scholarly journals Some Polynomial Sequence Relations

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 750 ◽  
Author(s):  
Chan-Liang Chung
Keyword(s):  
Generating Functions ◽  
The Other ◽  
Polynomial Sequence ◽  
Elementary Identity

We give some polynomial sequence relations that are generalizations of the Sury-type identities. We provide two proofs, one based on an elementary identity and the other using the method of generating functions.

Duels, Truels, Gruels, and Survival of the Unfittest

2017 ◽  
Author(s):  
Dominic Lanphier
Keyword(s):  
Generating Functions ◽  
The Other ◽  
Main Issue ◽  
Basic Facts ◽  
Combinatorial Methods

This chapter considers two generalizations of duels. The first generalization is to n players, like n-uels. The other generalization is an iterative duel. The main issue for both of these duel-type games is to determine the likelihood that a player will survive the game. Section 1 reviews the main definitions, rules, and results about (sequential) duels and truels. Section 2 introduces duel-type games called gruels, giving some simple examples and establishing certain basic facts concerning such games. Section 3 introduces the second generalization, iterative duels. It applies certain combinatorial methods to investigate these games. In particular, it applies generating functions and thereby obtains arithmetic results about numbers associated to such duels. Section 4 applies more delicate analysis. This section is more mathematically challenging and follows methods from Stanley and Wilf to study which player in a sequential iterative duel is most likely to win.

A GENERAL RELATION BETWEEN SUMS OF SQUARES AND SUMS OF TRIANGULAR NUMBERS

2005 ◽  
Vol 01 (02) ◽  
pp. 175-182 ◽  
Author(s):  
CHANDRASHEKAR ADIGA ◽  
SHAUN COOPER ◽  
JUNG HUN HAN
Keyword(s):  
Generating Functions ◽  
General Relation ◽  
The Other ◽  
Sums Of Squares ◽  
Triangular Numbers

Let rk(n) and tk(n) denote the number of representations of n as a sum of k squares, and as a sum of k triangular numbers, respectively. We give a generalization of the result rk(8n + k) = cktk(n), which holds for 1 ≤ k ≤ 7, where ck is a constant that depends only on k. Two proofs are provided. One involves generating functions and the other is combinatorial.

scholarly journals Generating functions for two-variable polynomials related to a family of Fibonacci type polynomials and numbers

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 969-975 ◽  
Author(s):  
Gulsah Ozdemir ◽  
Yilmaz Simsek
Keyword(s):  
Infinite Series ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Functional Equations ◽  
The Other ◽  
The Family

The purpose of this paper is to construct generating functions for the family of the Fibonacci and Jacobsthal polynomials. Using these generating functions and their functional equations, we investigate some properties of these polynomials. We also give relationships between the Fibonacci, Jacobsthal, Chebyshev polynomials and the other well known polynomials. Finally, we give some infinite series applications related to these polynomials and their generating functions.

CLASSES OF MRM-APPLICABLE MEASURES FOR THE POWER FUNCTION OF GENERAL ORDER

2013 ◽  
Vol 16 (04) ◽  
pp. 1350028
Author(s):  
IZUMI KUBO ◽  
HUI-HSIUNG KUO ◽  
SUAT NAMLI
Keyword(s):  
Orthogonal Polynomials ◽  
Power Function ◽  
Generating Functions ◽  
The Other ◽  
Large Classes ◽  
Beta Distributions ◽  
General Order ◽  
Multiplicative Renormalization ◽  
Renormalization Method ◽  
Special Classes

The authors have previously studied multiplicative renormalization method (MRM) for generating functions of orthogonal polynomials. In particular, they have determined all MRM-applicable measures for renormalizing functions h(x) = ex, h(x) = (1 - x)-κ, κ = 1/2, 1, 2. For the cases h(x) = ex and (1 - x)-1, there are very large classes of MRM-applicable measures. For the other two cases κ = 1/2, 2, MRM-applicable measures belong to special classes of a certain kind of beta distributions. In this paper, we determine all MRM-applicable measures for h(x) = (1 - x)-κ with κ ≠ 0, 1, 1/2.

scholarly journals Signed Words and Permutations II; The Euler-Mahonian Polynomials

10.37236/1879 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Dominique Foata ◽  
Guo-Niu Han
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
Statistical Study ◽  
Record Values ◽  
The Other ◽  
Length Function ◽  
Major Index ◽  
Signed Permutations ◽  
Inversion Number ◽  
The One

As for the symmetric group of ordinary permutations there is also a statistical study of the group of signed permutations, that consists of calculating multivariable generating functions for this group by statistics involving record values and the length function. Two approaches are here systematically explored, using the flag-major index on the one hand, and the flag-inversion number on the other hand. The MacMahon Verfahren appears as a powerful tool throughout.

scholarly journals A Analysis on Queuing System with Setup Time in Revamp Process

2019 ◽  
Vol 9 (1S4) ◽  
pp. 967-969
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Setup Time ◽  
Repair Process ◽  
The Other ◽  
Queuing Model ◽  
Second Stage ◽  
Time Stage ◽  
Set Up ◽  
Two Stages

In this paper, we study about a M/G/1 Queuing model with single stage of service. Service interrupts during the time of service. The server does not get into the repair process immediately. It gets into a Set up time stage for the prior work to be done. On completion of set up stage service, the server will get into the repair process consisting of two stages, in which first stage is compulsory and the second stage of service is optional. For the model defined, we get the steady state results in closed form in terms of the probability generating functions and all the other execution performance measures of the model defined.

scholarly journals On a functional equation arising from two processors with coupled inputs

2015 ◽  
Vol 4 (3) ◽  
pp. 264
Author(s):  
EL-Sayed El-Hady ◽  
Wolfgang Forg-Rob
Keyword(s):  
Functional Equation ◽  
Boundary Value Problem ◽  
Analytic Continuation ◽  
Generating Functions ◽  
Functional Equations ◽  
Elliptic Functions ◽  
Symmetric Case ◽  
The Other ◽  
Interesting Class

<p>During the last few decades, a certain interesting class of functional equations arises when obtaining the generating functions of many system distributions. Such a class of equations has numerous applications in many modern disciplines like wireless networks and communications. This paper has been motivated by an issue considered by Paul E. Wright in [Advances in applied probability, (1992), 986 􀀀 1007]. The functional equation obtained there has been solved using elliptic functions and analytic continuation, which in turn lead to the determination of the main unknown. Unfortunately that solution seems to be a bit too general with many technical assumptions. In this paper on one hand, we introduce a solution in the symmetric case using boundary value problem approach. On the other hand, we investigate the potential singularities of the unknowns of the functional equation giving one possible application, and we compute some expectation of interest using the corresponding generating function.</p>

Partial Differentiation

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Nineteenth Century ◽  
Generating Functions ◽  
The Other ◽  
Legendre Transform ◽  
Mathematical Tool ◽  
Partial Differentiation ◽  
Thermodynamic Potentials ◽  
One To One

This short chapter discusses the Legendre transform, which is used in mechanics to convert between the Lagrangian and the Hamiltonian formulations. The Legendre transform is a mathematical tool that can be used to convert the variables of a function through the methods of partial differentiation in a one-to-one fashion. Developed by Adrien-Marie Legendre in the nineteenth century, it is also central to converting between action principles, generating functions and thermodynamic potentials. By using the Legendre transform, two variables can be expressed in four different ways, via the idea of conjugate pairs; it just depends on what differential quantity is subtracted. Variables that are not considered in the transformation are called passive variables, whiles the important ones are the active variables. The information in this chapter provides the background for many of the other chapters in this book.

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.

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