The Bernstein polynomial in one variable

The lemma on L-functions is a result due to I.N. Bernstein about the existe nce of certain differential operators with polynomial coefficients. In this paper we give an elementary and constructive proof of this result that wor ks well in one variable. Our method results in a simple formula for the Ber nstein polynomial b(lambda) and a recursive definition for a differential o perator d(lambda) that produces b(lambda). As an application we consider tw o consequences about the poles of certain meromorphic functions defined by the analytic continuation of distributions.