On the asymptotic expansions of products related to the Wallis, Weierstrass, and Wilf formulas. (English) Zbl 1411.33001
Summary: For all integers \(n\geq1\), let \[ W_n(p, q) = \prod_{j = 1}^n \left\{e^{- p / j}\left(1 + \frac{p}{j} + \frac{q}{j^2}\right) \right\} \] and \[ R_n(p, q) = \prod_{j = 1}^n \left\{e^{- p /(2 j - 1)}\left(1 + \frac{p}{2 j - 1} + \frac{q}{(2 j - 1)^2}\right) \right\}, \] 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 \(n\rightarrow\infty\). In addition, we also establish asymptotic expansions for the Wallis sequence.
MSC:
| 33B15 | Gamma, beta and polygamma functions |
| 40A25 | Approximation to limiting values (summation of series, etc.) |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
Keywords:
gamma function; psi function; Euler-Mascheroni constant; asymptotic expansion; Wallis sequenceOnline Encyclopedia of Integer Sequences:
Decimal expansion of Wilf’s formula: Product_{k>=1} exp(-1/k)*(1 + 1/k + 1/(2*k^2)) = exp(-gamma)*cosh(Pi/2)/(Pi/2).References:
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