Functional Equation for $\sum_{n=0}^{N}n^{k}$

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The discussion centers on the existence of a functional equation for the sum of powers, represented as Z(N, k) = ∑_{n=0}^{N} n^k, where both k and N are real numbers. Participants explore the implications of taking N to infinity, drawing connections to the Riemann zeta function. The challenge lies in understanding the behavior of Z(N, k) when N is finite, which remains less explored. The conversation references a Mathworld article for further context on the topic. Overall, the inquiry seeks to clarify the properties of this summation in both finite and infinite cases.
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is there a functional equation for

\sum_{n=0}^{N}n^{k}=Z(N,k)

where k and N are real numbers, in case N tends to infinite we could consider the functional equation of Riemann zeta but what happens in the case of N finite ??
 
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