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It is shown that the regular-at-infinity solution of the 1D Schrodingerequation with the hyperbolic Poschl-Teller (h-PT) potential with integerparameters is expressible in terms of a n-order Heun polynomial in y=thr at anarbitrary negative energy. It was proven that the Heun polynomials in questionform a subset of generally complex Lambe-Ward polynomials corresponding to zerovalue of the accessory parameter. Since the mentioned solution expressed in thenew variable y has an almost-everywhere holomorphic (AEH) form it can be usedas the factorization function (FF) for canonical Liouville-Darbouxtransformations (CLDTs) to construct a continuous family of shape-invariantrational potentials exactly-solvable by the Hp-seed (HpS) Heine polynomials.There are also two sequences of infinitely many rational potentials generatedusing CLDTs with nodeless regular-at-origin AEH FFs.

Heun-Polynomial Representation of Regular-at-Infinity Solutions for the  Basic SUSY Ladder of Hyperbolic P\"oschl-Teller Potentials Starting from the  Reflectionless Symmetric Potential Well