Special functions, infinite divisibility and transcendental equations

1979 ◽  
Vol 85 (3) ◽  
pp. 453-464 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
C. Ping May
Keyword(s):  
Bessel Functions ◽  
Special Functions ◽  
Infinite Divisibility ◽  
Probability Density Functions ◽  
Integral Representations ◽  
Density Functions ◽  
Modified Bessel Functions ◽  
Explicit Formulas ◽  
Linear Combinations ◽  
Transcendental Equations

AbstractWe establish integral representations for quotients of Tricomi ψ functions and of several quotients of modified Bessel functions and of linear combinations of them. These integral representations are used to prove the complete monotonicity of the functions considered and to prove the infinite divisibility of a three parameter probability distribution. Limiting cases of this distribution are the hitting time distributions considered recently by Kent and Wendel. We also derive explicit formulas for the Kent–Wendel probability density functions.

scholarly journals TURÁN TYPE INEQUALITIES FOR MODIFIED BESSEL FUNCTIONS

2010 ◽  
Vol 82 (2) ◽  
pp. 254-264 ◽  
Author(s):  
ÁRPÁD BARICZ
Keyword(s):  
Bessel Functions ◽  
Infinite Divisibility ◽  
Modified Bessel Functions ◽  
Explicit Formulas ◽  
Turán Type Inequalities

AbstractIn this paper our aim is to deduce some sharp Turán type inequalities for modified Bessel functions of the first and second kinds. Our proofs are based on explicit formulas for the Turánians of the modified Bessel functions of the first and second kinds and on a formula which is related to the infinite divisibility of the Student t-distribution.

A Comparison of Some Multivariate Weibull Distributions

2010 ◽  
Author(s):  
Bernt J. Leira
Keyword(s):  
Probability Density ◽  
Density Function ◽  
Statistical Models ◽  
Mechanical Design ◽  
Bending Moment ◽  
Probability Density Functions ◽  
Density Functions ◽  
Weibull Distributions ◽  
Linear Combinations ◽  
Simplified Procedure

Three different possible choices of statistical models for multivariate Weibull distributions are considered and compared. The concept of “a correlation field” is introduced and is subsequently applied for the purpose of comparing the different models. Linear combinations of Weibull distributed random variables are considered, and expressions for the corresponding probability density functions are established. Furthermore, a simplified procedure for approximating the resulting density function is described. Comparison is made between the statistical moments of increasing order for the specific case of two Weibull components. This example of application arises e.g. in connection with mechanical design of a column which is subjected to a bi-axial bending moment.

scholarly journals Analytic approximations for special functions, applied to the modified Bessel functions I2(x) and I2/3(x)

2018 ◽  
Vol 11 ◽  
pp. 1028-1033 ◽  
Author(s):  
Pablo Martin ◽  
Jorge Olivares ◽  
Fernando Maass ◽  
Elvis Valero
Keyword(s):  
Bessel Functions ◽  
Special Functions ◽  
Modified Bessel Functions ◽  
Analytic Approximations

scholarly journals NUMERICAL CALCULATION OF BESSEL FUNCTIONS

2012 ◽  
Vol 23 (12) ◽  
pp. 1250084 ◽  
Author(s):  
CHARLES SCHWARTZ
Keyword(s):  
Numerical Calculation ◽  
Bessel Functions ◽  
Special Functions ◽  
Computational Procedure ◽  
Mathematical Physics ◽  
Trapezoidal Rule ◽  
Integral Representations ◽  
The Many

A new computational procedure is offered to provide simple, accurate and flexible methods for using modern computers to give numerical evaluations of the various Bessel functions. The trapezoidal rule, applied to suitable integral representations, may become the method of choice for evaluation of the many special functions of mathematical physics.

scholarly journals Extended Mellin integral representations for the absolute value of the gamma function

Analysis ◽  
2018 ◽  
Vol 38 (1) ◽  
pp. 11-20
Author(s):  
Nicolas Privault
Keyword(s):  
Fourier Transform ◽  
Gamma Function ◽  
Bessel Functions ◽  
Planck Equation ◽  
Integral Representations ◽  
Modified Bessel Functions ◽  
Absolute Value ◽  
Fokker Planck Equation ◽  
The Absolute ◽  
Fokker Planck

AbstractWe derive Mellin integral representations in terms of Macdonald functions for the squared absolute value{s\mapsto|\Gamma(a+is)|^{2}}of the gamma function and its Fourier transform when{a<0}is non-integer, generalizing known results in the case{a>0}. This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of a Fokker–Planck equation.

Calculating exit times for series Jackson networks

1987 ◽  
Vol 24 (1) ◽  
pp. 226-234 ◽  
Author(s):  
William A. Massey
Keyword(s):  
Bessel Functions ◽  
Transition Probabilities ◽  
Special Functions ◽  
Exit Time ◽  
Modified Bessel Functions ◽  
Exit Times ◽  
Jackson Network ◽  
Jackson Networks ◽  
Positive Orthant ◽  
New Family

We define a new family of special functions that we call lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N = 1, and the exponential function when N = 0. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at 0) can be solved in terms of modified Bessel functions. In this paper, we use lattice Bessel functions to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow us to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit-time distribution.

Calculating exit times for series Jackson networks

1987 ◽  
Vol 24 (01) ◽  
pp. 226-234 ◽  
Author(s):  
William A. Massey
Keyword(s):  
Bessel Functions ◽  
Transition Probabilities ◽  
Special Functions ◽  
Exit Time ◽  
Modified Bessel Functions ◽  
Exit Times ◽  
Jackson Network ◽  
Jackson Networks ◽  
Positive Orthant ◽  
New Family

We define a new family of special functions that we call lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N = 1, and the exponential function when N = 0. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at 0) can be solved in terms of modified Bessel functions. In this paper, we use lattice Bessel functions to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow us to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit-time distribution.

scholarly journals On Applications of Some Special Functions in Statistics

2020 ◽  
Vol 8 (6) ◽  
pp. 1902-1908
Keyword(s):  
Characteristic Function ◽  
Laplace Transform ◽  
Bessel Functions ◽  
Special Functions ◽  
Probability Distributions ◽  
Laguerre Polynomials ◽  
Real Life ◽  
Moment Generating Function ◽  
Cumulative Density Function ◽  
Modified Bessel Functions

In this paper we will introduce some probability distributions with help of some special functions like Gamma, kGamma functions, Beta, k-Beta functions, Bessel, modified Bessel functions and Laguerre polynomials and in mathematical analysis used Laplace transform. We will also obtain their cumulative density function, expected value, variance, Moment generating function and Characteristic function. Some characteristics and real life applications will be computed in tabulated for these distributions

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