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Paris","submitted_at":"2015-11-30T09:40:23Z","abstract_excerpt":"For all integers $n\\geq1$, let \\begin{align*}\n  W_n(p,q)=\\prod_{j=1}^{n}\\left\\{e^{-p/j}\\left(1+\\frac{p}{j}+\\frac{q}{j^2}\\right)\\right\\} \\end{align*} and \\begin{align*} R_n(p, q)=\\prod_{j=1}^{n}\\left\\{e^{-p/(2j-1)}\\left(1+\\frac{p}{2j-1}+\\frac{q}{(2j-1)^2}\\right)\\right\\}, \\end{align*} where $p$, $q$ are complex parameters. The infinite product $W_{\\infty}(p,q)$ 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 $W_n(p,q)$ and $R_n(p, q)$ as $n\\to\\infty$. 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Chen, R.B. Paris","submitted_at":"2015-11-30T09:40:23Z","abstract_excerpt":"For all integers $n\\geq1$, let \\begin{align*}\n  W_n(p,q)=\\prod_{j=1}^{n}\\left\\{e^{-p/j}\\left(1+\\frac{p}{j}+\\frac{q}{j^2}\\right)\\right\\} \\end{align*} and \\begin{align*} R_n(p, q)=\\prod_{j=1}^{n}\\left\\{e^{-p/(2j-1)}\\left(1+\\frac{p}{2j-1}+\\frac{q}{(2j-1)^2}\\right)\\right\\}, \\end{align*} where $p$, $q$ are complex parameters. The infinite product $W_{\\infty}(p,q)$ 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 $W_n(p,q)$ and $R_n(p, q)$ as $n\\to\\infty$. 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