E-transforms (II)

The following class of inte gral transform pairs is established(2)The kernel in the tra nsform ( 1) is Mac Robert's E•func tion and integration is performed with res pect to the arg ume nt of thi s fun ction.In the inversion formula (2), the kernel is like wise a n E-fun ction, but the integration is performed with respect to its parameters.Known spec ial cases of thi s ge neral transform pair is the Kantorovich-Lebedev transform s pair:r tKixfy)ffy)dy , 7r 0 fIx) = f o oo Ki y(X)g(y)dy , and the ge neralized Mehler trans form pair g(x) = -;sinh (7rx)r(!-k+ iX)r(!-k -iX) 1. 00 pkix _t/2 fy)ffy)dy , fIx) = 10 00 P~y-t/2 (x)g(y) dy.