Elementary specialization of a hypergeometric solution

Special parameter choices can identify a hypergeometric solution with an elementary solution of the same second-order differential equation by matching its local normalization.

For compatible local branches near $x=0$,It follows by transforming the harmonic oscillator equation $y^{\prime \prime }+k^{2}y=0$ into the Gauss hypergeometric equation and selecting the solution with value one and derivative zero at the origin.

For $k\ne 0$ and compatible local branches near $x=0$,This is obtained from the second local hypergeometric solution at zero and its normalization.