We use the operator method to evaluate a class of integrals involving Bessel or Besseltype functi... more We use the operator method to evaluate a class of integrals involving Bessel or Besseltype functions. The technique we propose is based on the formal reduction of these family of functions to Gaussians.
We reformulate the theory of Legendre polynomials using the method of integral transforms, which ... more We reformulate the theory of Legendre polynomials using the method of integral transforms, which allow us to express them in terms of Hermite polynomials. We show that this allows a self consistent point of view to their relevant properties and the possibility of framing generalized forms like the Humbert polynomials within the same framework. The multi-index multi-variable case is touched on.
ABSTRACT We introduce general set of relativistic Laguerre polynomials (RLP) by generalizing the ... more ABSTRACT We introduce general set of relativistic Laguerre polynomials (RLP) by generalizing the hypergeometric differential equation by means of which the RLP are generalized in a paper by P. Natalini [Rend. Mat. Appl., VII. Ser. 16(2), 299–313 (1996; Zbl 0866.33010)].
We combine the Lie algebraic methods and the technicalities associated with the monomialty princi... more We combine the Lie algebraic methods and the technicalities associated with the monomialty principle to obtain new results concerning Legendre polynomial expansions.
The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials a... more The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials and the so called Appel numbers. The relevant formula generalizes both the Euler-MacLaurin quadrature rule and a similar rule using Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the endpoints of the considered interval. In the general case, the remainder term is expressed in terms of Appel numbers, and all derivatives appear. A numerical example is also included. operators.
We show that the combination of the formalism underlying the principle of monomiality and of meth... more We show that the combination of the formalism underlying the principle of monomiality and of methods of an algebraic nature allows the solution of different families of partial differential equations. Here we use different realizations of the Heisenberg–Weyl algebra and show that a Sheffer type realization leads to the extension of the method to finite difference and integro-differential equations.
Journal of Computational and Applied Mathematics, 1994
There is a set of orthogonal polynomials {g n (x)} which plays a relevant role in the treatment o... more There is a set of orthogonal polynomials {g n (x)} which plays a relevant role in the treatment of the case of anisotropic scattering in neutron-transport and radiative-transfer theories. They appear also in the spherical harmonics treatment of the isotropic scattering. These polynomials are orthogonal with respect to a weight function which is continuos in the interval [−1, +1] and has a finite number of symmetric Dirac masses. Although some other structural properties of these polynomials (e.g. the three-term recurrence relation) as well as some properties of their zeros have been published, much more need to be known. In particular neither the second order differential equation nor the density of zeros (i.e. the number of zeros per unit of interval) of the polynomial g n (x) have been found. Here we obtain the second order differential equation in the case that these polynomials are hypergeometric, so remaining open the general case. Furthermore, the exact expressions of the moments around the origin of the density of zeros of gn (x) are given in the general case. The asymptotic density of zeros is also pointed out. Finally, these polynomials are shown to belong to the Nevai's class.
International Journal of Mathematics and Mathematical Sciences, 2012
ABSTRACT By using the integral transform method, we introduce some non-Sheffer polynomial sets. F... more ABSTRACT By using the integral transform method, we introduce some non-Sheffer polynomial sets. Furthermore, we show how to compute the connection coefficients for particular expressions of Appell polynomials.
ABSTRACT We discuss the properties of Laguerre orthogonal functions by means of an operational po... more ABSTRACT We discuss the properties of Laguerre orthogonal functions by means of an operational point of view, which is shown to be a fairly powerful tool to investigate the relevant properties and further generalizations.
ABSTRACT We show that under appropriate conditions the Hermite polynomials, with more than two va... more ABSTRACT We show that under appropriate conditions the Hermite polynomials, with more than two variables, belong to bi-orthogonal sets. We extend the result to the case of Bell-type polynomials.
A representation formula (by means of the generalized Lucas Polynomials of first kind) for the mo... more A representation formula (by means of the generalized Lucas Polynomials of first kind) for the moments of the density of zeros of Orthogonal Polynomial Sets defined by a threeterm recurrence relation is used to derive information directly from the entries of the Jacobi matrix. Numerical computations are explicitly developed in some particular case.
We extend the technique of the evolution operator to matrix differential equations. It is shown t... more We extend the technique of the evolution operator to matrix differential equations. It is shown that the combined use of the Cayley-Hamilton theorem and of nonstandard special functions may provide a new form of solutions for a wide family of this type of equation.Q
We use Laguerre and Hermite polynomials to show that the monomiality principle can be exploited t... more We use Laguerre and Hermite polynomials to show that the monomiality principle can be exploited to study the properties of the polynomials and of the associated biorthogonal functions.
The formalism underlying the concept of negative derivatives is proved to be a powerful tool to s... more The formalism underlying the concept of negative derivatives is proved to be a powerful tool to study new series expansions of special functions defined by integral representations. Its importance is also discussed within the context of multiple integrations and of integral equations.
In the framework of the ‘monomiality principle’, we introduce a class of Bessel-type functions wh... more In the framework of the ‘monomiality principle’, we introduce a class of Bessel-type functions which can be derived by applying the properties of an isomorphism, related to the so called Laguerre-type exponentials. Further extensions and possible generalizations to the multivariable case are mentioned.
