TY  - JOUR
A1  - Apelblat, Alexander
A1  - Consiglio, Armando
A1  - Mainardi, Francesco
T1  - The Bateman functions revisited after 90 years — a survey of old and new results
JF  - Mathematics
N2  - The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature.
KW  - bateman functions
KW  - havelock functions
KW  - integral-bateman functions
KW  - confluent hypergeometric functions
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-240970
SN  - 2227-7390
VL  - 9
IS  - 11
ER  - 
TY  - JOUR
A1  - Mainardi, Francesco
A1  - Consiglio, Armando
T1  - The Wright Functions of the Second Kind in Mathematical Physics
JF  - Mathematics
N2  - In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices, we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind.
KW  - fractional calculus
KW  - Wright functions
KW  - Green’s functions
KW  - diffusion-wave equation
KW  - Laplace transform
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-207782
SN  - 2227-7390
VL  - 8
IS  - 6
ER  -