On Wikipedia I came across the following equation for the central binomial coefficients: $$ \binom{2n}{n}=\frac{4^n}{\sqrt{\pi n}}\left(1-\frac{c_n}{n}\right) $$ for some $1/9<c_n<1/8$.
Does anyone know of a better reference for this fact than wikipedia or planet math? Also, does the equality continue to hold for positive real numbers $x$ instead of the integer $n$ if we replace the factorials involved in the definition of the binomial coefficient by Gamma functions?
It appears to be true for $x > .8305123339$ approximately: $c_x \to 0$ as $x \to 0+$.
One can find a number of estimates of the central binomial coefficient in the following papers and closely-related references therein.
@article {MR4127765, AUTHOR = {Popov, A. Yu.}, TITLE = {Two-sided estimates of the central binomial coefficient}, JOURNAL = {Chelyab. Fiz.-Mat. Zh.}, FJOURNAL = {Chelyabinski\u{\i} Fiziko-Matematicheski\u{\i} Zhurnal}, VOLUME = {5}, YEAR = {2020}, NUMBER = {1}, PAGES = {56-69}, ISSN = {2500-0101}, MRCLASS = {05A10 (05A15)}, MRNUMBER = {4127765}, DOI = {10.24411/2500-0101-2020-15105}, URL = {https://doi.org/10.24411/2500-0101-2020-15105}, }
@article {MR4127766, AUTHOR = {Tikhonov, I. V. and Sherstyukov, V. B. and Tsvetkovich, D. G.}, TITLE = {Comparative analysis of two-sided estimates of the central binomial coefficient}, JOURNAL = {Chelyab. Fiz.-Mat. Zh.}, FJOURNAL = {Chelyabinski\u{\i} Fiziko-Matematicheski\u{\i} Zhurnal}, VOLUME = {5}, YEAR = {2020}, NUMBER = {1}, PAGES = {70-95}, ISSN = {2500-0101}, MRCLASS = {05A10 (33B15 40A15)}, MRNUMBER = {4127766}, DOI = {10.24411/2500-0101-2020-15106}, URL = {https://doi.org/10.24411/2500-0101-2020-15106}, }
@article {MR1702663, AUTHOR = {Sasv'{a}ri, Zolt'{a}n}, TITLE = {Inequalities for binomial coefficients}, JOURNAL = {J. Math. Anal. Appl.}, FJOURNAL = {Journal of Mathematical Analysis and Applications}, VOLUME = {236}, YEAR = {1999}, NUMBER = {1}, PAGES = {223--226}, ISSN = {0022-247X}, MRCLASS = {26D15 (05A10)}, MRNUMBER = {1702663}, DOI = {10.1006/jmaa.1999.6420}, URL = {https://doi.org/10.1006/jmaa.1999.6420}, }
Let \begin{align*} \mathbb{Z}&=\{0,\pm1,\pm2,\dotsc\}, & \mathbb{N}&=\{1,2,\dotsc\},\\ \mathbb{N}_0&=\{0,1,2,\dotsc\}, & \mathbb{N}_-&=\{-1,-2,\dotsc\}. \end{align*} The extended binomial coefficient $\binom{z}{w}$ for $z,w\in\mathbb{C}$ is defined by \begin{equation}\label{Gen-Coeff-Binom} \binom{z}{w}= \begin{cases} \dfrac{\Gamma(z+1)}{\Gamma(w+1)\Gamma(z-w+1)}, & z\not\in\mathbb{N}_-,\quad w,z-w\not\in\mathbb{N}_-;\\ 0, & z\not\in\mathbb{N}_-,\quad w\in\mathbb{N}_- \text{ or } z-w\in\mathbb{N}_-;\\ \dfrac{\langle z\rangle_w}{w!},& z\in\mathbb{N}_-, \quad w\in\mathbb{N}_0;\\ \dfrac{\langle z\rangle_{z-w}}{(z-w)!}, & z,w\in\mathbb{N}_-, \quad z-w\in\mathbb{N}_0;\\ 0, & z,w\in\mathbb{N}_-, \quad z-w\in\mathbb{N}_-;\\ \infty, & z\in\mathbb{N}_-, \quad w\not\in\mathbb{Z}, \end{cases} \end{equation} where \begin{equation*}%\label{falling-Factorial} \langle\alpha\rangle_n=\prod_{k=0}^{n-1}(\alpha-k) = \begin{cases} \alpha(\alpha-1)\dotsm(\alpha-n+1), & n\in\mathbb{N}\\ 1, & n=0 \end{cases} \end{equation*} is called the falling factorial.
Equation (10) on page 116 in the paper [1] below reads that the double inequality \begin{equation}\label{Merkle-gamma-ineq}\tag{1} 6^x<\frac{\Gamma(2(1+x))}{\Gamma^2(1+x)}<(1+x)3^x \end{equation} holds for $x\in(0,1)$ and its reversed inequality is valid for $x>1$. We can reformulate the double inequality \eqref{Merkle-gamma-ineq} as follows: the double inequality \begin{equation}\label{Gen-Central-Binom-Coeff-bounds-Eq}\tag{2} \frac{6^x}{2x+1}>\binom{2x}{x}>\frac{(x+1)3^x}{2x+1} \end{equation} is valid for $x>1$ and its reversed version holds for $0<x<1$.
By the way, I have verified the following double inequalities:
I am looking for a suitable outlet for publishing, among other closely related things, the double inequalities \eqref{centr-binom-(2x+1)ineq} and \eqref{binom(2x+1)(x)-ineq}.
References