scholarly journals Applications of Generalized q-Difference Equations for General q-Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1222
Author(s):  
Zeya Jia ◽  
Bilal Khan ◽  
Qiuxia Hu ◽  
Dawei Niu
Keyword(s):  
Generating Function ◽  
Difference Equations ◽  
Generating Functions ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Fifth Order

Andrews gave a remarkable interpretation of the Rogers–Ramanujan identities with the polynomials ρe(N,y,x,q), and it was noted that ρe(∞,−1,1,q) is the generation of the fifth-order mock theta functions. In the present investigation, several interesting types of generating functions for this q-polynomial using q-difference equations is deduced. Besides that, a generalization of Andrew’s result in form of a multilinear generating function for q-polynomials is also given. Moreover, we build a transformation identity involving the q-polynomials and Bailey transformation. As an application, we give some new Hecke-type identities. We observe that most of the parameters involved in our results are symmetric to each other. Our results are shown to be connected with several earlier works related to the field of our present investigation.

scholarly journals Generating functions and congruences for some partition functions related to mock theta functions

2019 ◽  
Vol 16 (02) ◽  
pp. 423-446 ◽  
Author(s):  
Nayandeep Deka Baruah ◽  
Nilufar Mana Begum
Keyword(s):  
Generating Functions ◽  
Theta Functions ◽  
Partition Functions ◽  
Mock Theta Functions ◽  
Third Order

Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third-order mock theta functions [Formula: see text] and [Formula: see text]. In this paper, we find several new exact generating functions for those partition functions as well as the associated smallest part functions and deduce several new congruences modulo powers of 5.

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.

On two fifth order mock theta functions

2009 ◽  
Vol 20 (2) ◽  
pp. 207-214 ◽  
Author(s):  
Sander Zwegers
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions ◽  
Fifth Order

scholarly journals Generalizations of Dyson’s rank on overpartitions

2021 ◽  
pp. 1-13
Author(s):  
Alice X. H. Zhao
Keyword(s):  
Generating Function ◽  
Theta Functions ◽  
Mock Theta Functions

We introduce a statistic on overpartitions called the [Formula: see text]-rank. When there are no overlined parts, this coincides with the [Formula: see text]-rank of a partition introduced by Garvan. Moreover, it reduces to the D-rank of an overpartition when [Formula: see text]. The generating function for the [Formula: see text]-rank of overpartitions is given. We also establish a relation between the generating function of self-3-conjugate overpartitions and the tenth-order mock theta functions [Formula: see text] and [Formula: see text].

Bilateral series and Ramanujan radial limits of mock (false) theta functions

2019 ◽  
Vol 30 (04) ◽  
pp. 1950023
Author(s):  
Bin Chen
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Third Order ◽  
Radial Limits ◽  
The Third ◽  
False Theta Functions ◽  
Even Order ◽  
Fifth Order ◽  
Vector Valued

Ramanujan gave a list of seventeen functions which he called mock theta functions. For one of the third-order mock theta functions [Formula: see text], he claimed that as [Formula: see text] approaches an even order [Formula: see text] root of unity [Formula: see text], then [Formula: see text] He also pointed at the existence of similar properties for other mock theta functions. Recently, [J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci, Bilateral series and Ramanujan’s radial limits, Proc. Amer. Math. Soc. 143(2) (2014) 479–492] presented some similar Ramanujan radial limits of the fifth-order mock theta functions and their associated bilateral series are modular forms. In this paper, by using the substitution [Formula: see text] in the Ramanujan’s mock theta functions, some associated false theta functions in the sense of Rogers are obtained. Such functions can be regarded as Eichler integral of the vector-valued modular forms of weight [Formula: see text]. We find two associated bilateral series of the false theta functions with respect to the fifth-order mock theta functions are special modular forms. Furthermore, we explore that the other two associated bilateral series of the false theta functions with respect to the third-order mock theta functions are mock modular forms. As an application, the associated Ramanujan radial limits of the false theta functions are constructed.

Partitions associated with two fifth-order mock theta functions and Beck-type identities

2019 ◽  
Vol 16 (04) ◽  
pp. 841-855
Author(s):  
Runqiao Li ◽  
Andrew Y. Z. Wang
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions ◽  
Partition Identity ◽  
Fifth Order

In this paper, we first generalize Ramanujan’s partition identity derived from the fifth order mock theta functions [Formula: see text] and [Formula: see text]. Then we establish Beck-type identities based on our general partition identity. All the results are proved both analytically and combinatorially.

Ranks of overpartitions modulo 4 and 8

2020 ◽  
Vol 16 (10) ◽  
pp. 2293-2310
Author(s):  
Su-Ping Cui ◽  
Nancy S. S. Gu ◽  
Chen-Yang Su
Keyword(s):  
Generating Functions ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
First Occurrence ◽  
Rank Differences

An overpartition of [Formula: see text] is a partition of [Formula: see text] in which the first occurrence of a number may be overlined. Then, the rank of an overpartition is defined as its largest part minus its number of parts. Let [Formula: see text] be the number of overpartitions of [Formula: see text] with rank congruent to [Formula: see text] modulo [Formula: see text]. In this paper, we study the rank differences of overpartitions [Formula: see text] for [Formula: see text] or [Formula: see text] and [Formula: see text]. Especially, we obtain some relations between the generating functions of the rank differences modulo [Formula: see text] and [Formula: see text] and some mock theta functions. Furthermore, we derive some equalities and inequalities on ranks of overpartitions modulo [Formula: see text] and [Formula: see text].

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