Derivative of binomial coefficients

Question

I obtained the following formula in Mathematica:

$$\frac{d}{dn}\ln\binom{n}{k} = H_{n} - H_{n-k}$$

where $H_n$ are the harmonic numbers ($H_n = \sum_{i=1}^n 1/i$). But I have no idea how to prove it. Can someone help me? Or at least provide a reference to a textbook/paper?

Answer 1

\begin{eqnarray*} \binom{n}{k} &=& \frac{n(n-1) \cdots(n-k+1)}{k!} \\ \ln \binom{n}{k} &=&\ln n + \ln(n-1) + \cdots +\ln(n-k+1) -\ln(k!) \\ \frac{d}{dn} \ln \binom{n}{k} &=& \frac{1}{n} + \frac{1}{n-1} +\cdots + \frac{1}{n-k+1} =\color{red}{H_n-H_{n-k}}. \end{eqnarray*}

Answer 2

$\frac{d}{dn}\ln{n\choose k}=\frac{d}{dn}[\ln(n!)-\ln((n-k)!)-\ln(k!)]=\frac{d}{dn}\ln(n!)-\frac{d}{dn}\ln((n-k)!)$ Can you take it from here?

(Hint: $\ln(\Pi a_i)=\sum_i \ln(a_i)$)

Answer 3

$$k!\binom{n}k=n(n-1)(n-2)\cdots(n-k+1).$$ Taking the logarithmic derivative with respect to $n$ gives $$\frac1n+\frac1{n-1}+\frac1{n-2}+\cdots+\frac1{n-k+1}$$ (as logarithmic derivative converts products into sums).

Answer 4

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} &\bbox[10px,#ffd]{\ds{\totald{}{n}\ln\pars{n \choose k}}} = \totald{}{n}\bracks{\ln\pars{\Gamma\pars{n + 1}} - \ln\pars{\Gamma\pars{n - k + 1}}}\quad\pars{~\Gamma:\ Gamma Function~} \\[5mm] = &\ \Psi\pars{n + 1} - \Psi\pars{n + 1 - k}\qquad\pars{~\Psi:\ Digamma Function~} \\[5mm] = &\ \underbrace{\bracks{\Psi\pars{n + 1} + \gamma}} _{\ds{H_{n}}}\ -\ \underbrace{\bracks{\Psi\pars{n - k + 1} + \gamma}}_{\ds{H_{n - k}}} \qquad \pars{~\gamma:\ Euler\!-\!Mascheroni\ Constant~} \\[5mm] = &\ \bbx{H_{n} - H_{n - k}} \end{align}