Laguerre transform

The integral transform

where is the Laguerre polynomial (cf. Laguerre polynomials) of degree . The inversion formula is

if the series converges. If is continuous, is piecewise continuous on and , , , then

If and are continuous, is piecewise continuous on and , , , then

If is piecewise continuous on and , , , then for

and for ,

Suppose that and are piecewise continuous on and that

Then

The generalized Laguerre transform is

where is the generalized Laguerre polynomial (see [4]).

References

[1]	A.G. Zemanian, "Generalized integral transformations" , Interscience (1968)
[2]	J. McCully, "The Laguerre transform" SIAM Rev. , 2 : 3 (1960) pp. 185–191
[3]	L. Debnath, "On Laguerre transform" Bull. Calcutta Math. Soc. , 52 : 2 (1960) pp. 69–77
[4]	Yu.A. Brychkov, A.P. Prudnikov, "Operational calculus" Progress in Math. , 1 (1968) pp. 1–74 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 7–82