In 1994 Anderson proved power series identities for exponentials of Drinfeld modules and harmonic sums, which were inspired by formulas of Thakur on zeta values.  We will discuss a new proof of Anderson's result in the case of Drinfeld modules for coordinate rings of elliptic curves, which follows Thakur's method and relies on formulas for harmonic sums in terms of the shtuka function for the elliptic curve.  Joint with N. Green.