An Extended Version of Faulhaber’s Formula

Raphael Schumacher Department of Mathematics ETH Zurich Rämistrasse 101 8092 Zurich Switzerland

                       raphschu@ethz.ch

Abstract This paper presents an extended version of the well-known Faulhaber formula, which is used to compute the sum of the m-th powers of the first n natural numbers, where m and n are two natural numbers. Our expression is analogous to Faulhaber’s formula, but sums the m-th powers of the natural numbers ≤ x for any non-negative real number x.

1 Introduction

For two natural numbers m, n ∈ ℕ_{0}, the Faulhaber formula [1], which was found by Jacob Bernoulli around 1700, provides a very efficient way to compute the sum of the m-th powers of the first n natural numbers. It is given by

∑_{k=0}^{n} k^{m} = \frac{1}{m + 1} ∑_{k=0}^{m}(−1)^{k}\binom{m + 1}{k}B_{k}n^{m−k+1},

where the B_{k}’s are the Bernoulli numbers.

In this paper, we will prove the analogous expression for the sum ∑_{k=0}^{⌊x⌋} k^{m}, where x ∈ ℝ_{0}^{+} and m ∈ ℕ_{0}, in terms of Bernoulli polynomials B_{k}(x) instead of Bernoulli numbers B_{k}. This expression is given by Theorem 1. (extended Faulhaber formula) For any x ∈ ℝ_{0}^{+} we have that

∑_{k=0}^{⌊x⌋} k^{m} = \frac{1}{m + 1}x^{m+1} + (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=1}^{m+1}(−1)^{k}\binom{m + 1}{k}B_{k} (\{x\}) x^{m−k+1}.

We have searched this version of Faulhaber’s formula in the literature, but we have not found it and therefore we believe that this result is new.

2 Definitions

As usual, we denote the floor of x by ⌊x⌋ and the fractional part of x by {x}.

Definition 2. For k ∈ ℕ_{0} we define the k-th Bernoulli polynomial B_{k}(x) via the following exponential generating function [2]:

\frac{te^{xt}}{e^{t} − 1} = ∑_{k=0}^{∞} \frac{B_{k}(x)}{k!}t^{k} ∀t ∈ ℂ with |t| < 2π.

Definition 3. The k-th Bernoulli number B_{k} is defined as the value of the k-th Bernoulli polynomial B_{k}(x) at x = 0 [2], that is

B_{k} := B_{k}(0).

Moreover, we get from the definition of the Bernoulli polynomials [1] that

\frac{t}{e^{t} − 1} = ∑_{k=0}^{∞} \frac{B_{k}}{k!} t^{k} ∀t ∈ ℂ with |t| < 2π.

3 Proof of the extended Faulhaber formula

In this section we will prove our extended version of Faulhaber’s formula.

Proof. Let m, n ∈ ℕ_{0} be two natural numbers. Starting from [1] the usual Faulhaber formula

∑_{k=0}^{n} k^{m} = \frac{1}{m + 1} ∑_{k=0}^{m}(−1)^{k}B_{k}\binom{m + 1}{k}n^{m−k+1},

we obtain

∑_{k=0}^{n} k^{m} = (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=0}^{m+1}(−1)^{k}B_{k}\binom{m + 1}{k}n^{m−k+1}.

Setting here n := ⌊x⌋ = x − {x} for some x ∈ ℝ_{0}^{+}, we get

∑_{k=0}^{⌊x⌋} k^{m} = (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=0}^{m+1}(−1)^{k}B_{k}\binom{m + 1}{k} (x − \{x\})^{m−k+1}

= (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=0}^{m+1}(−1)^{k}B_{k}\binom{m + 1}{k} ∑_{l=0}^{m−k+1}(−1)^{m−k−l+1}\binom{m − k + 1}{l}x^{l}\{x\}^{m−k−l+1}

= (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=0}^{m+1} B_{k}\binom{m + 1}{k} ∑_{l=0}^{m−k+1}(−1)^{m−l+1}\binom{m − k + 1}{l}x^{l}\{x\}^{m−k−l+1},

where we have used the binomial theorem

(a + b)^{n} = ∑_{l=0}^{n} \binom{n}{l}a^{l}b^{n−l}

for a := x, b := {x} and n := m − k + 1.

