Self-Reciprocal Functions Involving Laguerre Polynomials

The object of this note is to prove that the function

x^-1/2e^-1/2x2L^1/2r-1/2_n(1/2x²)

is self-reciprocal for J_r transforms, where n is a positive integer.

Adopting a similar procedure, I also obtain the result that the function

x^r+2n+1/2e^-1/2x2L^n+r_n(1/2x²),

where n is a positive integer, is self-reciprocal for J_r transforms, a theorem proved otherwise by Howell.