Explicit estimates for the sum $\sum_{k=0}^{n} k! {n\choose k}^2 (-1)^{k}$
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909289620176896 |
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| author | Ernvall-Hytönen, Anne-Maria Matala-aho, Tapani |
| author_facet | Ernvall-Hytönen, Anne-Maria Matala-aho, Tapani |
| contents | We are interested in finding an explicit estimate to the binomial sum $Q_n(x)=\sum_{k=0}^{n} k! {n\choose k}^2 (-x)^{k}$ at $x=1$ for $n=0,1,2,\ldots$. Despite of its own interest the polynomial $Q_n(x)$ is important as the denominator in the Padé identity of the Euler's factorial series $E(x) = \sum_{k=0}^{\infty} k! x^k$ as well as its close connection to a classical Laguerre polynomial $L_n(x) = \frac{1}{n!} e^x \left(\frac{d}{dx}\right)^n (e^{-x}x^n)$. Our main result is the explicit bound $$\left|L_n(1)-\sqrt{\frac{e}π}\cdot \frac{\cos (2\sqrt{n}-\fracπ{4})}{n^{1/4}} +\frac{17}{48}\sqrt{\frac{e}π}\frac{\sin(2\sqrt{n}-\fracπ{4})}{n^{3/4}}\right|<\frac{0.51}{n}$$ for all $n=0,1,2,\ldots$, which replaces the Fejér's asymptotic formula from 1909. As a corollary of this, one also gets a new proof for the bound $|Q_{n}(1)| \le n!$, and even more. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11468 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Explicit estimates for the sum $\sum_{k=0}^{n} k! {n\choose k}^2 (-1)^{k}$ Ernvall-Hytönen, Anne-Maria Matala-aho, Tapani Number Theory We are interested in finding an explicit estimate to the binomial sum $Q_n(x)=\sum_{k=0}^{n} k! {n\choose k}^2 (-x)^{k}$ at $x=1$ for $n=0,1,2,\ldots$. Despite of its own interest the polynomial $Q_n(x)$ is important as the denominator in the Padé identity of the Euler's factorial series $E(x) = \sum_{k=0}^{\infty} k! x^k$ as well as its close connection to a classical Laguerre polynomial $L_n(x) = \frac{1}{n!} e^x \left(\frac{d}{dx}\right)^n (e^{-x}x^n)$. Our main result is the explicit bound $$\left|L_n(1)-\sqrt{\frac{e}π}\cdot \frac{\cos (2\sqrt{n}-\fracπ{4})}{n^{1/4}} +\frac{17}{48}\sqrt{\frac{e}π}\frac{\sin(2\sqrt{n}-\fracπ{4})}{n^{3/4}}\right|<\frac{0.51}{n}$$ for all $n=0,1,2,\ldots$, which replaces the Fejér's asymptotic formula from 1909. As a corollary of this, one also gets a new proof for the bound $|Q_{n}(1)| \le n!$, and even more. |
| title | Explicit estimates for the sum $\sum_{k=0}^{n} k! {n\choose k}^2 (-1)^{k}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2310.11468 |