### Abstract

In this paper the body‐centered cubic lattice Green's function,where *l* _{1}, *l* _{2}, and *l* _{3} are all even, or all odd, is studied. A complete analytic continuation for *P*(*z*) ≡ *P*(**0**, *z*) is derived of the form,where |1 − *z* ^{2}| < 1. Explicit formulas, recurrence relations, and asymptotic expansions are established for the coefficients *B*_{n} and *C*_{n} . A similar analytic continuation in powers of 1 − *z* is also investigated. The generalized Watson integral,where *m* ≥ 0 and *n* ≥ 0, is evaluated in closed form. Using this result, we show that *P*(**I**, 1) can, in principle, be evaluated for arbitrary **I**. Exact expressions and numerical values for *P*(**I**, 1) are given for 0 ≤ *l* _{1} ≤ *l* _{2} ≤ *l* _{3} ≤ 8. Detailed applications of the above results are made in the theory of random walks on a body‐centered cubic lattice. In particular, a new asymptotic expansion for the expected number of distinct lattice sites visited during an *n*‐step random walk is obtained. The closely related Green's function,where ξ_{0} is real, is expressed in terms of complete elliptic integrals for *all* ξ_{0} > 0, and evaluated numerically in the range 0 < ξ_{0} ≤ 1. The behavior of this Green's function in the neighborhood of the singularities at ξ_{0} = 0 and 1 is also discussed. *No attempt* is made, in the present paper, to discuss *P*(**I**, *z*) for the general case **I** ≠ 0 *and z* ≠ 1.