_{1}

^{*}

Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order
m
under the form of differential operators acting on monomials. It follows that (d/dn-d/dz) applied on a power sum has a meaning and is exactly equal to the Bernoulli polynomial of the same order. From this new property we get the formula giving powers sums in term of sums of successive derivatives of Bernoulli polynomial
multiplied
with
primitives of the same order of n. Then by changing the two arguments z,n
into
Z=z(z-1),
λ
where
λ
designed the 1^{st} order power sums and proving that Bernoulli polynomials of odd order vanish for arguments equal to 0
, 1/2, 1, we obtain easily the Faulhaber formula for powers sums in term of polynomials in λ
having coefficients depending on Z. These coefficients are found to be derivatives of odd powers sums on integers expressed in Z. By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibl
y the Bernoulli numbers, the Bernoulli polynomials, the powers sums and the Faulhaber formula for powers sums.

The problem of calculating the sums of the m^{th} powers of n first integers

is investigated from antiquity by mathematicians around the world.

We learn for examples in the thesis of Coen [^{th} century ibn al-Haytham had developed the formulae

∑ n = n 2 2 + n 2 , ∑ n 2 = n 3 3 + n 2 2 + n 6 , ∑ n 3 = n 4 4 + n 3 2 − n 2 4 (1.2)

In 15^{th} century his successors had found

About two centuries quietly passed until the day in 1631 when Faulhaber [

After Faulhaber, in 1636 French mathematicians Fermat utilizing the figurate numbers and in 1656 Pascal utilizing results of arithmetic triangle, found also recurrence formulae for calculating ∑ n m from lower-order sums [

Then in 1713 in his posthumous Ars conjectandi, Jacob Bernoulli [

This famous conjectured formula of Bernoulli was proven in 1755 based on the calculus of finite difference by Euler [

Returning to the Faulhaber conjecture saying that S m ( n ) is a polynomial in S 1 ( n ) for all m

we know that Jacobi [

d d n ∑ n m = B m ( n ) = m ∑ n m − 1 + B m (1.8)

Long years passed until Edwards [

Following Coen, we know the existence of the work Bernoulli numbers: bibliography (1713-1990) of Dilcher [

Concerning the more general problem of powers sums on arithmetic progressions

we remark the recent formula given by Dattoli, Cesarano, Lorenzutta [

and the formula of Chen, Fu, Zhang [

which are not easy to apply.

After these authors we have proposed a method leading to the formula [

where Z = z ( z − 1 ) and S ^ m ( Z ) ≡ S m ( z ) .

The Faulhaber formula is thus obtained but we see that the method for obtaining it is cumbersome and the practice calculations of S ^ m ( Z ) ≡ S m ( z ) for obtaining the Faulhaber coefficients fastidious.

Rethinking the problem, we observe that an arithmetic progression is a matter of translation, that there is a somehow symmetry between n , z and S 1 ( n ) , Z and ∂ n , ∂ z so that finally we found a more concise method for resolving the problem and theoretically and practically that we will expose in the following paragraphs.

Let

be the powers sums on an arithmetic progression and

the powers sums of the first integers.

We may utilize the shift or translation operator

to get the differential representation

which gives directly the relations

and the generating function

Because (2.4) is valid for all integers n it is also valid for all real and complex values so that we may write

Defining now the set of polynomials B m ( z ) by the differential representation

we see that B m ( z ) verify

and have the generating function

allowing the identification of them with Bernoulli polynomials defined by Euler [

B 0 ( z ) = 1 , B 1 ( z ) = z − 1 2 . (2.13)

The Bernoulli polynomials are linked to powers sums according to formulae (2.5), (2.8), (2.9) by the relations

which lead to the followed beautiful formula where the second member does not depend on n

Besides it leads also to the formula

which jointed with (2.15) gives rise to the historic Jacobi conjectured formula [

From (2.18) we see that B m ( 0 ) are identifiable with Bernoulli numbers B m .

