Abstract
We are interested in finding an explicit estimate to the binomial sum Qn(x)=∑nk=0k!(nk)2(−x)k at x=1 for n=0,1,2,…. Despite of its own interest the polynomial Qn(x) is important as the denominator in the Padé identity of the Euler's factorial series E(x)=∑∞k=0k!xk as well as its close connection to a classical Laguerre polynomial Ln(x)=1n!ex(ddx)n(e−xxn). Our main result is the explicit bound
∣∣∣Ln(1)−eπ−−√⋅cos(2n−−√−π4)n1/4+1748eπ−−√sin(2n−−√−π4)n3/4∣∣∣<0.51n
for all n=0,1,2,…, which replaces the Fejér's asymptotic formula from 1909. As a corollary of this, one also gets a new proof for the bound |Qn(1)|≤n!, and even more.
Original language | English |
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Pages (from-to) | 679-692 |
Number of pages | 14 |
Journal | Proceedings of the American Mathematical Society |
Volume | 153 |
Issue number | 2 |
Early online date | 12 Dec 2024 |
DOIs | |
Publication status | Published - Feb 2025 |
MoE publication type | A1 Journal article-refereed |