I am curious about whether a closed form expression of $$\sum_{k=1}^{n}k^\alpha $$ for $\alpha \in \mathbb{R}$ exists in terms of special functions. Clearly for the natural number case we have the Faulhaber formulae, and when $\alpha = -1$ it is known that we can make use of the digamma function $\psi(x) := \frac{d}{dx}\log(\Gamma(x))$ to create an analogue of $\int\frac{1}{x} dx = \log(x)$ allowing us to sum expressions of the form:
$$\sum_{k=1}^{n}\frac{1}{k+p} = \psi(n+p+1) - \psi(1+p) = \psi(n+p+1) + \gamma - H_p$$
Where $\gamma$ is the Euler-Mascheroni constant and $H_p$ the $p$'th Harmonic number (the formula above is prettier when expressed using the methods of discrete calculus, but I don't want this question to become too niche, see Concrete Mathematics or this nice pdf for the curious).
Is there a slightly more general idea like this? I am more interested in $\sum\limits_{k=1}^{n}\sqrt{k}$, $\sum\limits_{k=1}^{n}\frac{1}{k^m}, m \in \mathbb{N}$ rather than $\sum\limits_{k=1}^{n}\frac{1}{(k+p)^m}$ or arbitrary reals, but I suppose if such a generalisation exists it probably goes the full mile.
I have seen some interesting identities involving polygamma functions and the Hurwitz Zeta, but not quite to the level (at least from what I can glance) of giving a closed form.
$s=\Sigma^n_{k=1}k^a=1 ^a+2^a+3^a+ . . . k^a$
$e-1=\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+ . . .+\frac{1}{k!}+ ...$
Term by term multiplication of two series gives:
$s . (e-1)=\Sigma^n_{k=1}k^a.\Sigma^n_{k=1}\frac{1}{k!}=\Sigma^n_{k=1}k^a.\frac{1}{k!}=\frac{1^a}{1!}+\frac{2^a}{2!}+\frac{3^a}{3!}+ . . .$
Also:
$e^{e^x}=e[1+x+x^2+(5/6)x^3 +(5/8)x^4+ . . .(c_a) x^a]$
We can see that the coefficients of $x^a$, i.e $(e.c_a)$ in this expansion is:
$e.c_a=\frac{1}{a!}.[\frac{1^a}{1!}+\frac{2^a}{2!}+\frac{3^a}{3!}+ . . .]$
Therefore :
$e. c_a.a!=s(e-1)$
⇒ $s=\Sigma^n_{k=1}k^a=\frac{a!.e.c_a}{e-1}$