Annales Mathematicae et Informaticae 42 (20. 0 htt://ami.ektf.hu Averaging sums of owers of integers and Faulhaber olynomials José Luis Cereceda a a Distrito Telefónica Madrid Sain jl.cereceda@movistar.es Submitted May 20 Acceted Setember 29 20 Abstract As an alication of Faulhaber s theorem on sums of owers of integers and the associated Faulhaber olynomials in this article we rovide the solution to the following two questions: ( when is the average of sums of owers of integers itself a sum of the first n integers raised to a ower? and (2 when is the average of sums of owers of integers itself a sum of the first n integers raised to a ower times the sum of the first n squares? In addition to this we derive a family of recursion formulae for the Bernoulli numbers. Keywords: sums of owers of integers Faulhaber olynomials matrix inversion Bernoulli numbers MSC: C08 B68. Introduction Recently Pfaff [] investigated the solutions of the equation n i= ia + ( n n c i= ib = i (. 2 i= for ositive integers a b and c and found that the only solution (a b c to (. with a b is ( 4 (the remaining solutions being the trivial one ( and the well-known solution ( 2. Furthermore Pfaff rovided some necessary 0
06 J. L. Cereceda conditions for n i= ia + n i= ia2 + + n i= iam m ( n am = i (.2 to hold. Secifically by assuming that a a 2 a m < a m = = a m (with a a 2... a m ositive integers Pfaff showed that any solution to (.2 must fulfil the condition m = a m. (. 2am There are infinitely many solutions to (.. For examle for a m = 8 the set of solutions to (. is given by (a m m = (8 6 with. However as Pfaff himself ointed out [] it is not known if any given solution to (. also yields a solution to (.2 so that solving this roblem for m > will require some other aroach. In this article we show that for any given value of a m there is indeed a unique solution to equation (.2 on the understanding that the fraction m is given in its lowest terms. Interestingly this is done by exloiting the roerties of the coefficients of the so-called Faulhaber olynomials [2 4 6]. Although there exist more direct ways to arrive at the solution of equation (.2 (for examle by means of the binomial theorem or by mathematical induction our edagogical aroach here will serve to introduce the (relatively lesser known toic of the Faulhaber olynomials to a broad audience. In addition to the equation (.2 considered by Pfaff we also give the solution to the closely related equation n i= ia + n i= ia2 + + n i= iam m i= ( n ( n am = i i 2 (.4 where now a m 0. Obviously for a m = 0 we have the trivial solution a = a 2 = = a m = 2. In general it turns out that all the owers a a 2... a m on the left-hand side of (.4 must be even integers whereas those aearing in the left-hand side of (.2 must be odd integers. This is a straightforward consequence of the following theorem. 2. Faulhaber s theorem on sums of owers of integers Let us denote by S r the sum of the first n ositive integers each raised to the integer ower r 0 S r = n i= ir. The key ingredient in our discussion is an old result concerning the S r s which can be traced back to Johann Faulhaber (80 6 an early German algebraist who was a close friend of both Johannes Keler and René Descartes. Faulhaber discovered that for even owers r = 2k (k S 2k can be ut in the form [ (2k S 2k = S 2 F 0 + F (2k S + F (2k 2 S 2 + + F (2k ] k Sk (2. i= i=
Averaging sums of owers of integers and Faulhaber olynomials 0 whereas for odd owers r = 2k + (k S 2k+ can be exressed as S 2k+ = S 2 [ (2k+ F 0 + F (2k+ S + F (2k+ 2 S 2 + + F (2k+ k S k ] (2.2 where {F (2k j } and {F (2k+ j } j = 0... k are sets of numerical coefficients. Equations (2. and (2.2 can be rewritten in comact form as S 2k = S 2 F (2k (S (2. S 2k+ = S 2 F (2k+ (S (2.4 where both F (2k (S and F (2k+ (S are olynomials in S of degree k. Following Edwards [2] we refer to them as Faulhaber olynomials and by extension we call F (2k j and F (2k+ j the Faulhaber coefficients. Next we quote the first instances of S 2k and S 2k+ in Faulhaber form as S 2 = S 2 S = S 2 S 4 = S 2 [ + 6 S ] S = S 2 [ + 4 S ] S 6 = S 2 [ 6 S + 2 S2 ] S = S 2 [ 4 S + 2S 2 ] [ S 8 = S 2 + 6 S 8 S2 + 8 ] S S 9 = S[ 2 + 2 S 4S 2 + 6 ] S [ S 0 = S 2 0 S + 68 S2 80 S + 48 ] S4 S = S[ 2 20 S + 4 S2 2 S + 6 ] S4. Note that from the exressions for S and S we quickly get that S+S 2 = S 4. Let us now write the equations (.2 and (.4 using the notation S r for the sums of owers of integers S a + S a2 + + S am m = S am (2. and S a + S a2 + + S am m = S 2 S am. (2.6 Since S = n(n + /2 and S 2 = (2n + S / from (2. and (2.2 we retrieve the well-known result that S r is a olynomial in n of degree r +. From this result it in turn follows that the maximum index a m on the left-hand side of (2. is given by a m = 2a m a condition already established in []. In fact in order for equation (2. to hold it is necessary that all the indices a a 2... a m aearing in the left-hand side of (2. be odd integers. To see this suose on the contrary
08 J. L. Cereceda that one of the indices is even say a j. Then from (2. and (2.4 the left-hand side of (2. can be exressed as follows: L(S S 2 = S 2F (aj (S + SP 2 (S m where F (aj (S and P (S are olynomials in S. On the other hand for nonnegative integers u and v it is clear that S 2 S u S v irresective of the values of u and v as S 2 S u (S v is a olynomial in n of odd (even degree. This means that S 2 F (aj (S cannot be reduced to a olynomial in S from which we conclude in articular that L(S S 2 S am. Similarly using (2. and (2.4 it can be seen that in order for equation (2.6 to hold all the indices a a 2... a m in the left-hand side of (2.6 have to be even integers the maximum index a m being given by a m = 2a m + 2.. Faulhaber s coefficients The sets of coefficients {F (2k j } and {F (2k+ j } satisfy several remarkable roerties a number of which will be described below. As it haens with the binomial coefficients and the Pascal triangle the roerties of the Faulhaber coefficients are better areciated and exlored when they are arranged in a triangular array. In Table we have dislayed the set {F (2k+ j } for k = 2... 0 while the corresonding coefficients {F (2k j } (also for k = 2... 0 are given in Table 2. The numeric arrays in Tables and 2 also can be viewed as lower triangular matrices with the rows being labelled by k and the columns by j. The following list of roerties of the Faulhaber coefficients are readily verified for the coefficients shown in Tables and 2. They are however comletely general.. The Faulhaber coefficients are nonzero rational numbers. 2. The entries in a row have alternating signs the sign of the leading coefficient (which is situated on the main diagonal being ositive.. The sum of the entries in a row is equal to unity k j=0 F (2k j =. k j=0 F (2k+ j = 4. The entries on the main diagonal are given by F (2k+ 2 k 2k+ k = 2k k+ and F (2k k = and the entries in the j = 0 column are F (2k+ 0 = 2(2k + B 2k and F (2k 0 = 6B 2k where B 2k denotes the 2k-th Bernoulli number. Furthermore the entries in the j = column are connected to those in the j = 0 column by the simle relations F (2k+ = 4F (2k+ 0 = 8(2k + B 2k and F (2k = 6F (2k 0 = 6B 2k.
Averaging sums of owers of integers and Faulhaber olynomials 09 k\j 0 2 4 6 8 9 2 4 4 2 4 20 6 69 0 40 8 6 486 9 2 468 2 0 2222 2 4 2 0 944 2 4468 4948 488908 0684 6 4 2 8 04 004 0 2066 6 492 0 80 448 2404 96 9 242 6 80688 6 408 448 42 442 26 9 6944 024 44096 040 6 Table : The set of coefficients {F (2k+ j } for k 0. 26 68 024. There exists a relation between F (2k+ j and F (2k j namely F (2k+ 2(2k + j = (j + 2 F (2k j j = 0... k. (. This formula allows us to obtain the k-th row in Table from the k-th row in Table 2 and vice versa. 6. The entries F (2k+ k 2 F (2k+ k... F (2k+ 0 within the k-th row in Table can be successively obtained by the rule q ( k + j 2 j F (2k+ k j = 0 q k (.2 2q + 2j j=0 given the initial condition F (2k+ k = 2k k+. By alying the bijection (. to the coefficients F (2k+ k j one gets the corresonding rule for the entries in the k-th row in Table 2.. For any given k and for each j = 0... k we have k ( k odd(r F (2k r j = 0 (. r r= where odd(r restricts the summation to odd values of r i.e. odd(r = (0 for odd (even r. Similarly by alying the bijection (. to the coefficients F (2k r j it can be seen that for any given k and for each j = 0... k k+ ( k + 2 even(r (2k + rf (2k+2 r j = 0 (.4 r + r=0