1) Powers sums in terms of Bernoulli polynomials and powers of n

From the Equation (2.16) and the boundary condition S m ( z , 0 ) = 0 we get immediately the solution of (2.16)

S m ( z , n ) = 1 ∂ n − ∂ z B m ( z ) = ∑ k = 0 m ∂ z k ∂ n − k B m ( z ) n (3.1)

which may be put under the algorithmic form very easy to remember

or taking into account (2.11)

S m ( z , n ) = B m ( z ) n + ⋯ + m ( m − 1 ) ⋯ ( m − k + 1 ) B m − k ( z ) n k + 1 ( k + 1 ) ! + ⋯ + m ! B 0 ( z ) n m + 1 ( m + 1 ) !

Putting z = 0 in (3.2) and replacing B m − k ( 0 ) with B m − k we recognize the famous Bernoulli formula [

2) The Faulhaber formula on powers sums

In S m ( z , n ) instead of utilizing z and n for arguments let us utilize

Z = z ( z − 1 ) and λ = S 1 ( z , n ) = ( z − 1 2 ) n + n 2 2 (3.4)

Because

we have

∂ n − ∂ z ≡ B 1 ( z ) ∂ λ − 2 B 1 ( z ) ∂ Z ≡ B 1 ( z ) ( ∂ λ − 2 ∂ Z ) (3.8)

Equation (2.16) becomes

Happily from the definition of B m ( z ) by differential operators (2.9) we may write down

B m ( − z ) = − ∂ z e − ∂ z − 1 ( − z ) m = e ∂ z − ∂ z 1 − e ∂ z ( − z ) m = ( − ) m B m ( z + 1 ) (3.10)

which jointed with (2.10) leads to the important properties

As an polynomial of order 2k having z 0 , z 1 for roots may be put by identification of coefficients under the form of a homogeneous polynomial of order k in ( z − z 0 ) ( z − z 1 ) we obtain the important property:

B 1 − 1 ( z ) B 2 k + 1 ( z ) is a homogeneous polynomial of order k in Z = z ( z − 1 ) k > 0 (3.14)

By the way we notify that because S 2 k + 1 ( n ) = 0 for n = 0 , 1 it is also a homogeneous polynomial of order ( k + 1 ) in u = n ( n − 1 ) as conjectured Faulhaber and proven somehow by Jacobi.

Moreover the calculations of the quoted polynomials in Z or in u may be done thank to the hereinafter

Jointed (3.14) with the formula came from the Jacobi formula (2.18)

we may define a polynomial S ^ k ( Z ) of order k depending on Z such that

The definition of S ^ k ( Z ) by (3.16) differs a little in index with its definition in [

Thank to these considerations we get the solution of (3.9) corresponding to the boundary condition S m ( z , 0 ) = 0 under the form

S 2 k + 1 ( z , n ) = ( ∂ λ − 2 ∂ Z ) − 1 2 S ^ ′ k + 1 ( Z ) = ∑ j = 0 k ∂ Z j + 1 S ^ k + 1 ( Z ) ∂ 2 λ − j ( 2 λ ) (3.18)

or under the algorithmic form

where we see that the Faulhaber coefficients are successive derivations beginning from the first one of the power sums on integers writing under the form S ^ k + 1 ( Z ) ≡ S 2 k + 1 ( z ) .

Curiously by replacing z with n and consequently Z with u = n ( n − 1 ) in the definition S ^ k + 1 ( Z ) ≡ S 2 k + 1 ( z ) we get the very important and very interesting formula linking the Faulhaber powers sums on integers and on arithmetic progressions

For examples

For calculating B m we remark that from

B m ( z + y ) = e z ∂ y B m ( y ) = B m ( y ) + ⋯ + 1 k ! z k B m ( k ) ( y ) + ⋯ + z m B 0 (y)

and (2.10) we get the recursion relation

which may be written under the matrix form

| 1 ⋯ 1 2 ⋯ 1 3 3 ⋯ 1 4 6 4 ⋯ ⋮ ⋮ ⋱ ⋱ ⋱ ( m + 1 0 ) ( m + 1 1 ) ⋯ ( m + 1 m ) | | B 0 B 1 B 2 B 3 ⋮ B m | = | 1 0 0 0 ⋮ 0 | (4.3)

This matrix equation may be resolved by doing by hand or by program linear combinations over lines from the second one in order to replace them with lines each containing only some non-zero rational numbers.

