Following Torre (2011b), we will see in Section 10.7 that the preceding can be obtained by composing Appell transformations with scalings and shifts of both variables. This is a restatement in the optical context of the result established in Leutwiler (1988) (see also Shimomura (2000)) in relation to the symmetry transformations of the n-dimensional HE.
First, we will clarify the meaning of the optical Appell transformation, which, being well known in the theory of HE (Appell 1892, Widder 1962, 1975), has been recently introduced in the optical context as well (Torre 2011a).
10.4 OPTICAL APPELL TRANSFORMATION
As shown in Torre (2011a), the optical Appell transformation connects wavefunctions, whose source functions are Fourier related. In ...