On polynomial transformations preserving purely imaginary zeros*

In this present work polynomial transformations are identified that preserve the property of the polynomials having all zeros lying on the imaginary axis. Existence results concerning families of polynomials whose generalized Mellin transforms have zeros all lying on the critical line are then derived. Inherent structures are identified from which a simple proof relating to the Gegenbauer family of orthogonal polynomials is subsequently deduced. Some discussion about the choice of generalized Mellin transform is also given.