Existuje obecný vzorec pro součet mocnin přirozených čísel?
Rozhledy matematicko-fyzikální, Tome 90 (2015) no. 3, pp. 1-13
The article derives the formula for the sum of the $k$-th powers of positive integers from $1$ to $n$ in the form of a polynomial in the variable $n$. The determination of the coefficients $a_{k,j}$ of the polynomial (a two-parametric problem) is converted into the determination of the members of a progression $B_p% , so called Bernoulli numbers (a one-parametric problem), and a recurrent formula for these numbers is derived. Then, mutual divisibility of the polynomials is examined for different values of $k$, and Nikomachos theorem is mentioned as a special case.
The article derives the formula for the sum of the $k$-th powers of positive integers from $1$ to $n$ in the form of a polynomial in the variable $n$. The determination of the coefficients $a_{k,j}$ of the polynomial (a two-parametric problem) is converted into the determination of the members of a progression $B_p% , so called Bernoulli numbers (a one-parametric problem), and a recurrent formula for these numbers is derived. Then, mutual divisibility of the polynomials is examined for different values of $k$, and Nikomachos theorem is mentioned as a special case.
@article{RMF_2015_90_3_a0,
author = {Tomsa, Jan},
title = {Existuje obecn\'y vzorec pro sou\v{c}et mocnin p\v{r}irozen\'ych \v{c}{\'\i}sel?},
journal = {Rozhledy matematicko-fyzik\'aln{\'\i}},
pages = {1--13},
year = {2015},
volume = {90},
number = {3},
language = {cs},
url = {http://geodesic.mathdoc.fr/item/RMF_2015_90_3_a0/}
}
Tomsa, Jan. Existuje obecný vzorec pro součet mocnin přirozených čísel?. Rozhledy matematicko-fyzikální, Tome 90 (2015) no. 3, pp. 1-13. http://geodesic.mathdoc.fr/item/RMF_2015_90_3_a0/