991 research outputs found
The Ramanujan master theorem and its implications for special functions
We study a number of possible extensions of the Ramanujan master theorem,
which is formulated here by using methods of Umbral nature. We discuss the
implications of the procedure for the theory of special functions, like the
derivation of formulae concerning the integrals of products of families of
Bessel functions and the successive derivatives of Bessel type functions. We
stress also that the procedure we propose allows a unified treatment of many
problems appearing in applications, which can formally be reduced to the
evaluation of exponential- or Gaussian-like integrals.Comment: 12 page
Definite integrals and operational methods
An operatorial method, already employed to formulate a generalization of the
Ramanujan master theorem, is applied to the evaluation of integrals of various
type. This technique provide a very flexible and powerful tool yielding new
results encompassing various aspects of the special function theory.Comment: 9 pages; minor changes to match published versio
Operator solutions for fractional Fokker-Planck equations
We obtain exact results for fractional equations of Fokker-Planck type using
evolution operator method. We employ exact forms of one-sided Levy stable
distributions to generate a set of self-reproducing solutions. Explicit cases
are reported and studied for various fractional order of derivatives, different
initial conditions, and for different versions of Fokker-Planck operators.Comment: 4 pages, 3 figure
Spectral density of generalized Wishart matrices and free multiplicative convolution
We investigate the level density for several ensembles of positive random
matrices of a Wishart--like structure, , where stands for a
nonhermitian random matrix. In particular, making use of the Cauchy transform,
we study free multiplicative powers of the Marchenko-Pastur (MP) distribution,
, which for an integer yield Fuss-Catalan
distributions corresponding to a product of independent square random
matrices, . New formulae for the level densities are derived
for and . Moreover, the level density corresponding to the
generalized Bures distribution, given by the free convolution of arcsine and MP
distributions is obtained. We also explain the reason of such a curious
convolution. The technique proposed here allows for the derivation of the level
densities for several other cases.Comment: 10 latex pages including 4 figures, Ver 4, minor improvements and
references updat
- …