<i>q</i>-Functions and Distributions, Operational and Umbral Methods

Abstract

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.

Keywords

• operators theory 44A99, 47B99, 47A62
• umbral methods 05A40
• special functions 33C52, 33C65, 33C99, 33B10, 33B15
• Hermite polynomials 33C45
• calculus Tsallis 60E99
• Student’s distribution 60E05