scholarly journals COMPLEXITY OF SHORT GENERATING FUNCTIONS

2018 ◽  
Vol 6 ◽  
Author(s):  
DANNY NGUYEN ◽  
IGOR PAK
Keyword(s):  
Generating Functions ◽  
Complexity Analysis ◽  
Theta Functions ◽  
Polynomial Factor

We give complexity analysis for the class of short generating functions. Assuming #P$\not \subseteq$FP/poly, we show that this class is not closed under taking many intersections, unions or projections of generating functions, in the sense that these operations can increase the bit length of coefficients of generating functions by a super-polynomial factor. We also prove that truncated theta functions are hard for this class.

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.

scholarly journals Siegel modular forms of genus 2 and level 2

2015 ◽  
Vol 26 (05) ◽  
pp. 1550034 ◽  
Author(s):  
Fabien Cléry ◽  
Gerard van der Geer ◽  
Samuel Grushevsky
Keyword(s):  
Modular Forms ◽  
Generating Functions ◽  
Theta Functions ◽  
Siegel Modular Forms ◽  
Genus 2 ◽  
Vector Valued ◽  
Level 2

We study vector-valued Siegel modular forms of genus 2 on the three level 2 groups Γ[2] ◁ Γ1[2] ◁ Γ0[2] ⊂ Sp(4, ℤ). We give generating functions for the dimension of spaces of vector-valued modular forms, construct various vector-valued modular forms by using theta functions and describe the structure of certain modules of vector-valued modular forms over rings of scalar-valued Siegel modular forms.

scholarly journals Isomorphism and Symmetries in Random Phylogenetic Trees

2009 ◽  
Vol 46 (04) ◽  
pp. 1005-1019 ◽  
Author(s):  
Miklós Bóna ◽  
Philippe Flajolet
Keyword(s):  
Phylogenetic Tree ◽  
Gaussian Distribution ◽  
Generating Functions ◽  
Phylogenetic Trees ◽  
Singularity Analysis ◽  
Square Root ◽  
Analytic Combinatorics ◽  
Large Size ◽  
Polynomial Factor

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.

scholarly journals Statistical Properties of Lambda Terms

10.37236/8491 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Maciej Bendkowski ◽  
Olivier Bodini ◽  
Sergey Dovgal
Keyword(s):  
Generating Functions ◽  
Complexity Analysis ◽  
Parameter Distribution ◽  
Random Generation ◽  
Average Case ◽  
Case Complexity ◽  
Average Case Complexity ◽  
Novel Method ◽  
De Bruijn ◽  
Index Value

We present a quantitative, statistical analysis of random lambda terms in the De Bruijn notation. Following an analytic approach using multivariate generating functions, we investigate the distribution of various combinatorial parameters of random open and closed lambda terms, including the number of redexes, head abstractions, free variables or the De Bruijn index value profile. Moreover, we conduct an average-case complexity analysis of finding the leftmost-outermost redex in random lambda terms showing that it is on average constant. The main technical ingredient of our analysis is a novel method of dealing with combinatorial parameters inside certain infinite, algebraic systems of multivariate generating functions. Finally, we briefly discuss the random generation of lambda terms following a given skewed parameter distribution and provide empirical results regarding a series of more involved combinatorial parameters such as the number of open subterms and binding abstractions in closed lambda terms.

scholarly journals Isomorphism and Symmetries in Random Phylogenetic Trees

2009 ◽  
Vol 46 (4) ◽  
pp. 1005-1019 ◽  
Author(s):  
Miklós Bóna ◽  
Philippe Flajolet
Keyword(s):  
Phylogenetic Tree ◽  
Gaussian Distribution ◽  
Generating Functions ◽  
Phylogenetic Trees ◽  
Singularity Analysis ◽  
Square Root ◽  
Analytic Combinatorics ◽  
Large Size ◽  
Polynomial Factor

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.

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.

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].

RANK GENERATING FUNCTIONS FOR ODD-BALANCED UNIMODAL SEQUENCES, QUANTUM JACOBI FORMS, AND MOCK JACOBI FORMS

2020 ◽  
Vol 109 (2) ◽  
pp. 157-175
Author(s):  
MICHAEL BARNETT ◽  
AMANDA FOLSOM ◽  
WILLIAM J. WESLEY
Keyword(s):  
Generating Functions ◽  
Theta Functions ◽  
Laurent Polynomial ◽  
Roots Of Unity ◽  
Jacobi Forms ◽  
Unimodal Sequences ◽  
Partial Theta Functions

Let $\unicode[STIX]{x1D707}(m,n)$ (respectively, $\unicode[STIX]{x1D702}(m,n)$) denote the number of odd-balanced unimodal sequences of size $2n$ and rank $m$ with even parts congruent to $2\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ (respectively, $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$) and odd parts at most half the peak. We prove that two-variable generating functions for $\unicode[STIX]{x1D707}(m,n)$ and $\unicode[STIX]{x1D702}(m,n)$ are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single $C^{\infty }$ function in $\mathbb{R}\times \mathbb{R}$ to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables $w$ and $q$, how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size $2n$ with even parts congruent to $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ and odd parts at most half the peak.

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