scholarly journals Asymptotics of Coefficients of Multivariate Generating Functions: Improvements for Smooth Points

10.37236/813 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Alexander Raichev ◽  
Mark C. Wilson
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Power Series ◽  
Generating Functions ◽  
Numerical Approximation ◽  
Smooth Point ◽  
Singularity Analysis ◽  
Lattice Paths ◽  
Singular Variety ◽  
Quantum Random Walks

Let $\sum_{\beta\in{\Bbb N}^d} F_\beta x^\beta$ be a multivariate power series. For example $\sum F_\beta x^\beta$ could be a generating function for a combinatorial class. Assume that in a neighbourhood of the origin this series represents a nonentire function $F=G/H^p$ where $G$ and $H$ are holomorphic and $p$ is a positive integer. Given a direction $\alpha\in{\Bbb N}_+^d$ for which the asymptotics are controlled by a smooth point of the singular variety $H = 0$, we compute the asymptotics of $F_{n \alpha}$ as $n\to\infty$. We do this via multivariate singularity analysis and give an explicit uniform formula for the full asymptotic expansion. This improves on earlier work of R. Pemantle and the second author and allows for more accurate numerical approximation, as demonstrated by our our examples (on lattice paths, quantum random walks, and nonoverlapping patterns).

scholarly journals Interlacing polynomials and the veronese construction for rational formal power series

2019 ◽  
Vol 150 (1) ◽  
pp. 1-16
Author(s):  
Philip B. Zhang
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Generating Functions ◽  
Ehrhart Polynomial ◽  
Polynomial Sequence ◽  
Real Zeros ◽  
Alternative Approach ◽  
Formal Power ◽  
Interlacing Property

AbstractFixing a positive integer r and $0 \les k \les r-1$, define $f^{\langle r,k \rangle }$ for every formal power series f as $ f(x) = f^{\langle r,0 \rangle }(x^r)+xf^{\langle r,1 \rangle }(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle }(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\, h(x) := ( (1+x+\cdots +x^{r-1})^{n} h(x) )^{\langle r,k \rangle }$ has only non-positive zeros for any $r \ges \deg h(x) -k$ and any positive integer n. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension n, which states that $U^{n}_{r,0}\,h(x)$ has only negative, real zeros whenever $r\ges n$. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence $( h^{\langle r,r-i \rangle }(x))_{1\les i \les r}$ is interlacing, so is $( U^{n}_{r,r-i}\, h(x) )_{1\les i \les r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for coloured permutations. Besides, we derive a Carlitz identity for refined coloured permutations.

Binomial subsampling

2006 ◽  
Vol 462 (2068) ◽  
pp. 1181-1195 ◽  
Author(s):  
Carsten Wiuf ◽  
Michael P.H Stumpf
Keyword(s):  
Mathematical Biology ◽  
Power Series ◽  
Generating Functions ◽  
Binomial Distribution ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient ◽  
Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X ,  p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X ,  p ) to mean that given X = x ,  Z is a draw from the binomial distribution Bi( x ,  p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals Lattice Paths, Sampling Without Replacement, and Limiting Distributions

10.37236/156 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
M. Kuba ◽  
A. Panholzer ◽  
H. Prodinger
Keyword(s):  
Generating Functions ◽  
Probability Distributions ◽  
Random Variable ◽  
Phase Changes ◽  
Lattice Paths ◽  
Sampling Without Replacement ◽  
Sample Paths ◽  
Quarter Plane ◽  
Explicit Formulae ◽  
Weighted Lattice Paths

In this work we consider weighted lattice paths in the quarter plane ${\Bbb N}_0\times{\Bbb N}_0$. The steps are given by $(m,n)\to(m-1,n)$, $(m,n)\to(m,n-1)$ and are weighted as follows: $(m,n)\to(m-1,n)$ by $m/(m+n)$ and step $(m,n)\to(m,n-1)$ by $n/(m+n)$. The considered lattice paths are absorbed at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$. We provide explicit formulæ for the sum of the weights of paths, starting at $(m,n)$, which are absorbed at a certain height $k$ at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$, using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, obtaining a total of five phase changes.

scholarly journals ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

2008 ◽  
Vol 78 (1) ◽  
pp. 129-140 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Quadratic Forms ◽  
Triangular Numbers ◽  
Representations Of Integers

AbstractGenerating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

A General Asymptotic Scheme for the Analysis of Partition Statistics

2014 ◽  
Vol 23 (6) ◽  
pp. 1057-1086 ◽  
Author(s):  
PETER J. GRABNER ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER
Keyword(s):  
Generating Function ◽  
Asymptotic Expansions ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Higher Moments ◽  
Integer Partitions ◽  
Classical Singularity ◽  
Random Integer ◽  
Partition Statistics ◽  
New Statistics

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

The random walk associated by the game of roulette

1981 ◽  
Vol 18 (04) ◽  
pp. 931-936
Author(s):  
James M. Hill ◽  
Chandra M. Gulati
Keyword(s):  
Random Walk ◽  
Power Series ◽  
Generating Functions ◽  
The Right ◽  
Lagrange's Theorem ◽  
Explicit Expressions

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.

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