Pierpaolo Natalini

Some Inequalities for Modified Bessel Functions

We denote by and the Bessel functions of the first and third kinds, respectively. Motivated by the relevance of the function , , in many contexts of applied mathematics and, in particular, in some elasticity problems Simpson and Spector (1984), we establish new inequalities for . The results are based on the recurrence relations for and and the Turán-type inequalities for such functions. Similar investigations are developed to establish new inequalities for .

Generalizations of the Bernoulli and Appell polynomials

We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampe de Feriet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.

On some Turán-type inequalities

We prove Turán-type inequalities for some special functions by using a generalization of the Schwarz inequality.

The Dirichlet problem for the Laplace equation in a starlike domain of a Riemann surface

We consider the Dirichlet problem for the Laplace equation in a starlike domain, i.e. a domain which is normal with respect to a suitable polar co-ordinates system. Such a domain can be interpreted as a non-isotropically stretched unit circle. We write down the explicit solution in terms of a Fourier series whose coefficients are determined by solving an infinite system of linear equations depending on the boundary data. Numerical experiments show that the same method works even if the considered starlike domain belongs to a two-fold Riemann surface.

An extension of the bell polynomials

Abstract After recalling the most important properties and applications of the Bell polynomials, we introduce an extension of this special class of functions. More precisely, we consider the case of multicomposite functions, and we show connections with the ordinary Bell polynomials.

Symbolic Computation of Newton Sum Rules for the Zeros of Polynomial Eigenfunctions of Linear Differential Operators

A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients.

Supplements to known monotonicity results and inequalities for the gamma and incomplete gamma functions

We denote by and the gamma and the incomplete gamma functions, respectively. In this paper we prove some monotonicity results for the gamma function and extend, to, a lower bound established by Elbert and Laforgia (2000) for the function, with, only for.

Fourier solution of the wave equation for a star-like-shaped vibrating membrane

The Fourier solution of the wave equation for a circular vibrating membrane is generalized to a star-like-shaped structure. We show that the classical solution can be used in this more general case, provided that a suitable change of variables in the spherical co-ordinate system is performed.

Laguerre-type Bell polynomials

We develop an extension of the classical Bell polynomials introducing the Laguerre-type version of this well-known mathematical tool. The Laguerre-type Bell polynomials are useful in order to compute the nth Laguerre-type derivatives of a composite function. Incidentally, we generalize a result considered by L. Carlitz in order to obtain explicit relationships between Bessel functions and generalized hypergeometric functions.

Particular solutions for a class of ODE related to the L-exponential functions

Abstract Particular solutions of a class of higher order ordinary differential equations, with non-constant coefficients, are determined by using the properties of the Laguerre exponentials functions introduced by G. Dattoli and P. E. Ricci in [Georgian Math. J. 10: 481–494, 2003]