Mellin transforms with only critical zeros: Legendre functions

Abstract: We consider the Mellin transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line Re $s=1/2$. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. These polynomials possess the functional equation $p_n(s)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s)$. Other hypergeometric representations are presented, as well as certain Mellin transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory.