On a particular form of a symmetric P\"oschl-Teller potential

We show that solutions of the Schr\"odinger equation with a symmetric P\"oschl-Teller potential of a particular form can be expressed in terms of a closed combination (not series) of trigonometric functions. Using some properties of the eigenfunctions of the Schr\"odinger equation and their inner product we determine a new exact representation of the hypergeometric function with certain values of parameters in terms of a closed combination of trigonometric functions. We also obtain new results in an explicit closed form for integrals with the hypergeometric function and with the specific combination of trigonometric functions.