Wikipedia:Reference desk/Archives/Mathematics/2016 November 25

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November 25

Polynomial coefficients generated by a sum of powers

Is there a general algorithm to generate the coefficients of the expansion of $f_{k}(n)=\sum _{i=1}^{n}i^{k}$? For example, $f_{7}(n)={\frac {n^{8}}{8}}+{\frac {n^{7}}{2}}+{\frac {7n^{6}}{12}}-{\frac {7n^{4}}{24}}+{\frac {n^{2}}{12}}$. I could use polynomial interpolation on the first $k+2$ terms of the series, but that gets impractical quickly if $k$ is large. 24.255.17.182 (talk) 21:47, 25 November 2016 (UTC)

One interesting thing I noticed is that ${\frac {n^{k-i}(i+1)!}{\operatorname {C} (k,i-1)}}$ is invariant with respect to $k$, and starts out $1,{\frac {1}{2}},0,-1,0,20,0,-1512,0,302400,0,-131345280,0,108972864000,0,157661939635200,0,371498581149696000...$, but I don't see an obvious pattern here and this series isn't in OEIS. 24.255.17.182 (talk) 22:13, 25 November 2016 (UTC)

(ec)Use Binomial_coefficient#Binomial_coefficients_as_a_basis_for_the_space_of_polynomials and the Hockey-stick identity.

$\sum _{i=1}^{n}i^{k}=\sum _{i=1}^{n}\sum _{m=0}^{k}a_{m}{\binom {i}{m}}=\sum _{m=0}^{k}a_{m}\sum _{i=1}^{n}{\binom {i}{m}}=\sum _{m=0}^{k}a_{m}{\binom {n+1}{m+1}}$

Bo Jacoby (talk) 22:24, 25 November 2016 (UTC).

Sorry if it's obvious but could you explain how $a_{m}$ is to be computed? 24.255.17.182 (talk) 23:03, 25 November 2016 (UTC)

I think what you're looking for is Faulhaber's formula. A generalization is the Euler–Maclaurin formula. --RDBury (talk) 01:43, 26 November 2016 (UTC)

The formula is found in the link I gave you.

$a_{m}=\sum _{i=0}^{m}(-1)^{m-i}{\binom {m}{i}}i^{k}.$

Bo Jacoby (talk) 07:18, 26 November 2016 (UTC).