Applied Mathematics, 2019, 10, 100-112 http://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385 The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked Do Tan Si The HoChiMinh-City Physical Association, HoChiMinh-City, Vietnam How to cite this paper: Si, D.T. (2019) The Abstract Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked. Applied Mathemat- Utilizing translation operators we get the powers sums on arithmetic pro- ics, 10, 100-112. gressions and the Bernoulli polynomials of order m under the form of diffe- https://doi.org/10.4236/am.2019.103009 rential operators acting on monomials. It follows that (ddnz− dd) applied Received: February 18, 2019 on a power sum has a meaning and is exactly equal to the Bernoulli poly- Accepted: March 19, 2019 nomial of the same order. From this new property we get the formula giving Published: March 22, 2019 powers sums in term of sums of successive derivatives of Bernoulli polynomi- Copyright © 2019 by author(s) and al multiplied with primitives of the same order of n. Then by changing the Scientific Research Publishing Inc. two arguments zn, into Z= zz( −1) , λ where λ designed the 1st order This work is licensed under the Creative Commons Attribution International power sums and proving that Bernoulli polynomials of odd order vanish for License (CC BY 4.0). arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for pow- http://creativecommons.org/licenses/by/4.0/ ers sums in term of polynomials in λ having coefficients depending on Z. Open Access 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 possibly the Ber- noulli numbers, the Bernoulli polynomials, the powers sums and the Faulha- ber formula for powers sums. Keywords Bernoulli Numbers, Bernoulli Polynomials, Powers Sums, Faulhaber Conjecture, Shift Operator, Operator Calculus 1. Introduction The problem of calculating the sums of the mth powers of n first integers n ∑∑nkmm= (1.1) k =1 DOI: 10.4236/am.2019.103009 Mar. 22, 2019 100 Applied Mathematics D. T. Si is investigated from antiquity by mathematicians around the world. We learn for examples in the thesis of Coen [1] as so as in Edwards [2] that at the beginning of the 11th century ibn al-Haytham had developed the formulae nn2 nnn32 nnn432 ∑ n = + , ∑ n2 =++, ∑ n3 =+− (1.2) 22 3 26 424 In 15th century his successors had found nnn543 n ∑ n4 =++− (1.3) 5 2 3 30 About two centuries quietly passed until the day in 1631 when Faulhaber [3] published at Ulm the prodigious results for sums of odd powers from ∑ n un- til ∑ n17 in term of powers of ∑ n . One may find more details on the works of Faulhaber in the reference [4]. 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 [5]. Then in 1713 in his posthumous Ars conjectandi, Jacob Bernoulli [6], men- tioning Faulhaber, published the lists of ten first ∑ n m . It is plausible that from m this list he observed that ∑ n may be written in terms of the numbers Bk which are the same for all m m m 1 j m +1 mj+−1 ∑∑n =( −) Bnj . (1.4) m +1 j=0 j This famous conjectured formula of Bernoulli was proven in 1755 based on the calculus of finite difference by Euler [7], a researcher working with the Ber- noulli brothers at Zurich. As informed by Raugh [8], Euler had followed de Moivre in given Bk the denomination Bernoulli numbers, introduced the Ber- noulli polynomials in 1738 via the generating function t ∞ 1 zt = m t e ∑ Bm ( zt) (1.5) e1− m=0 m! Returning to the Faulhaber conjecture saying that Snm ( ) is a polynomial in Sn1 ( ) for all m m (m) j Snmj( ) = ∑ ASn1 ( ) (1.6) j=1 we know that Jacobi [9] has the merit of giving the right proof for this conjecture (m) and moreover calculating the first six Faulhaber coefficients Aj although he did not get a formula for obtaining all of them. Another merit of Jacobi consists in pioneering the use of the derivative with respect to n when observing that m 1 ∑ n=( Bnmm++11( +−10) B( )) (1.7) m +1 d mm−1 ∑∑n= Bnmm( ) = mn + B (1.8) dn Long years passed until Edwards [10] showed how to obtain the Faulhaber DOI: 10.4236/am.2019.103009 101 Applied Mathematics D. T. Si coefficients by matrix inversion and Knuth [4] by identification of coefficients of odd powers of n in Bernoulli formula with those in Faulhaber conjecture. Nev- ertheless the methods