Is there a formula for sums of consecutive powers, where the powers are non integer?

Question

I mean like the formula for $\sum^n_{j=1}j$, or $\sum^n_{j=1}j^2$ extended to things like $\sum^n_{j=1}j^{1.5}$, etc.

Answer 1

There is no "simple" formula for an exact evaluation of the sum for non-integers like there is for the integer cases, by which I mean there is no polynomial in $n$ with rational coefficients that yields the desired sum, like there are for the integer cases.

However we know the approximate behaviour of the value of the sum quite well. By the Euler-Maclaurin summation formula (or you could think of the following as just an application of the trapezoidal approximation to integrals) we have:

$$ \sum_{k=1}^n k^s = \frac{n^{s+1} }{s+1} + \frac{n^s}{2} + \mathcal{O}(n^{s-1} ) $$

where $ s>0.$

Answer 2

The generalized harmonic numbers

$$H_n^{(p)}=\sum_{k=1}^n \frac1{k^p}$$

can be expressed in terms of certain special functions.

One can represent them in terms of Hurwitz zeta functions:

$$H_n^{(p)}=\zeta (p)-\zeta (p,k+1)$$

or equivalently in terms of polygamma functions. See this for details.

Answer 3

The bernoulli-polynomials, or better: their replacement by the use of $\small \zeta() $-terms (call them $\small \zeta()$-polynomials) can be generalized to fractional arguments at the $\small \zeta() $-s . However, while the bernoulli-/zeta-polynomials have finitely many terms at integer exponents due to the vanishing binomial-cofactors, that cofactors do not vanish if the exponent is noninteger, so we get infinite series for fractional exponents.

Define the m'th $\small \zeta()$-polynomial (which sums the powers of exponent m)
$$ z_m(x)=\sum_{c=0}^{\infty} \zeta(-(m-c))\binom{m}{c} x^c $$ . For integer m this reduces to a finite number of terms $$ z_m(x)=\sum_{c=0}^{m} \zeta(-(m-c))\binom{m}{c} x^c - {1 \over m+1} x^{m+1} $$ and is identical to the integrals of the bernoulli-polynomials (the Faulhaber's-and Bernoulli's version of the sum-of-like-powers problem). The requested sum of consecutive powers s(a,b,m) is then $\small s(a,b,m)= z_m(a)- z_m(b+1) $

(This answer crossed just with J.M.'s answer)