On the asymptotic expansions of products related to the Wallis, Weierstrass, and Wilf formulas

For all integers n ≥ 1, let Wn(p,q)=∏j=1n{e−p/j(1+pj+qj2)}and Rn(p,q)=∏j=1n{e−p/(2j−1)(1+p2j−1+q(2j−1)2)},where p, q are complex parameters. The infinite product W∞(p, q) includes the Wallis and Wilf formulas, and also the infinite product definition of Weierstrass for the gamma function, as special cases. In this paper, we present asymptotic expansions of Wn(p, q) and Rn(p, q) as n → ∞. In addition, we also establish asymptotic expansions for the Wallis sequence.