Abstract
We prove that all the Faulhaber coefficients of a sum of odd power of elements of an arithmetic progression may simply be calculated from only one of them which is easily calculable from two Bernoulli polynomials as so as from power sums of integers. This gives two simple formulae for calculating them. As for sums related to even powers, they may be calculated simply from those related to the nearest odd one’s.
Highlights
The problem of showing the sum of mth power of integers, denoted in this work by Sm (n), as a polynomial function of n has fascinated mathematicians from the antiquity until nowadays as we can see for examples in a master thesis of Coen (1996), in Beery (2010), etc.As for the problem of expressing Sm (n) as a sum of powers of S1(n), Johann Faulhaber (1631) is certainly the most celebrated for his conjecture saying that a sum of odd power of integers are homogeneous polynomials in sum of them, appeared in a booklet entitled Academia Algebrœ published in 1631, rediscovered surprisingly only in 1981 by Edwards (1986) at Cambridge university
As for the problem of expressing Sm (n) as a sum of powers of S1(n), Johann Faulhaber (1631) is certainly the most celebrated for his conjecture saying that a sum of odd power of integers are homogeneous polynomials in sum of them, appeared in a booklet entitled Academia Algebrœ published in 1631, rediscovered surprisingly only in 1981 by Edwards (1986) at Cambridge university
Continuing the researches on Faulhaber conjecture on power sums on arithmetic progressions we propose in this work firstly to improve by simplification the quoted proof given by Chen, Yang and Zhang; secondly to obtain a simple method for calculating the Faulhaber coefficients related to sums of odd powers of integers S2k+1(n) ; thirdly to obtain those related to power sums on arithmetic progressions S2k+1(z, n) and S2k (z, n)
Summary
The problem of showing the sum of mth power of integers, denoted in this work by Sm (n) , as a polynomial function of n has fascinated mathematicians from the antiquity until nowadays as we can see for examples in a master thesis of Coen (1996), in Beery (2010), etc. About the problem of representing sums of mth power of the elements of an arithmetic progression z , z +1 ,..., z + n −1 when m is odd, denoted by S2k+1(z, n) , in term of polynomial in n we have the formulae of Dattoli, Cesarano, Lorenzutta (2002) and of Do (Do T.S.2017b). As for the Faulhaber problem of representing S2k+1(z, n) as homogeneous polynomial in S1(z, n) , we notice the very recent publication of Chen, Fu, Zhang (2009) These authors utilized the Dattoli, Cesarano, Lorenzutta formula that links S2k+1(z, n) to the Bernou lli po lyno mials Bm (z + n) , th e add ition th eo rem for Bernou lli p o lyn omials and th e property B2k+1( 1 / 2 ) = 0 in order to prove the Faulhaber conjecture concerning S2k+1(z, n). We get (B(z) + n)m+1 := ((B + z) + n)m+1 := (B + (z + n))m+1 which proves directly the formula of Dattoli, Cesarano, Lorenzutta (2002)
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