THE LEGENDRE POLYNOMIALS AS A BASIS FOR BESSEL FUNCTIONS

Abstract: By using the concepts and the formalism of the Monomiality Principle, we introduce a generalization of Bessel functions. The starting point is represented by Legendre polynomials, presented as particular case of generalized Laguerre polynomials of two variables. From the well known properties of the ordinary Bessel functions, we derive similar relations for this family of Legendre-Bessel functions, in particular, by exploiting the monomiality properties of Legendre polynomials, we discuss some differential rules of this family of generalized Bessel functions.


Introduction
In a number of previous investigations ( [1], [2], [3]) we have deeply discussed the concepts and the related formalism of the Monomiality Principle; in particular we have shown that it is possible to derive many operational identities for a wide class of orthogonal polynomials and special functions.The Monomiality Principle allows also us to state a different point of view regarding the properties of some families of polynomials, as for example Hermite ( [4]) and Laguerre ( [5]) polynomials and, moreover, to study isospectral problems as in the case of Hermite-Bessel functions and Laguerre-Bessel functions ( [6]).
We remind that the generalized Hermite polynomials H n (x, y) and twovariable Laguerre polynomials L n (x, y) can be viewed as quasi-monomials with the following operators ( [2]): Where we have denoted with D−1 x the inverse of the derivative operator ( [7]).It has been shown that the Legendre polynomials ( [8]) can be introduced directly by using the multiplication and derivative operators acting on Hermite and Laguerre polynomials, in the sense that we can combine the above operators (eq.( 1)) to define new operators that can be used to view Legendre polynomials as quasi-monomials and at same time to define them.It is known that there exists a class of generalized Laguerre polynomials ( [8], [9]) with the following explicit form: where: By introducing the operators: after noting that: we can immediately write that: where we have used the identity: These generalized Laguerre polynomials can be reduced to the ordinary Legendre polynomials: by setting: By using the formalism of Monomiality Principle ( [1]), we can also derive the generating function of the above polynomials.Since we have observed that 2 L 0 (x, y) = 1, we immediately get: where t is a continuous parameter.By exploiting the exponential on the above relation, we have: Since the operators ty and 2t D−1 x ∂ ∂y do not commute, we have to compute their commutator: and then equation ( 10) could be written as: or, in a more convenient form: From the fact that: we finally obtain: that is the generating function of generalized Laguerre polynomials 2 L n (x, y).
It is immediately noted that ( [7]): to outline the common nature of the monomiality operators used, relatively to Hermite and Laguerre polynomials families.The Monomiality Principle operational techniques exploited for other classes of polynomials, as reminder for Hermite and Laguerre families, suggest to state further relations for the polynomials 2 L n (x, y).Directly from equations (3), we have: to allow us to state the following differential equation solved by polynomials 2 L n (x, y): or in differential form: that gives: Furthermore, we can also observe that the Tricomi function of order zero, of a suitable variable, reads: and then, from equation (15), we can write a different form of the generating function of polynomials 2 L n (x, y): and, in terms of ordinary Legendre polynomials: The Legendre polynomials presented in this section in terms of the special class of the generalized Laguerre polynomials of type 2 L n (x, y), could be used to introduce a further generalization of the p-based Bessel functions as discussed in the previous papers ( [6], [10]).In the following sections we will discuss the aspects and the related properties of the Legendre-Bessel functions.

Legendre-Bessel Functions
It has been shown that, in the framework of the monomiality principle, it is possible to construct a class of Bessel-type functions, by using different families of polynomials, recognized as quasi-monomial.In particular, we have discussed the properties of the Laguerre-Bessel functions ( [10]): and we have obtained some useful operational results, as for example the following differential equation, solved by the Bessel functions of type p J n (x): In the previous section, we have introduced a generalized family of Laguerre polynomials that could be treated as ordinary Legendre polynomials see eq. (...); we use these polynomials to introduce a further class of Bessel-type functions, by following an operational approach in according with the definition and the related techniques of monomiality principle.
We start to consider the classical first kind Bessel function ( [5]), defined as follows: where r is an integer, and the generating function is given by: By applying the differential isomorphism to the two-variable Laguerre polynomials of type 2 L n (x, y), we have: where M = y + 2 D−1 x ∂ ∂y and P = ∂ ∂y .
From relation (28) 1 , by substituting the isospectral values, we formally get: where L J n (x, y) denotes the Legendre-Bessel function to be defined.
By expliciting the exponential on the above equation, we have: that is: Since the two operators do not commute, it is necessary to compute their commutator ( [11]): By expliciting the r.h.s. of equation (39), we have: and, after setting m = n − 2s, we get: that allows us to state the explicit form of generalized Bessel function on Legendre basis: or, that is the same (eq.(...)): We can also deduce the explicit form of the above Legendre-Bessel function, by expanding directly their generating function, derived above (see eq. ( 39)).We have, in fact: and then: that is, after rearranging the indexes: which allows us to state the following recurrence relation for the generalized Legendre-Bessel functions: The recurrence relations stated above can be used to derive further interesting operational results for this kind of generalized Bessel functions, based on Legendre polynomials, that we will discuss in a forthcoming paper.It could also be noted that Legendre-Bessel functions when x = 0 reduce to (see equation ( 45)): and by noting the value of the polynomials of type 2 L n (x, y) when x = 0, allow us to explore different operational relations involving this kind of Bessel functions.The concepts and the related formalism exploited in this paper to construct interested generalizations of Bessel functions has been used to derive similar results by using as basis the Hermite polynomials (see [12], [10]), but in general it is possible to extend the procedure focusing on other families of polynomials, recognized as quasi-monomials as for instance generalized classes of Hermite polynomials (see [3], [13], [14]) or families belonging to the class of Chebyshev polynomials (see [15], [16], [17], [18]).