of Edwards and Knuth are not easy to apply. Following Coen, we know the existence of the work Bernoulli numbers: bibli- ography (1713-1990) of Dilcher [11] which contained 1956 references by 839 authors! Concerning the more general problem of powers sums on arithmetic progres- sions m m m Sm ( zn,1) = z ++( z) +++− ( z( n 1)) (1.9) we remark the recent formula given by Dattoli, Cesarano, Lorenzutta [12] in term of Bernoulli polynomials 1 Szn( , ) =( BznBz++( +−) ( )) (1.10) mm +1 mm11 and the formula of Chen, Fu, Zhang [13] in term of sums of powers of S1 ( zn, ) m j 2m 1 2 2mS21m−−( zn ,) =∑ B22mj(2, S 1( zn) z+− B1( z)) B2m( z) (1.11) j=0 2 j 2 which are not easy to apply. After these authors we have proposed a method leading to the formula [14] k +1 1 ˆ( j) j S21kk++( zn,) = ∑ S21( Z)(2, S 1( zn)) (1.12) j=1 j! ˆ where Z= zz( −1) and SZmm( ) ≡ Sz( ) . The Faulhaber formula is thus obtained but we see that the method for ob- ˆ taining it is cumbersome and the practice calculations of SZmm( ) ≡ Sz( ) 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 nz, and SnZ1 ( ), and ∂∂nz, 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. 2. Representation of a Power Sums and Bernoulli Polynomials by Differential Operators Let m m m Sm ( zn,1) = z ++( z) +++− ( z( n 1)) n−1 m (2.1) =+∈∑( zk) , zC k =0 be the powers sums on an arithmetic progression and mm m Snmm( ) ≡ S(0, n) = 0 +++− 1 (n 1) (2.2) the powers sums of the first integers. DOI: 10.4236/am.2019.103009 102 Applied Mathematics D. T. Si We may utilize the shift or translation operator ea∂z fz( ) = fz( + a) (2.3) to get the differential representation n∂z ∂ (n−∂1) e1− = +z + +z mm = ∀∈ Sm ( zn,) ( 1e e ) z∂ z, n N (2.4) e1z − which gives directly the relations n∂z e1− m ∂=∂zmS( zn, ) z z (2.5) e1∂z − ∂=zmS( zn,,) mSm−1 ( zn) (2.6) and the generating function ∞∞m n∂z m m nt t e−− 1zt e1zt S( zn,e) = = (2.7) ∑∑m ∂z t mm=00mm!!e1− = e1− Because (2.4) is valid for all integers n it is also valid for all real and complex values so that we may write n∂z ∂ z e m ∂=nmS( zn, ) z (2.8) e1∂z − Defining now the set of polynomials Bzm ( ) by the differential representa- tion ∂ z m Bzm ( ) = z (2.9) e1∂z − we see that Bzm ( ) verify m Bzm( +−1) Bz mz( ) =∂ z (2.10) Bmm′ ( z) = mB−1 ( z) (2.11) and have the generating function ∞ ttm 1 t =zt =zt = −+ + + ∑ Bm ( z) t e e1( 1zt ) (2.12) m=0 m!2e1− t 1++ 2 allowing the identification of them with Bernoulli polynomials defined by Euler [7] and giving the first of them 1 Bz( ) = 1, Bz( ) = z − . (2.13) 0 1 2 The Bernoulli polynomials are linked to powers sums according to formulae (2.5), (2.8), (2.9) by the relations ∂ nn∂∂zzz m ∂zmSzn( ,) =( e1 −) ∂ z =( e1 −) Bzm( ) = Bzn m( +−) Bzm( ) (2.14) e1z − z∂n ∂=nmS( zn,e) Bm( n) =+ B m( z n) (2.15) which lead to the followed beautiful formula where the second member does not depend on n DOI: 10.4236/am.2019.103009 103 Applied Mathematics D. T. Si (∂n −∂ zm) S( zn, ) = B m( z) (2.16) Besides it leads also to the formula ∂=nmS( zn,,) mSm−1 ( zn) + Bm( z) (2.17) which jointed with (2.15) gives rise to the historic Jacobi conjectured formula [9] ∂nmS( n) = mSm−1 ( n) += Bm(0) B m( n) (2.18) From (2.18) we see that Bm (0) are identifiable with Bernoulli numbers Bm . 3. The Powers Sums 1) Powers sums in terms of Bernoulli polynomials and powers of n From the Equation (2.16) and the boundary condition Szm ( ,0) = 0 we get immediately the solution of (2.16) 1 Smm( zn, ) = B( z) ∂nz −∂ (3.1) m kk− =∑ ∂∂znB m( zn) k =0 which may be put under the algorithmic form very easy to remember 21m+ (1) nn(m) S( zn, ) = B( z) n + B( z) ++ B( z) (3.2) m mm2! m(m +1) ! or taking into account (2.11) nk +1 S( zn,) = B( z) n + + mm( −11) ( m −+ k) B− ( z) m m mk (k +1!) nm+1 ++ mB! ( z) 0 (m +1!) Putting z = 0 in (3.2) and replacing Bmk− (0) with Bmk− we recognize the famous Bernoulli formula [6] m nnkm++11 Sm( n) = Bn m ++ Bmk− ++B0 (3.3) k km++11 2) The Faulhaber formula on powers sums In Sm ( zn, ) instead of utilizing z and n for arguments let us utilize 1 n2 Z= zz( −1) and λ =S1 ( zn, ) =−+ z n (3.4) 22 Because dZZ ddλλ d =2Bz11( ) , ==+= 0, Bz( ) n, n (3.5) dz dd nn d z we have ∂≡n (Bz1 ( ) + n) ∂λ (3.6) ∂≡zZ2Bz1 ( ) ∂+∂ nλ (3.7) DOI: 10.4236/am.2019.103009 104 Applied Mathematics D.