0 J. L. Cereceda k\j 0 2 4 6 8 9 2 6 6 2 4 6 8 0 6 69 4 42 8 6 8 486 9 26202 0 46 446 4 888 9 202 8 9896 04666 0068 2 8 68 80 46 2 000668 66 028 9 48 28 240 22 96 226 2 4484 96 448 224 406 66 2668 9 4928 92 22048 6060 84 828 9 4608 9 Table 2: The set of coefficients {F (2k j } for k 0. 68 9 6 840 2 where even(r = (0 for even (odd r icks out the even ower terms. Informally we may call the roerty embodied in equations (. and (.4 the sum-to-zero column roerty as the coefficients F (2k r j [F (2k+2 r j ] entering the summation in (. [(.4] ertain to a given column j. This is to be distinguished from the sum-to-zero row roerty in equation (.2 where the coefficients F (2k+ k j belong to a given row k. 8. For comleteness next we write down the exlicit formula for F (2k+ j which was originally obtained in [ Section 2]. Adating the notation in [] to ours we have that F (2k+ j = ( j 2j+2 j + 2 j/2 r=0 ( ( 2j + 2r 2k + B 2k 2r (. j + 2r + for j = 0... k and where j/2 denotes the floor function of j/2 namely the largest integer not greater than j/2. The set of coefficients {F (2k j } can then be found through relation (.. We shall use relation (. in Section 6 to derive a family of recursion formulae for the Bernoulli numbers. 4. Averaging sums of owers of integers Interestingly enough the sum-to-zero column roerty in equations (. and (.4 rovides the solution to the roblem of averaging sums of owers of integers in equations (2. and (2.6. For the sake of brevity next we focus on the connection between (. and (2.. An analogous reasoning can be made to establish the link between (.4 and (2.6. To gras the meaning of equation (. consider a concrete
Averaging sums of owers of integers and Faulhaber olynomials examle where k =. Then the column index j takes the values j = 0 2 4 and (. gives rise to the following five equalities: ( ( F 0 + ( ( F 0 + ( (9 F 0 + ( ( F 0 = 0 ( ( F + ( ( F + ( (9 F + ( ( F = 0 ( ( F 2 + ( ( F 2 + ( (9 F 2 + ( ( F 2 = 0 ( ( F + ( ( F + ( (9 F = 0 ( ( F 4 + ( ( F 4 = 0. For a reason that will become clear in just a moment we add to this list of equalities a last one to include the value of ( ( F namely ( ( F = 2 6. Furthermore we multily the first equality by S 0 the second equality by S the third equality by S 2 and so on that is ( ( F 0 S 0 + ( ( F 0 S 0 + ( (9 F 0 S 0 + ( ( F 0 S 0 = 0 ( ( F S + ( ( F S + ( (9 F S + ( ( F S = 0 ( ( F 2 S 2 + ( ( F 2 S 2 + ( (9 F 2 S 2 + ( ( F 2 S 2 = 0 ( ( F S + ( ( F S + ( (9 F S = 0 ( ( F 4 S 4 + ( ( F 4 S 4 = 0 ( ( F S = 2 6 S. Now we can see that the sum of the entries in the first column is just ( times the Faulhaber olynomial F ( (S the sum of the entries in the second column is ( times F ( (S the sum of the third column is ( times F (9 (S and the sum of the fourth column is ( times F ( (S. Then we have ( F ( (S + ( F ( (S + ( F (9 (S + ( F ( (S = 2 6 S. Next we multily both sides of this equation by S 2 and divide them by 2 6. Thus taking into account (2.4 we finally obtain ( S + ( S9 + ( S + ( S 2 6 = S. (4. Since ( ( + ( + ( + = 2 6 the identity (4. constitutes the solution to (2. for the articular case a m =. In this case we have that m = 2 in accordance 6 with condition (.. In general for an arbitrary exonent a m the solution to equation (2. is given by am r= odd(r ( a mr S2am r 2 am = S am. (4.2 A few comments are in order concerning the solution in (4.2. In the first lace by the constructive rocedure we have used to obtain the solution (4. for the