For instance for calculating successively

Matrix equation for calculating B m .

We remark that the last line of this matrix has replaced the line

B 0 = 1 , B 0 + 2 B 1 = 0 , B 1 + 3 B 2 = 0 , − B 2 + 7 B 6 = 0

For calculating Bernoulli polynomials, we remark that (2.11) and the Jacobi formula (2.18) give rise to the relations

which, knowing

For calculating S m ( z , n ) we utilize the algorithmic formula (3.2) and get the results shown in

Practically for transforming S 2 k + 1 ( z ) into S ^ k + 1 ( Z ) from which one obtains the Faulhaber coefficients let us remark that

so that we may replace the first term z 2 k + 2 of S 2 k + 1 ( z ) with Z k + 1 minus a polynomial in z. The polynomial in z so obtained must begin with a term in z 2 k and not z 2 k + 1 so that we may continue to replace in it z 2 k with a term in Z k

B m B m ( z ) = m ∫ 0 z B m − 1 ( z ) + B m S m ( n ) = ∫ 0 n B m ( n ) ¯ _ B 0 = 1 B 0 ( z ) = 1 S 0 ( n ) = n 1 B 1 = − 1 2 B 1 ( z ) = z − 1 2 S 1 ( n ) = n 2 2 − n 2 B 2 = 1 6 B 2 ( z ) = z 2 − z + 1 6 S 2 ( n ) = n 3 3 − n 2 2 + n 6 B 3 = 0 B 3 ( z ) = z 3 − 3 z 2 2 + z 2 S 3 ( n ) = n 4 4 − n 3 2 + n 2 4 B 4 = − 1 30 B 4 ( z ) = z 4 − 2 z 3 + z 2 − 1 30 S 4 ( n ) = n 5 5 − n 4 2 + n 3 3 − n 30 B 5 = 0 B 5 ( z ) = z 5 − 5 2 z 4 + 5 3 z 3 − z 6 S 5 ( n ) = n 6 6 − n 5 2 + 5 n 4 12 − n 2 12 B 6 = 1 42 B 6 ( z ) = z 6 − 3 z 5 + 5 2 z 4 − z 2 2 + 1 42 S 6 ( n ) = n 7 7 − n 6 2 + n 5 12 − n 3 36 + n 42 B 7 = 0 B 7 ( z ) = z 7 − 7 2 z 6 + 7 2 z 5 − 7 6 z 3 + 1 42 z S 7 ( n ) = n 8 8 − n 7 2 + 7 n 6 12 − 7 n 4 24 + n 2 84 _

B m ( z ) S m ( z , n ) = B m ( z ) n + B m ' ( z ) n 2 2 ! + ... + B m ( m ) ( z ) n m + 1 ( m + 1 ) ! ¯ _ B 0 ( z ) = 1 S 0 ( z , n ) = n B 1 ( z ) = z − 1 2 S 1 ( z , n ) = ( z − 1 2 ) n + n 2 2 ! B 2 ( z ) = z 2 − z + 1 6 S 2 ( z , n ) = ( z 2 − z + 1 6 ) n + ( 2 z − 1 ) n 2 2 ! + 2 n 3 3 ! B 3 ( z ) = z 3 − 3 z 2 2 + z 2 S 3 ( z , n ) = ( z 3 − 3 z 2 2 + z 2 ) n + ( 3 z 2 − 3 z + 1 2 ) n 2 2 ! + ( 6 z − 3 ) n 3 3 ! + 6 n 4 4 ! _

minus a polynomial in z. The operations continue so on until finish.

