Referee’s report on ” Faulhaber’s Theorem for Arithmetic Progressions ” by Chen, Fu and Zhang

Faulhaber’s 17th century formula was the precursor of the discovery of Bernoulli numbers and Bernoulli’s formula for the power sums of consecutive integers:

∑ n^{m} := ∑_{k=1}^{n} k^{m} = \frac{1}{m + 1}(B_{m+1}(n + 1) − B_{m+1}(1)).  (1)

Today it’s routine to derive Faulhaber’s formula starting from the latter, that’s why Knuth gave a proof of Faulhaber’s formula by using a technique available at in 17th century.

The authors of this paper essentially considered shifted versions of classical sums. To make this clearer, I will illustrate this point by taking (1) as example. Note that, when x is a nonnegative integer, we can write

∑(x + n)^{m} = ∑_{k=1}^{x+n} k^{m} − ∑_{k=1}^{x} k^{m},

which implies by (1) that

∑(x + n)^{m} = \frac{1}{m + 1}(B_{m+1}(x + n + 1) − B_{m+1}(x + 1)).  (2)

Now, Eq. (2) is valid for any complex number x because the two sides are polynomials of x. Thus, although (2) is more general than (1), but it follows straightforwardly.

The authors of this paper proved in Theorem 2.1 that ∑(x + n)^{2m−1} can be written as a polynomial of λ = n(n + 2x + 1), but this follows from Faulhaber’s formula in exactly the same way as (2) follows from (1). Other formulas are of the same nature. Consequently all the formulas proved in this paper are trivial from algebraic point of view. Now, from computational point of view these sums are covered by the classical Euler-Maclaurin formula discovered in the 18th century. Finally, the authors have not shown any combinatorial interest of their formulas.