A finite dam with exponential release

1974 ◽  
Vol 11 (1) ◽  
pp. 122-133 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Differential Equation ◽  
Shot Noise ◽  
Generating Functions ◽  
Series Solution ◽  
Recurrence Relations ◽  
Order Differential Equation ◽  
Geiger Counter ◽  
Special Cases ◽  
Instantaneous Release ◽  
Finite Dam

This paper considers a finite dam with independently and identically distributed (i.i.d.) inputs occurring in a Poisson process; the special cases where the inputs are (i) deterministic and (ii) negative exponentially distributed are considered in detail. The instantaneous release trate is proportional to the content, i.e., there is an exponential fall in conten except when inputs occur. This model may arise in several other situations such as a geiger counter or integrated shot noise. The distribution of the number of inputs, and of the time, to first overflowing is obtained in terms of generating functions; in Case (i) the solution is obtained through recurrence relations involving iterated integrals which can be evaluated numerically, and in Case (ii) using a series solution of a second order differential equation. Numerical results, in particular for the first two moments, are obtained for various values of the parameters of the model, and compared with a large number of simulations. Some remarks are also made about the infinite dam.

A finite dam with exponential release

1974 ◽  
Vol 11 (01) ◽  
pp. 122-133 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Differential Equation ◽  
Shot Noise ◽  
Generating Functions ◽  
Series Solution ◽  
Recurrence Relations ◽  
Order Differential Equation ◽  
Geiger Counter ◽  
Special Cases ◽  
Instantaneous Release ◽  
Finite Dam

This paper considers a finite dam with independently and identically distributed (i.i.d.) inputs occurring in a Poisson process; the special cases where the inputs are (i) deterministic and (ii) negative exponentially distributed are considered in detail. The instantaneous release trate is proportional to the content, i.e., there is an exponential fall in conten except when inputs occur. This model may arise in several other situations such as a geiger counter or integrated shot noise. The distribution of the number of inputs, and of the time, to first overflowing is obtained in terms of generating functions; in Case (i) the solution is obtained through recurrence relations involving iterated integrals which can be evaluated numerically, and in Case (ii) using a series solution of a second order differential equation. Numerical results, in particular for the first two moments, are obtained for various values of the parameters of the model, and compared with a large number of simulations. Some remarks are also made about the infinite dam.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

scholarly journals The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

Orthogonal Polynomials With Weight Function (1 - x)α( l + x)β + Mδ(x + 1) + Nδ(x - 1)

1984 ◽  
Vol 27 (2) ◽  
pp. 205-214 ◽  
Author(s):  
Tom H. Koornwinder
Keyword(s):  
Differential Equation ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Hypergeometric Functions ◽  
Order Differential Equation ◽  
Second Order Differential Equations ◽  
Special Cases ◽  
Delta Functions ◽  
Fourth Order Differential Equation

AbstractWe study orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and — 1. These polynomials can be expressed as 4F3 hypergeometric functions and they satisfy second order differential equations. They include Krall’s Jacobi type polynomials as special cases. The fourth order differential equation for the latter polynomials is derived in a more simple way.

scholarly journals On Generating Functions

1930 ◽  
Vol 2 (2) ◽  
pp. 71-82 ◽  
Author(s):  
W. L. Ferrar
Keyword(s):  
Differential Equation ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Three Term Recurrence Relation ◽  
Sequence Of Functions

It is well known that the polynomial in x,has the following properties:—(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;(B) it satisfies the three-term recurrence relation(C) it is the solution of the second order differential equation(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),i.e. whenSeveral other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

Asymptotics of linear recurrences

2014 ◽  
Vol 12 (04) ◽  
pp. 463-484 ◽  
Author(s):  
R. Wong
Keyword(s):  
Differential Equation ◽  
Transition Point ◽  
Recurrence Relations ◽  
Turning Point ◽  
Order Differential Equation ◽  
Second Order ◽  
Asymptotic Approximations ◽  
Linear Difference Equations ◽  
Asymptotic Results ◽  
Linear Recurrences

In this paper, we review the asymptotic results that are now available for second-order linear difference equations containing a parameter (or, equivalently, three-term recurrence relations). These include asymptotic expansions for solutions to these equations when the parameter is fixed or varying in an interval containing a turning point or a transition point. Also presented is a method for deriving asymptotic approximations for solutions when the initial values are given. These results are particularly useful when a given system of orthogonal polynomials (i) does not satisfy any second-order differential equation, (ii) does not have any integral representation, and (iii) is not even associated with a unique (or any) weight function.

scholarly journals The Number of Ways to Assemble a Graph

10.37236/2644 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Andrew Vince ◽  
Miklós Bóna
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Explicit Formulas ◽  
Future Directions ◽  
Macromolecular Assembly ◽  
Open Questions ◽  
Asymptotic Formulae ◽  
Special Cases ◽  
Assembly Problem ◽  
Precise Asymptotic

Motivated by the question of how macromolecules assemble,the notion of an assembly tree of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies to the macromolecular assembly problem. Explicit formulas or generating functions are provided for the number of assembly trees of several families of graphs, in particular for what we call $(H,\phi)$-graphs.  In some natural special cases, we use a powerful recent result of Zeilberger and Apagodu to provide recurrence relations for the diagonal of the relevant multivariate generating functions, and we use a result of Wimp and Zeilberger to find very precise asymptotic formulae for the coefficients of these diagonals.  Future directions for reseach, as well as open questions, are suggested.

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