2 J. L. Cereceda case a m = it should be clear that the solution (4.2 is unique for each a m the quotient m characterizing the solution being determined (when exressed in lowest terms by the relation m = am 2. Secondly for odd (even a am m the numerator of (4.2 involves am+ 2 ( am 2 different sums S j with j being an odd integer ranging in a m j 2a m (a m + j 2a m. Furthermore the binomial coefficients fulfil the identity a m r= odd(r ( a mr = 2 a m thus ensuring that the overall number of terms aearing in the numerator of (4.2 equals 2 am. For examle for a m = from (4.2 we get the solution S+S 4 = S which was also found in []. For a m = 4 noting that m = 4 8 = 2 we get the solution S +S 2 = S 4 which as we saw corresonds to the one denoted as (a b c = ( 4 in []. More sohisticated examles are for instance S 9 + 6S + 26S + 84S + 9S 26 = S 9 and ( S2 + 28S 2 + 86S 2 + 980S 2 + 4990S 29 02 + 4990S + 980S + 86S + 28S + S 9 = S 20. On the other hand starting with equation (.4 and making an analysis similar to that leading to equation (4.2 one can deduce the following general solution to equation (2.6 namely am+ a m+2 r=0 even(r ( a m+2 r+ (2am + rs 2am+2 r = S 2 S am 2 am. (4. m Now for each a m 0 the quotient characterizing the solution (4. turns out to be m = 2am+ 2. Further for odd (even a am m the numerator of (4. involves am+ 2 ( am 2 + different sums S j with j being an even integer ranging in a m + j 2a m + 2 (a m + 2 j 2a m + 2. Moreover the following identity holds a m+ ( am + 2 even(r (2a m + r = 2 am (a m + 2 r + r=0 and then the overall number of terms in the numerator of (4. is 2 am. For examle for a m = from equation (4. we find ( S8 + 89S 20 + 4692S 22 + 00S 24 + 0406S 26 926 + 40998S 28 + 8222S 0 + 2096S 2 + 8S 4 + S 6 = S2 S. Finally we note that by combining (4.2 and (4. we obtain the double identity (with a m :
Averaging sums of owers of integers and Faulhaber olynomials (a m + 2 a m r= odd(r = 6 ( am a m+ r=0. Matrix inversion S 2 S 2am r r ( am + 2 even(r r + = 2 am (a m + 2S 2 S am. (2a m + rs 2am+2 r It is worth ointing out that for any given a m we can equally obtain S am (S 2 S am by inverting the corresonding triangular matrix formed by the Faulhaber coefficients in Table (Table 2. This method was originally introduced by Edwards [] (see also [2] to obtain the Faulhaber coefficients themselves by inverting a matrix related to Pascal s triangle. As a concrete examle illustrating this fact consider the equation (2.2 written in matrix format u to k = 6: 0 S S S S 9 S S = S 4 4 2 0 2 6 4 6 20 69 0 2 0 944 2 4 2 492 0 80 0 S 2 S S 4 S S 6 S. (. Let us call the square matrix of (. F. Clearly F is invertible since all the elements in its main diagonal are nonzero. Then to evaluate the column vector on the right of (. we re-multily by the inverse matrix of F on both sides of (. to get S 2 0 S S 4 4 S S 4 0 S = 2 2 0 S 0 6 8 6 S 9 S 6 6 8 6 0 S from which we obtain the owers S 2 S S 4... exressed in terms of the odd ower sums S S S.... Of course the resulting formula for S agrees with that in equation (4.. Conversely by inverting the matrix F we get the corresonding Faulhaber coefficients. Note that the elements of F are nonnegative and that the sum of the elements in each of the rows is equal to one. In fact the row elements of F are given by the corresonding coefficients ( a mr /2 a m aearing in the left-hand side of (4.2. 2 S
4 J. L. Cereceda Similarly writting the equation (2. in matrix format u to k = 6 we have S 2 S 4 S 6 S 8 S 0 S 2 = 0 S 2 6 S 2 S 6 2 0 S 2 S 2 6 8 8 S 2 S. (.2 68 80 48 0 S 2 S 4 S 2 S 0 69 4 446 4 888 9 28 240 Let us call the square matrix of (.2 G. Then re-multilying both sides of (.2 by the inverse matrix of G we obtain S 2 0 S 2 S 2 S 6 6 S 4 S 2 S 2 S 6 S 2 S = S 2 S 4 S 2 S 0 2 0 24 48 96 96 2 0 2 8 8 9 2 48 0 96 96 from which we can determine S 2 times the owers S S 2 S... in terms of the even ower sums S 2 S 4 S 6.... Likewise we can see that the elements of G are nonnegative and that the sum of the elements in each row is equal to one. In view of this examle it is clear that the formulae (4.2 and (4. can be regarded as a rule for calculating the inverse of the triangular matrices in Tables and 2 resectively. Moreover the uniqueness of the solutions in (4.2 and (4. follows ultimately from the uniqueness of the inverse of such triangular matrices. 6. A family of recursion formulae for the Bernoulli numbers As a last imortant remark we note that the sum-to-zero column roerty allows us to derive a family of recursive relationshis for the Bernoulli numbers. Consider initially the equation (. for the column index j = 0. So recalling that F (2k+ 0 = 2(2k + B 2k we will have (for odd r that F (2k r 0 = 2(2k rb 2k r and then equation (. becomes (for j = 0 r= S 8 S 0 S 2 k ( k odd(r (2k rb 2k r = 0 (6. r which holds for any given k. For examle for k = from (6. we obtain B 2 + 90B 4 + 9B 6 + 208B 8 + 209B 20 + 06B 22 + 2B 24 = 0
