On the asymptotics of products related to generalizations of the Wilf and Mortini problems. (English) Zbl 1350.33001
The authors determine the asymptotic expansions as \(n\to\infty\) of the products determined by \(P_n= \prod^n_{j=1} (1+ p_1/j+\cdots+ p_m/j^m)\) and \(Q_n= \prod^n_{j=1} (1+ p_1/(2j- 1)+\cdots+ p_m/(2j-1)^m)\), where \(p_1,\dots,p_m\) are certain constants. As application, they present the connection between the generalized Wilf and Mortini problems.
Reviewer: József Sándor (Cluj-Napoca)
MSC:
| 33B15 | Gamma, beta and polygamma functions |
Software:
DLMFOnline 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:
| [1] | DOI: 10.2307/2974795 · doi:10.2307/2974795 |
| [2] | Choi J, Indian J Pure Appl Math 30 pp 649– (1999) |
| [3] | Mortini R, Amer Math Monthly 116 pp 747– (2009) |
| [4] | Geupel O, Amer Math Monthly 118 pp 185– (2011) |
| [5] | DOI: 10.2996/kmj/1050496647 · Zbl 1040.11062 · doi:10.2996/kmj/1050496647 |
| [6] | DOI: 10.1080/10652469.2012.693081 · Zbl 1275.33003 · doi:10.1080/10652469.2012.693081 |
| [7] | DOI: 10.1080/10652469.2014.885965 · Zbl 1303.33001 · doi:10.1080/10652469.2014.885965 |
| [8] | Olver FWJ, NIST handbook of mathematical functions (2010) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
