related to statistical physics and partition theory, the [Formula: see text]-bracket of Bloch–Okounkov. Second, we find [Formula: see text] to extend in [Formula: see text] to the entire complex plane minus the unit circle, and give a finite formula for this universal mock theta function at roots of unity, that is simple by comparison to other such formulas in the literature; we also indicate similar formulas for other [Formula: see text]-hypergeometric series. Finally, we look at interesting “quantum” behaviors of mock theta functions inside, outside, and on the unit circle.

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