Averaging sums of owers of integers and Faulhaber olynomials and so knowing B 2 B 4 B 6 B 8 B 20 and B 22 we can get B 24. On the other hand from equation (. we obtain F (2k+ 2 = 4 (2k + [ 0B 2k + k(2k B 2k 2 ] from which we in turn deduce that for odd r F (2k r 2 = 40(2k rb 2k r + 4 ( 2k r B 2k r. Therefore recalling (6. from equation (. with j = 2 we obtain the recurrence relation k ( ( k 2k r odd(r B 2k r = 0 (6.2 r r= which holds for any given k. On the other hand from equation (. we obtain F (2k+ 4 = 6 4 (2k + [ 80B 2k + 20k(2k B 2k 2 + k(k (2k (2k B 2k 4 ] from which we in turn deduce that for odd r ( F (2k r 2k r 4 = 44(2k rb 2k r + 224 B 2k r + 2 ( 2k r B 2k r. Thus taking into account (6. and (6.2 we see that for j = 4 equation (. yields the recurrence relation k ( ( k 2k r odd(r B 2k r = 0 (6. r r= which holds for any given k. The attern is now clear. Indeed by assuming that F (2k r 2s (for odd r is of the form s ( F (2k r 2k r 2s = f q (2k r B 2k r 2q 2q + q=0 with the f q (2k r s being nonzero rational coefficients from equation (. one readily gets the following general recurrence relation for the Bernoulli numbers: k ( ( k 2k r odd(r B 2k r 2s = 0 (6.4 r 2s + r= which holds for any given k 2s+ with s = 0 2.... Formulae (6. (6.2 and (6. are articular cases of the recurrence (6.4 for s = 0 and 2 resectively.
6 J. L. Cereceda Similarly starting from equation (.4 it can be shown that k+ ( ( k + 2 2k + r even(r B 2k+2 r 2s = 0 (6. r + 2s + r=0 which holds for any given k 2s + with s = 0 2.... It is easy to see that relations (6.4 and (6. are equivalent to each other. Moreover we note that the recurrence (6.4 is essentially equivalent to the one given in [8 Theorem.].. Conclusion In this article we have tackled the roblem of averaging sums of owers of integers as considered by Pfaff []. For this urose we have exressed the S r s in the Faulhaber form and then we have used certain roerties of the coefficients of the Faulhaber olynomials. Indeed as we have seen the sum-to-zero column roerty in equations (. and (.4 constitutes the skeleton of the solutions dislayed in (4.2 and (4.. It is to be noted on the other hand that the formulae (4.2 and (4. can be obtained in a more straightforward way by a roer alication of the binomial theorem (for a derivation of the counterart to the formulae (4.2 and (4. using this method see [ Subsections.2 and.]. Furthermore a demonstration by mathematical induction of the identities in (4.2 and (4. (although exressed in a somewhat different manner already aeared in [9]. We believe however that our aroach here is worthwhile since it introduces an imortant toic concerning the sums of owers of integers that may not be widely known namely the Faulhaber theorem and the associated Faulhaber olynomials. We invite the interested reader to rove some of the roerties listed above and to ursue the subject further [ 4 6 0 2]. References [] Pfaff T. J. Averaging sums of owers of integers College Math. J. 42 (20 402 404. [2] Edwards A. W. F. Sums of owers of integers: a little of the history Math. Gaz. 66 (982 22 28. [] Edwards A. W. F. A quick route to sums of owers Amer. Math. Monthly 9 (986 4 4. [4] Knuth D. E. Johann Faulhaber and sums of owers Math. Com. 6 (99 2 294. [] Kotiah T. C. T. Sums of owers of integers A review Int. J. Math. Edu. Sci. Technol. 24 (99 86 84. [6] Beardon A. F. Sums of owers of integers Amer. Math. Monthly 0 (996 20 2.
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