Fractional calculus and generalized Rodrigues formula
The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers. The two generalizations of the Rodrigues formula of the Laguerre polynomials L"@a^@b(x)=(x^-^@b/n!)e^xD^@ae^-^xx^n^+^@b, and L"@a^@b(x)=((x^-^@be^x)/@C([email protected]))D^@ae^-^xx^@a^+^@b, are defined in [Math. Sci. Res. Hot-line 1 (10) (1997) 7; Appl. Math. Comput. 106 (1) (1999) 51] and some of their properties are proved. Here we define the new special function L"@a^@b(@c,a;x) based on a generalization of the Rodrigues formula, then we study some of its properties, some recurrence relations and prove that the set of functions {L"@a^@b(@c,a;x),@[email protected]?R} is continuous as a function of @[email protected]?R. The continuation as @a,@c->n and a=1 to the Rodrigues formula of the Laguerre polynomials L"n^@b(x) are proved. Also the confluent hypergeometric representation will be given.