Twin Nicomachean q-identities and conjectures for the associated discriminants, polynomials, and inequalities

2020 ◽  
pp. 1-25
Author(s):  
Seon-Hong Kim ◽  
Kenneth B. Stolarsky
Keyword(s):  
Generating Functions ◽  
Rational Generating Functions ◽  
Root Distributions ◽  
Molien Series

We insert additional variables into Warnaar’s [Formula: see text]-analogue of Nicomachus’ identity and other related identities, and compute discriminants with respect to [Formula: see text]. Factorization of these discriminants reveals pairs of partitions that conjecturally relate in the manner of Wheatstone. The factorization also yields, conjecturally, families of polynomials with relations to various Molien series, remarkable rational generating functions, and notable root distributions. For a [Formula: see text]-analogue of Nicomachus’ identity produced by Cigler, we provide proofs of the partition properties. We also state and in part prove tight inequalities for the elements of two interlacing sequences that led us to the “twin” of Warnaar’s [Formula: see text]-analogue.

scholarly journals Counting with rational generating functions

2008 ◽  
Vol 43 (2) ◽  
pp. 75-91 ◽  
Author(s):  
Sven Verdoolaege ◽  
Kevin Woods
Keyword(s):  
Generating Functions ◽  
Rational Generating Functions

scholarly journals PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALS

2015 ◽  
Vol 80 (2) ◽  
pp. 433-449 ◽  
Author(s):  
KEVIN WOODS
Keyword(s):  
Generating Functions ◽  
Counting Function ◽  
Order Theory ◽  
Characteristic Functions ◽  
Presburger Arithmetic ◽  
First Order ◽  
First Order Theory ◽  
Rational Generating Function ◽  
Rational Generating Functions

AbstractPresburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, ifp= (p1, . . . ,pn) are a subset of the free variables in a Presburger formula, we can define a counting functiong(p) to be the number of solutions to the formula, for a givenp. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

scholarly journals On a Class of Distributions Stable Under Random Summation

2012 ◽  
Vol 49 (02) ◽  
pp. 303-318 ◽  
Author(s):  
L. B. Klebanov ◽  
A. V. Kakosyan ◽  
S. T. Rachev ◽  
G. Temnov
Keyword(s):  
Normal Distribution ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Analytic Functions ◽  
Random Variable ◽  
Random Summation ◽  
The Stability ◽  
Rational Generating Functions ◽  
Family Of Distributions

We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

scholarly journals On poles of rational generating functions. Probabilities of the appearance of combinations

1965 ◽  
Vol 5 (4) ◽  
pp. 585-591
Author(s):  
V. A. Malyshev
Keyword(s):  
Generating Functions ◽  
Rational Generating Functions

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: В. А. Малышев, О полосах рациональных производящих функций. Вероятности появления комбинации V. A. Malyševas, Racionalinių generuojančių funkcijų polių klausimu. Kombinacijų pasirodymo tikimybės

scholarly journals Rational Generating Functions and Integer Programming Games

2011 ◽  
Vol 59 (6) ◽  
pp. 1445-1460 ◽  
Author(s):  
Matthias Köppe ◽  
Christopher Thomas Ryan ◽  
Maurice Queyranne
Keyword(s):  
Integer Programming ◽  
Generating Functions ◽  
Rational Generating Functions
Sign in / Sign up

Export Citation Format

Share Document