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We present a new six-parameter family of potentials whose solutions areexpressed in terms of the hypergeometric functions 3F2, 2F2 and 1F2. Both thescattering data and the bound states of these potentials are explicitlycomputed and the peculiar properties of the discrete spectrum are depicted in asuitable phase diagram. Our starting point is a third-order formal eigenvalueequation of the hypergeometric type (with a suitable solution known) which istransformed to the Schr\"odinger equation by applying the reduction of ordertechnique as the crucial first step. The general preconditions allowing for thereduction to Schr\"odinger form of an arbitrary eigenvalue equation of higherorder, are discussed at the end of the article, and two universal features ofthe potentials arising this way are also stated and discussed. In this generalscheme the Natanzon potentials are the simplest special case, those presentedhere the next ones, and so on for potentials arising from equations of fourthor higher order.

Quantum mechanical potentials exactly solvable in terms of higher hypergeometric functions. I: The third-order case