On the generalized Laguerre polynomials of arbitrary (fractional) orders and quantum mechanics

The generalized Laguerre polynomials (L) over bar(alpha)(beta)(x) of arbitrary order alpha is an element of R have been defined by the author (El-Sayed 1997 Math. Sci. Res. Not-line 17-14, 1999 to appear). In the latter reference it is proved that they are continuous as functions of alpha, alpha is an element of R, and some other properties that generalize (interpolate) those of the classical Laguerre polynomials L-n(beta) (x), n = 1, 2,... have been proved. Here we prove that (L) over bar(alpha)(beta)(x) alpha is an element of R are orthogonal in L-2(0, infinity) and are particular solutions of the differential equation x D-2 u (x) + (1 + beta - x) Du (x) + alpha u (x) = 0 generalizing the one for L-n(beta) (x), n = 1, 2,... Also some applications in quantum mechanics are discussed.