scholarly journals Degenerate Stirling Polynomials of the Second Kind and Some Applications

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1046 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Jongkyum Kwon
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Probability Distributions ◽  
Random Variables

Recently, the degenerate λ -Stirling polynomials of the second kind were introduced and investigated for their properties and relations. In this paper, we continue to study the degenerate λ -Stirling polynomials as well as the r-truncated degenerate λ -Stirling polynomials of the second kind which are derived from generating functions and Newton’s formula. We derive recurrence relations and various expressions for them. Regarding applications, we show that both the degenerate λ -Stirling polynomials of the second and the r-truncated degenerate λ -Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables.

Generalizations of the elementary renewal theorem to distributions defined by concave recurrence relations

1977 ◽  
Vol 14 (3) ◽  
pp. 516-526
Author(s):  
W. Reh
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Probability Distributions ◽  
Branching Processes ◽  
General Procedure ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Probability Generating Functions ◽  
Dependent Probability

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.

scholarly journals Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

2008 ◽  
Vol 45 (02) ◽  
pp. 531-541 ◽  
Author(s):  
A. G. Rossberg
Keyword(s):  
Laplace Transform ◽  
High Precision ◽  
Computational Efficiency ◽  
Generating Functions ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Random Variables ◽  
Numerical Examples ◽  
Pareto Distributions ◽  
The Laplace Transform

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

Generalizations of the elementary renewal theorem to distributions defined by concave recurrence relations

1977 ◽  
Vol 14 (03) ◽  
pp. 516-526
Author(s):  
W. Reh
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Probability Distributions ◽  
Branching Processes ◽  
General Procedure ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Probability Generating Functions ◽  
Dependent Probability

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.

scholarly journals Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

2008 ◽  
Vol 45 (2) ◽  
pp. 531-541 ◽  
Author(s):  
A. G. Rossberg
Keyword(s):  
Laplace Transform ◽  
High Precision ◽  
Computational Efficiency ◽  
Generating Functions ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Random Variables ◽  
Numerical Examples ◽  
Pareto Distributions ◽  
The Laplace Transform

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

scholarly journals The extended generalized half logistic distribution based on ordered random variables

2015 ◽  
Vol 46 (3) ◽  
pp. 245-256 ◽  
Author(s):  
Devendra Kumar
Keyword(s):  
Order Statistics ◽  
Generating Functions ◽  
Conditional Expectation ◽  
Recurrence Relations ◽  
Random Variables ◽  
Record Values ◽  
Joint Moment ◽  
Generalized Order Statistics ◽  
Logistic Distribution ◽  
Generalized Order

In this paper, we considered an extended generalized half logistic distribution and derived some explicit expressions and recurrence relations for marginal and joint moment generating functions of generalized order statistics from extended generalized half logistic distribution. The results for record values and order statistics are deduced from the result. Finally, we obtained the characterizing result of this distribution on using the conditional expectation of generalized order statistics.

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.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

scholarly journals n-Color partitions with weighted differences equal to minus two

1997 ◽  
Vol 20 (4) ◽  
pp. 759-768 ◽  
Author(s):  
A. K. Agarwal ◽  
R. Balasubrananian
Keyword(s):  
Explicit Expression ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Combinatorial Identities ◽  
Combinatorial Identity ◽  
Divisor Function ◽  
Open Problems

In this paper we study thosen-color partitions of Agarwal and Andrews, 1987, in which each pair of parts has weighted difference equal to−2Results obtained in this paper for these partitions include several combinatorial identities, recurrence relations, generating functions, relationships with the divisor function and computer produced tables. By using these partitions an explicit expression for the sum of the divisors of odd integers is given. It is shown how these partitions arise in the study of conjugate and self-conjugaten-color partitions. A combinatorial identity for self-conjugaten-color partitions is also obtained. We conclude by posing several open problems in the last section.

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