If f ( x ) f(x) and g ( x ) g(x) satisfy the equations \[ g ( x ) = d d x ∫ 0 ∞ 1 t f ( t ) k 1 ( x t ) d t , f ( x ) = d d x ∫ 0 ∞ 1 t g ( t ) k 1 ( x t ) d t , g(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}f(t){k_1}(xt)dt,\quad f(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}g(t){k_1}(xt)dt, \] then we call f and g a pair of k 1 {k_1} -transforms, where \[ k 1 = 1 2 π i ∫ 1 / 2 − i ∞ 1 / 2 + i ∞ K ( s ) 1 − s x 1 − s d s . k_1 = \frac {1}{2\pi i} \int _{1/2 - i\infty }^{1/2 + i\infty } \frac {K(s)}{1 - s} x^{1-s}\,ds. \] In this paper alternative sets of conditions are established for f and g to be k 1 {k_1} -transform provided K ( s ) K(s) is decomposable in a special way. These conditions involve simpler functions, which replace the kernel k 1 ( x ) {k_1}(x) . Results are proved for the function spaces L 2 {L^2} . The necessary and sufficient conditions are established for the two functions to be self-reciprocal. Conditions are given for generating pairs of transforms for a given kernel. Two examples are given at the end to illustrate the methods and the advantage of the results.