We use the operator method to evaluate a class of integrals involving Bessel or Besseltype functi... more We use the operator method to evaluate a class of integrals involving Bessel or Besseltype functions. The technique we propose is based on the formal reduction of these family of functions to Gaussians.
We reformulate the theory of Legendre polynomials using the method of integral transforms, which ... more We reformulate the theory of Legendre polynomials using the method of integral transforms, which allow us to express them in terms of Hermite polynomials. We show that this allows a self consistent point of view to their relevant properties and the possibility of framing generalized forms like the Humbert polynomials within the same framework. The multi-index multi-variable case is touched on.
ABSTRACT We introduce general set of relativistic Laguerre polynomials (RLP) by generalizing the ... more ABSTRACT We introduce general set of relativistic Laguerre polynomials (RLP) by generalizing the hypergeometric differential equation by means of which the RLP are generalized in a paper by P. Natalini [Rend. Mat. Appl., VII. Ser. 16(2), 299–313 (1996; Zbl 0866.33010)].
We combine the Lie algebraic methods and the technicalities associated with the monomialty princi... more We combine the Lie algebraic methods and the technicalities associated with the monomialty principle to obtain new results concerning Legendre polynomial expansions.
The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials a... more The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials and the so called Appel numbers. The relevant formula generalizes both the Euler-MacLaurin quadrature rule and a similar rule using Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the endpoints of the considered interval. In the general case, the remainder term is expressed in terms of Appel numbers, and all derivatives appear. A numerical example is also included. operators.
We show that the combination of the formalism underlying the principle of monomiality and of meth... more We show that the combination of the formalism underlying the principle of monomiality and of methods of an algebraic nature allows the solution of different families of partial differential equations. Here we use different realizations of the Heisenberg–Weyl algebra and show that a Sheffer type realization leads to the extension of the method to finite difference and integro-differential equations.
Journal of Computational and Applied Mathematics, 1994
There is a set of orthogonal polynomials {g n (x)} which plays a relevant role in the treatment o... more There is a set of orthogonal polynomials {g n (x)} which plays a relevant role in the treatment of the case of anisotropic scattering in neutron-transport and radiative-transfer theories. They appear also in the spherical harmonics treatment of the isotropic scattering. These polynomials are orthogonal with respect to a weight function which is continuos in the interval [−1, +1] and has a finite number of symmetric Dirac masses. Although some other structural properties of these polynomials (e.g. the three-term recurrence relation) as well as some properties of their zeros have been published, much more need to be known. In particular neither the second order differential equation nor the density of zeros (i.e. the number of zeros per unit of interval) of the polynomial g n (x) have been found. Here we obtain the second order differential equation in the case that these polynomials are hypergeometric, so remaining open the general case. Furthermore, the exact expressions of the moments around the origin of the density of zeros of gn (x) are given in the general case. The asymptotic density of zeros is also pointed out. Finally, these polynomials are shown to belong to the Nevai's class.
International Journal of Mathematics and Mathematical Sciences, 2012
ABSTRACT By using the integral transform method, we introduce some non-Sheffer polynomial sets. F... more ABSTRACT By using the integral transform method, we introduce some non-Sheffer polynomial sets. Furthermore, we show how to compute the connection coefficients for particular expressions of Appell polynomials.
ABSTRACT We discuss the properties of Laguerre orthogonal functions by means of an operational po... more ABSTRACT We discuss the properties of Laguerre orthogonal functions by means of an operational point of view, which is shown to be a fairly powerful tool to investigate the relevant properties and further generalizations.
ABSTRACT We show that under appropriate conditions the Hermite polynomials, with more than two va... more ABSTRACT We show that under appropriate conditions the Hermite polynomials, with more than two variables, belong to bi-orthogonal sets. We extend the result to the case of Bell-type polynomials.
A representation formula (by means of the generalized Lucas Polynomials of first kind) for the mo... more A representation formula (by means of the generalized Lucas Polynomials of first kind) for the moments of the density of zeros of Orthogonal Polynomial Sets defined by a threeterm recurrence relation is used to derive information directly from the entries of the Jacobi matrix. Numerical computations are explicitly developed in some particular case.
We extend the technique of the evolution operator to matrix differential equations. It is shown t... more We extend the technique of the evolution operator to matrix differential equations. It is shown that the combined use of the Cayley-Hamilton theorem and of nonstandard special functions may provide a new form of solutions for a wide family of this type of equation.Q
We use Laguerre and Hermite polynomials to show that the monomiality principle can be exploited t... more We use Laguerre and Hermite polynomials to show that the monomiality principle can be exploited to study the properties of the polynomials and of the associated biorthogonal functions.
The formalism underlying the concept of negative derivatives is proved to be a powerful tool to s... more The formalism underlying the concept of negative derivatives is proved to be a powerful tool to study new series expansions of special functions defined by integral representations. Its importance is also discussed within the context of multiple integrations and of integral equations.
In the framework of the ‘monomiality principle’, we introduce a class of Bessel-type functions wh... more In the framework of the ‘monomiality principle’, we introduce a class of Bessel-type functions which can be derived by applying the properties of an isomorphism, related to the so called Laguerre-type exponentials. Further extensions and possible generalizations to the multivariable case are mentioned.
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Papers by B. Germano