Is there a proof that the Harmonic numbers are not an elementary function? The Harmonic numbers $H_x = \sum_{n=1}^x 1/n$ are the sum of the reciprocals of the natural numbers up to a given number. The first few are $0, 1, 3/2, 11/6, \ldots$. $H_x$ can be defined for noninteger $x$ by the integral definition $H_x = \int_0^1 \frac{1-t^x}{1-t} \, dt$, ie $H_{1/2} = 2-2\ln 2$
$H_x$ is not a rational function because a rational function is asymptotic to $x^k$ for some integer $k$, and $H_x$ is asymptotic to $\ln x + \gamma$, where $\gamma\sim 0.577 \dots$.
Can you prove $H_x$ is not an elementary function using the integral definition? Take "elementary" to mean a finite composition of arithmetic operators $(+, -, \div, \times)$, exponents, logarithms, and known constants (algebraic numbers and algebraic compositions of proven transcendental numbers, like $\pi, e, e^\pi \dots$ etc).
 A: Your $H_x$ function is (up to the Euler constant $\gamma$) the digamma function $\psi(x+1)$ : $$\tag{1}H_x = \int_0^1 \frac{1-t^x}{1-t} dt= \psi(x+1)+\gamma$$
There is no closed form in the general case of $x\in\mathbb{R}$ (or $\mathbb{C}$) as we may deduce from the Hölder theorem for the $\Gamma$ function.
Elementary functions and their integrals (recursively) define the "Liouvillian functions" which are all solutions of algebraic differential equations. From the Hölder theorem we deduce that $\Gamma$ and thus $\;\psi(x)=(\log\Gamma(x))'$ are not solutions of any algebraic differential equation and thus can't be elementary ($\psi$ is a special function).
On the other side I must add that a closed form exists for any fixed fraction $x\in \mathbb{Q}$ from a nice formula from Gauss :
For two positive integers $r$ and $m$ with $r<m$ we have
$$\tag{2}\psi\left(\frac rm\right)=-\gamma-\log(2m)-\frac {\pi}2\cot\left(\frac{r\pi}m\right)+2\sum_{n=1}^{\lfloor(m-1)/2\rfloor}\cos\left(\frac{2\pi nr}m\right)\log\,\sin \left(\frac{\pi n}m\right)$$
For fractions out of $(0,1)$ we may then use the recurrence $$\tag{3}\psi(x+1)=\psi(x)+\dfrac 1x$$
ADDITION (from the comments by Rahul and Winther) :
I added fixed fraction in the previous text because using the previous formulas you will obtain a closed form for (say) $\psi(179/3)\;$ but I fear this won't be considered as a closed form for a general fraction $\;x=\dfrac rm\;$ since :

*

*you will have to add a number of terms depending of $\lfloor x\rfloor$


*the denominator of $x$ (i.e. $m$) will add further $m$ or $\left\lfloor\frac {m-1}2\right\rfloor$ terms
Setting an upper bound for $|x|$ and $m$ may be more satisfying for a closed form...
The digamma function could be considered as a mere rewriting of $H_n$ but the link with the $\Gamma$ function allows to get very useful expansions for large $n$ (and $x$) and numerous neat analytical results (polygamma and generalizations).