By this operation we observe that we may omit all odd powers terms in z before and during the transformation of S 2 k + 1 ( z ) into S ^ k + 1 ( Z ) for k > 0 . From these remarks we may establish

This algorithm for obtaining S ^ k + 1 ( Z ) gives rise astonishingly to the easy calculation of the Faulhaber formula for S 2 k + 1 ( z , n ) .

As examples we have

・ S 1 ( z ) = z 2 2 − z 2 → S 1 − ( z ) = z 2 2 → S ^ 1 ( Z ) = Z 2

・ S 3 ( n ) = n 4 4 − n 3 2 + n 2 4 → S 3 − ( z ) = z 4 4 → S ^ 2 ( Z ) = 1 4 Z 2

S 3 ( z , n ) = Z 2 2 λ 1 ! + 1 2 4 λ 2 2 ! = Z λ + λ 2 (4.9)

・ S 5 ( n ) = n 6 6 − n 5 2 + 5 n 4 12 − n 2 12 → S 5 − ( z ) = z 6 6 + 5 z 4 12

S ^ 3 ( Z ) = 1 6 Z 3 − 1 12 Z 2

S 5 ( n ) = 1 6 u 3 − 1 12 u 2

S 5 ( z , n ) = S ^ ′ 3 ( Z ) 2 λ 1 ! + S ^ ″ 3 ( Z ) 4 λ 2 2 ! + S ^ ‴ 3 ( Z ) 8 λ 3 3 ! = ( Z 2 − Z 3 ) λ + ( Z − 1 6 ) 2 λ 2 + 4 3 λ 3 (4.10)

・ B 7 ( z ) = z 7 − 7 2 z 6 + 7 2 z 5 − 7 6 z 3 + 1 42 z

S 7 ( n ) = n 8 8 − n 7 2 + 7 n 6 12 − 7 n 4 24 + n 2 84 → S 7 − ( z ) = z 8 8 + 7 z 6 12 − 7 z 4 24

S ^ 4 ( Z ) = 1 8 ( Z 4 − 6 Z 3 + 17 Z 2 ) + 7 12 ( Z 3 − 3 Z 2 ) − 7 24 Z 2 = 1 8 Z 4 − 1 6 Z 3 + 1 12 Z 2

S 7 ( n ) = 1 8 u 4 − 1 6 u 3 + 1 12 u 2

S 7 ( z , n ) = S ^ ′ 4 ( Z ) 2 λ 1 ! + S ^ ″ 4 ( Z ) 4 λ 2 2 ! + ( 3 Z − 1 ) 8 λ 3 3 ! + 3 16 λ 4 4 ! (4.11)

By derivation of S 2 k + 1 ( z , n ) with respect to z and remarking that

∂ z Z = 2 z − 1 , ∂ z λ = n (5.1)

we obtain the formulae giving S 2 k ( z , n ) in function of Z , λ , z , n

( 2 k + 1 ) S 2 k ( z , n ) = ∂ z S 2 k + 1 ( z , n ) = 2 ( B 1 ( z ) ∂ Z + n ∂ 2 λ ) S 2 k + 1 ( z ) (5.2)

= 2 ( B 1 ( z ) ∂ Z + n ∂ 2 λ ) ∑ j = 1 k + 1 S k + 1 ( j ) ( Z ) ( 2 λ ) j j ! (5.3)

= 2 B 1 ( z ) ( S k + 1 ( 2 ) ( Z ) ( 2 λ ) 1 1 ! + S k + 1 ( 3 ) ( Z ) ( 2 λ ) 2 2 ! + ⋯ ) + 2 n ( S k + 1 ( 1 ) ( Z ) ( 2 λ ) 0 0 ! + S k + 1 ( 2 ) ( Z ) ( 2 λ ) 1 1 ! + ⋯ ) (5.4)

