scholarly journals q-Functions and Distributions, Operational and Umbral Methods

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2664
Author(s):  
Giuseppe Dattoli ◽  
Silvia Licciardi ◽  
Bruna Germano ◽  
Maria Renata Martinelli
Keyword(s):  
Special Functions ◽  
Operational Calculus ◽  
Analytical Tool ◽  
Gaussian Distributions

The use of non-standard calculus means have been proven to be extremely powerful for studying old and new properties of special functions and polynomials. These methods have helped to frame either elementary and special functions within the same logical context. Methods of Umbral and operational calculus have been embedded in a powerful and efficient analytical tool, which will be applied to the study of the properties of distributions such as Tsallis, Weibull and Student’s. We state that they can be viewed as standard Gaussian distributions and we take advantage of the relevant properties to infer those of the aforementioned distributions.

From the hyper-Bessel operators of Dimovski to the generalized fractional calculus

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Differential Operators ◽  
Integral Transform ◽  
Special Functions ◽  
Operational Calculus ◽  
Integral Transforms ◽  
Bessel Operators ◽  
Generalized Fractional Calculus ◽  
Basic Facts ◽  
Fox’S H Function

AbstractIn 1966 Ivan Dimovski introduced and started detailed studies on the Bessel type differential operators B of arbitrary (integer) order m ≥ 1. He also suggested a variant of the Obrechkoff integral transform (arising in a paper of 1958 by another Bulgarian mathematician Nikola Obrechkoff) as a Laplace-type transform basis of a corresponding operational calculus for B and for its linear right inverse integral operator L. Later, the developments on these linear singular differential operators appearing in many problems of mathematical physics, have been continued by the author of this survey who called them hyper-Bessel differential operators, in relation to the notion of hyper-Bessel functions of Delerue (1953), shown to form a fundamental system of solutions of the IVPs for By(t) = λy(t). We have been able to extend Dimovski’s results on the hyper-Bessel operators and on the Obrechkoff transform due to the happy hint to attract the tools of the special functions as Meijer’s G-function and Fox’s H-function to handle successfully these matters. These author’s studies have lead to the introduction and development of a theory of generalized fractional calculus (GFC) in her monograph (1994) and subsequent papers, and to various applications of this GFC in other topics of analysis, differential equations, special functions and integral transforms.Here we try briefly to expose the ideas leading to this GFC, its basic facts and some of the mentioned applications.

scholarly journals An Analytic Exact form of Heaviside Function

2019 ◽  
Author(s):  
John Venetis
Keyword(s):  
Special Functions ◽  
Error Function ◽  
Operational Calculus ◽  
Delta Function ◽  
Step Function ◽  
Heaviside Function ◽  
Exact Form ◽  
Dirac Delta ◽  
Computational Procedures ◽  
Engineering Practices

In this paper, the author obtains an analytic exact form of Heaviside function, which is also known as Unit Step function and constitutes a fundamental concept of the Operational Calculus.In particulat, this function is explicitly expressed in a very simple manner by the aid of purely algebraic representations. The novelty of this work is that the proposed explicit formula is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Hence, it may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

scholarly journals An Explicit Form of Heaviside Step Function

2021 ◽  
Author(s):  
John Venetis
Keyword(s):  
Explicit Form ◽  
Special Function ◽  
Special Functions ◽  
Error Function ◽  
Beta Function ◽  
Operational Calculus ◽  
Step Function ◽  
Heaviside Step Function ◽  
Computational Procedures ◽  
Engineering Practices

In this paper, the author derives an explicit form of Heaviside Step Function, which evidently constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics.In particular, this special function is exhibited in a very simple manner as a summation of four inverse tangent functions. The novelty of this work is that the proposed exact formulae are not performed in terms of miscellaneous special functions, e.g. Bessel functions, Error function, Beta function etc and also are neither the limit of a function, nor the limit of a sequence of functions with point – wise or uniform convergence.Hence, this formula may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

LPCD framework: Analytical tool or psychological model?

2018 ◽  
Vol 41 ◽  
Author(s):  
David Danks
Keyword(s):  
Decision Making ◽  
Analytical Tool ◽  
Mathematical Framework ◽  
Bayesian Decision ◽  
Perceptual Decision Making ◽  
Psychological Model ◽  
Psychological Reality ◽  
Psychological Theories ◽  
Target Article ◽  
Bayesian Decision Making

AbstractThe target article uses a mathematical framework derived from Bayesian decision making to demonstrate suboptimal decision making but then attributes psychological reality to the framework components. Rahnev & Denison's (R&D) positive proposal thus risks ignoring plausible psychological theories that could implement complex perceptual decision making. We must be careful not to slide from success with an analytical tool to the reality of the tool components.

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