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I want to understand how do you prove by induction: $\sum_{k=0}^n{{(-1)^k}{\binom nk}{k^m}}=0$ where $0 \leq m < n$.

As far as I understood I have to:

  • Prove the initial case $n = 1$
  • Assume the hypothesis $\sum_{k=0}^n{{(-1)^k}{\binom nk}{k^m}}=0$
  • Find for $n+1$, and here it's where I am stuck. I do not understand how can I manipulate the expression in order to solve the problem.

Could someone help me understand how to achieve this or point me in the right direction?

Thank you in advance

ajr
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Alexandre Pires
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  • Have you heard abot the finite differences? See, e.g. here https://en.wikipedia.org/wiki/Finite_difference – richrow Apr 07 '22 at 19:25
  • On the other hand, you can try to prove by induction (on $n$) that $\sum_{k=0}^{n}(-1)^k\binom{n}{k}f(k)=0$ for any polynomial $f$ of the degree $\deg f<n$ (hint: consider $g(x)=f(x+1)-f(x)$). – richrow Apr 07 '22 at 19:26

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