We now interchange the order of summation and use the binomial identity

\binom{m + 1}{k}\binom{m − k + 1}{l} = \frac{(m + 1)!(m − k + 1)!}{k!l!(m + 1 − k)!(m − k − l + 1)!}

= \frac{(m + 1)!}{k!l!(m − k − l + 1)!}

= \frac{(m + 1)!(m − l + 1)!}{k!l!(m − l + 1)!(m − k − l + 1)!}

= \binom{m + 1}{l}\binom{m − l + 1}{k}

to obtain that

∑_{k=0}^{⌊x⌋} k^{m} = (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=0}^{m+1} B_{k}\binom{m + 1}{k} ∑_{l=0}^{m−k+1}(−1)^{m−l+1}\binom{m − k + 1}{l}x^{l}\{x\}^{m−k−l+1}

= (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{l=0}^{m+1}(−1)^{m−l+1}x^{l} ∑_{k=0}^{m−l+1} B_{k}\binom{m + 1}{k}\binom{m − k + 1}{l}\{x\}^{m−k−l+1}

= (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{l=0}^{m+1}(−1)^{m−l+1}x^{l} ∑_{k=0}^{m−l+1} B_{k}\binom{m + 1}{l}\binom{m − l + 1}{k}\{x\}^{m−k−l+1}

= (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{l=0}^{m+1}(−1)^{m−l+1}\binom{m + 1}{l}x^{l} ∑_{k=0}^{m−l+1} B_{k}\binom{m − l + 1}{k}\{x\}^{m−k−l+1}.

If we use now the following explicit formula [3, Proposition 23.2, p. 86] for the Bernoulli polynomials

B_{n}(x) = ∑_{k=0}^{n} B_{k}\binom{n}{k}x^{n−k}

for n := m − l + 1 and x := {x}, we get

\begin{aligned}∑_{k=0}^{⌊x⌋} k^{m} & = (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{l=0}^{m+1}(−1)^{m−l+1}\binom{m + 1}{l}x^{l} ∑_{k=0}^{m−l+1} B_{k}\binom{m − l + 1}{k}\{x\}^{m−k−l+1} \\ & = (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{l=0}^{m+1}(−1)^{m−l+1}\binom{m + 1}{l}B_{m−l+1} (\{x\}) x^{l}. \end{aligned}

In the above formula we can change variables according to l := m − k + 1 ⟺ k = m − l + 1 and use the symmetry of the binomial coefficients

\binom{m + 1}{m − k + 1} = \binom{m + 1}{k},

to conclude that

∑_{k=0}^{⌊x⌋} k^{m} = (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{l=0}^{m+1}(−1)^{m−l+1}\binom{m + 1}{l}B_{m−l+1} (\{x\}) x^{l}

= (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=0}^{m+1}(−1)^{k}\binom{m + 1}{m − k + 1}B_{k} (\{x\}) x^{m−k+1}

= (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=0}^{m+1}(−1)^{k}\binom{m + 1}{k}B_{k} (\{x\}) x^{m−k+1}.

Finally, if we use the fact that B_{0}(x) = 1 ∀x ∈ ℝ, we get our claimed formula

∑_{k=0}^{⌊x⌋} k^{m} = \frac{1}{m + 1}x^{m+1} + (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=1}^{m+1}(−1)^{k}\binom{m + 1}{k}B_{k} (\{x\}) x^{m−k+1}

for all x ∈ ℝ_{0}^{+}.

Remark 4. The ordinary Faulhaber formula follows by setting x := n ∈ ℕ_{0} in our developed extension, because

∑_{k=0}^{n} k^{m} = ∑_{k=0}^{⌊n⌋} k^{m}

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

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

= (−1)^{m} \frac{B_{m+1}}{m + 1} + \frac{1}{m + 1} ∑_{k=0}^{m+1}(−1)^{k}\binom{m + 1}{k}B_{k}n^{m−k+1}

= \frac{1}{m + 1} ∑_{k=0}^{m}(−1)^{k}\binom{m + 1}{k}B_{k}n^{m−k+1},

where we have used that B_{k} ({n}) = B_{k}(0) = B_{k} for all k ∈ ℕ_{0}.

4 Acknowledgment

The author thanks the referee for the valuable suggestions regarding the improvement of this paper.

References

[1] Kevin J. McGown and Harold R. Parks, The generalization of Faulhaber’s formula to sums of non-integral powers, J. Math. Anal. Appl. 330 (2007), 571–575.

[2] A. Bazsó, Á. Pintér, and H. M. Srivastava, A refinement of Faulhaber’s theorem concerning sums of powers of natural numbers, Appl. Math. Lett. 25 (2012), 486–489.

[3] Victor Kac and Pokman Cheung, Quantum Calculus, Springer, 2002.

2010 Mathematics Subject Classification: Primary 11B68; Secondary 34M30. Keywords: extended Faulhaber formula, floor function, sum of powers of the first few natural numbers, Bernoulli polynomial, Bernoulli number.

(Concerned with sequences \underline{A027641} and \underline{A027642}.)

Received November 7 2015; revised versions received November 9 2015; March 19 2016. Published in Journal of Integer Sequences, April 7 2016.

Return to Journal of Integer Sequences home page.