A quick route to sums of powers. (English) Zbl 0605.40004

Edwards, A. W. F.

Am. Math. Mon. 93, 451-455 (1986).

This paper is concerned with the expressions for the sums of the powers of the integers found by Johann Faulhaber in 1631: \(\sum^{n}_{\nu =1}\nu^ r\quad (r\quad even) =a\) polynomial in \(n(n+1)\), \(\sum^{n}_{\nu =1}\nu^ r\quad (r\quad odd) =(2n+1)\) (a polynomial in \(n(n+1))\). The author shows how the coefficients which recur in these ”Faulhaber polynomials” can be obtained by inverting certain matrices closely related to Pascal’s triangle, and then uses his results to explain the algorithms actually used by Faulhaber to compute the coefficients.

MSC:

40A25

Approximation to limiting values (summation of series, etc.)

Keywords:

Faulhaber polynomials; algorithms