This paper presents a brief overview of the life story and professional career of Prof. R. Gorenf... more This paper presents a brief overview of the life story and professional career of Prof. R. Gorenflo — a well-known mathematician, an expert in the field of Differential and Integral Equations, Numerical Mathematics, Fractional Calculus and Applied Analysis, an interesting conversational partner, an experienced colleague, and a real friend. Especially his role in the modern Fractional Calculus and its applications is highlighted.
In this expository paper we first survey the method of Mellin-Barnes
integrals to represent the ... more In this expository paper we first survey the method of Mellin-Barnes
integrals to represent the \alpha stable Le'vy distributions in probability theory
(0 < \alpha <= 2). These integrals are known to be useful for obtaining
convergent and asymptotic series representations of the corresponding probability
density functions. The novelty concerns the convolution between two stable
probability densities of different Levy index, which turns to be a
probability law of physical interest, even if it is no longer stable and self-similar.
A particular but interesting case of convolution is obtained combining the
Cauchy-Lorentz density (\alpha = 1) with the Gaussian density (\alpha = 2) that
yields the so-called Voigt profile. Our machinery can be applied to derive
the fundamental solutions of space-fractional diffusion equations of two
orders.
Studies of Nonlinear Phenomena in Life Science, 1999
From the point of view of the general theory of the hyper-Bessel operators, we consider a particu... more From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument �
A generalization of the canonical coherent states of a quantum harmonic oscillator has been perfo... more A generalization of the canonical coherent states of a quantum harmonic oscillator has been performed by requiring the conditions of normalizability, continuity in the label, and resolution of the identity operator with a positive weight function. Relying on this approach, in the present scenario, coherent states are generalized over the canonical or finite dimensional Fock space of the harmonic oscillator. A class of generalized coherent states is determined such that the corresponding distributions of the number of excitations depart from the Poisson statistics according to combinations of stretched exponential decays, power laws, and logarithmic forms. The analysis of the Mandel parameter shows that the generalized coherent states exhibit (non-classical) sub-Poissonian or super-Poissonian statistics of the number of excitations, based on the realization of determined constraints. Mittag-Leffler and Wright generalized coherent states are analyzed as particular cases.
This paper presents a brief overview of the life story and professional career of Prof. R. Gorenf... more This paper presents a brief overview of the life story and professional career of Prof. R. Gorenflo — a well-known mathematician, an expert in the field of Differential and Integral Equations, Numerical Mathematics, Fractional Calculus and Applied Analysis, an interesting conversational partner, an experienced colleague, and a real friend. Especially his role in the modern Fractional Calculus and its applications is highlighted.
In this expository paper we first survey the method of Mellin-Barnes
integrals to represent the ... more In this expository paper we first survey the method of Mellin-Barnes
integrals to represent the \alpha stable Le'vy distributions in probability theory
(0 < \alpha <= 2). These integrals are known to be useful for obtaining
convergent and asymptotic series representations of the corresponding probability
density functions. The novelty concerns the convolution between two stable
probability densities of different Levy index, which turns to be a
probability law of physical interest, even if it is no longer stable and self-similar.
A particular but interesting case of convolution is obtained combining the
Cauchy-Lorentz density (\alpha = 1) with the Gaussian density (\alpha = 2) that
yields the so-called Voigt profile. Our machinery can be applied to derive
the fundamental solutions of space-fractional diffusion equations of two
orders.
Studies of Nonlinear Phenomena in Life Science, 1999
From the point of view of the general theory of the hyper-Bessel operators, we consider a particu... more From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument �
A generalization of the canonical coherent states of a quantum harmonic oscillator has been perfo... more A generalization of the canonical coherent states of a quantum harmonic oscillator has been performed by requiring the conditions of normalizability, continuity in the label, and resolution of the identity operator with a positive weight function. Relying on this approach, in the present scenario, coherent states are generalized over the canonical or finite dimensional Fock space of the harmonic oscillator. A class of generalized coherent states is determined such that the corresponding distributions of the number of excitations depart from the Poisson statistics according to combinations of stretched exponential decays, power laws, and logarithmic forms. The analysis of the Mandel parameter shows that the generalized coherent states exhibit (non-classical) sub-Poissonian or super-Poissonian statistics of the number of excitations, based on the realization of determined constraints. Mittag-Leffler and Wright generalized coherent states are analyzed as particular cases.
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Papers by Francesco Mainardi
integrals to represent the \alpha stable Le'vy distributions in probability theory
(0 < \alpha <= 2). These integrals are known to be useful for obtaining
convergent and asymptotic series representations of the corresponding probability
density functions. The novelty concerns the convolution between two stable
probability densities of different Levy index, which turns to be a
probability law of physical interest, even if it is no longer stable and self-similar.
A particular but interesting case of convolution is obtained combining the
Cauchy-Lorentz density (\alpha = 1) with the Gaussian density (\alpha = 2) that
yields the so-called Voigt profile. Our machinery can be applied to derive
the fundamental solutions of space-fractional diffusion equations of two
orders.
integrals to represent the \alpha stable Le'vy distributions in probability theory
(0 < \alpha <= 2). These integrals are known to be useful for obtaining
convergent and asymptotic series representations of the corresponding probability
density functions. The novelty concerns the convolution between two stable
probability densities of different Levy index, which turns to be a
probability law of physical interest, even if it is no longer stable and self-similar.
A particular but interesting case of convolution is obtained combining the
Cauchy-Lorentz density (\alpha = 1) with the Gaussian density (\alpha = 2) that
yields the so-called Voigt profile. Our machinery can be applied to derive
the fundamental solutions of space-fractional diffusion equations of two
orders.