For examples

・ S ^ 2 ( Z ) = 1 4 Z 2 , S ^ ′ 2 ( Z ) = 1 2 Z , S ^ ″ 2 ( Z ) = 1 2

3 S 2 ( z , n ) = 2 B 1 ( z ) S ″ 2 ( Z ) 2 λ + 2 n S ′ 2 ( Z ) + 2 n S ″ 2 ( Z ) 2 λ = ( 2 z − 1 ) λ + ( Z + 2 λ ) n (5.5)

・ S ^ 3 ( Z ) = Z 3 6 − Z 2 12 , S ^ ′ 3 ( Z ) = Z 2 2 − Z 6 , S ^ ″ 3 ( Z ) = Z − 1 6 , S ^ ‴ 3 ( Z ) = 1

5 S 4 ( z , n ) = 2 B 1 ( z ) ( S ^ ″ 3 ( Z ) 2 λ + 2 λ 2 ) + 2 n ( S ^ ′ 3 ( Z ) + S ^ ″ 3 ( Z ) 2 λ + 2 λ 2 ) (5.6)

The main particularity of this work consists in obtaining S m ( z , n ) as the transform of z m by a differential operator, as so as the Bernoulli polynomial B m ( z ) , from which we deduce the new formula ( ∂ n − ∂ z ) S m ( z , n ) = B m ( z ) and get immediately S m ( z , n ) as polynomials in n. From which we get also the property saying that B 1 − 1 ( z ) B 2 k + 1 ( z ) is a homogeneous polynomial of order k in Z = z ( z − 1 ) .

The second particularity consists in performing the change of arguments from z into Z = z ( z − 1 ) and n into λ = S 1 ( z , n ) in order to get the relation ( ∂ n − ∂ z ) = B 1 ( z ) ( ∂ 2 λ − ∂ Z ) which gives rise to the Faulhaber formula of S m ( z , n ) . From this formula we see that the Faulhaber coefficients are successive derivatives of the function S ^ k + 1 ( Z ) ≡ S 2 k + 1 ( z ) where S 2 k + 1 ( n ) are powers sums on integers.

The relation S ^ k + 1 ( Z ) ≡ S 2 k + 1 ( z ) leads also to the Faulhaber formula S 2 k + 1 ( n ) ≡ S ^ k + 1 ( u ) where u = n ( n − 1 ) .

The third particularity of this work is proposing a method for obtaining easily the Bernoulli numbers B m from a matrix equation, then of

B m ( z ) = m ∫ 0 z B m − 1 ( z ) + B m

S m ( z ) = ∫ 0 z B m (z)

S m ( z , n ) = B m ( z ) n + B m ( 1 ) ( z ) n 2 2 ! + ⋯

S 2 k + 1 ( z , n ) = S ^ ′ k + 1 ( Z ) ( 2 λ ) 1 1 ! + S ^ ( 2 ) k + 1 ( Z ) ( 2 λ ) 2 2 ! + ⋯ .

S 2 k + 1 ( n ) = S ^ k + 1 (u)

As conclusion we think that the calculations of powers sums are greatly facilitated by the utilization of derivation operators ∂ z , ∂ n and the translation operator exp ( a ∂ z ) , parts of Operator Calculus.

Operator calculus, which is very different from Heaviside operational calculus, is thus merited to be known and introduced into Mathematical Analysis. Moreover it just had a solid foundation and many interesting applications for instance in the domains of Special functions, Differential equations, Fourier, Laplace and other transforms, quantum mechanics [

The author acknowledges the reviewer for comments leading to the clear proof of the important formula (3.14). The author also acknowledges Mrs. Beverly GUO for her perseverance in the refinement of the article representation.

He reiterates his gratitude toward his adorable wife for all the cares she devoted for him during the long months he performed this work.

The author declares no conflicts of interest regarding the publication of this paper.

Si, D.T. (2019) The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked. Applied Mathematics, 10, 100-112. https://doi.org/10.4236/am.2019.103009