A note on the coefficients of power sums of arithmetic progressions
aa r X i v : . [ m a t h . N T ] J a n A NOTE ON THE COEFFICIENTS OF POWER SUMS OF ARITHMETICPROGRESSIONS
JOS´E LUIS CERECEDA
Abstract.
In this note we exhibit a simple formula for the coefficients of the polynomial asso-ciated to the sums of powers of the terms of an arbitrary arithmetic progression. Our formulaconsists of a double sum involving only ordinary binomial coefficients and binomial powers.Arguably, this is the simplest formula that can probably be found for the said coefficients.Furthermore, we give an explicit formula for the Bernoulli polynomials involving the Stirlingnumbers of the first and second kind. Introduction
Consider the sum of k th powers of the terms of an arithmetic progression with first term r and common difference mS m,rk ( n ) = r k + ( m + r ) k + (2 m + r ) k + · · · + (( n − m + r ) k , where k, m, r, and n are assumed to be integer variables with m, n ≥ k, r ≥ S m,rk ( n )turns out to be a polynomial in n of degree k + 1 without constant term, that is, it can beexpressed in the form S m,rk ( n ) = P k +1 t =1 c m,rk,t n t for certain rational coefficients c m,rk,t . In [1], theauthors gave the following formula for c m,rk,t : c m,rk,t = k X j =0 m j W m,r ( k, j ) j + 1 S ( j + 1 , t ) , t = 1 , . . . , k + 1 , (1)where S ( j + 1 , t ) are the (signed) Stirling numbers of the first kind and W m,r ( k, j ) are the r -Whitney numbers of the second kind, which are defined by [2] W m,r ( k, j ) = 1 m j j ! j X i =0 ( − j − i (cid:18) ji (cid:19)(cid:0) mi + r (cid:1) k . On the other hand, Griffiths [3] obtained the following formula for c m,rk,t : c m,rk,t = m k k +1 X j = t k +1 X i = j i (cid:16) rm (cid:17) j − t (cid:18) jt (cid:19) S ( i, j ) S ( k, i − , t = 1 , . . . , k + 1 , (2)where S ( i, j ) are the (signed) Stirling numbers of the first kind and S ( k, i −
1) are the(unsigned) Stirling numbers of the second kind. For other explicit formulas concerning thepolynomial S m,rk ( n ) as a whole see, for example, [4, 5, 6].2. A simpler formula for c m,rk,t More recently, the present author derived the following explicit formula for S m,rk ( n ) ([7,Equation (A7)]): S m,rk ( n ) = m k k + 1 k +1 X j =0 j X i =0 ( − i j + 1 (cid:18) ji (cid:19) (cid:20)(cid:16) i + n + rm (cid:17) k +1 − (cid:16) i + rm (cid:17) k +1 (cid:21) , from which it follows that c m,rk,t = m k k + 1 (cid:18) k + 1 t (cid:19) k +1 X j =0 j X i =0 ( − i j + 1 (cid:18) ji (cid:19) (cid:16) i + rm (cid:17) k +1 − t , t = 1 , . . . , k + 1 . (3)Let us now recall the following combinatorial identity (see, for example, [8, Corollary 3.7]):Let w be any positive integer and P ( z ) be any complex polynomial of degree less than w . Then w X i =0 ( − i (cid:18) wi (cid:19) P ( i ) = 0 . This implies that k +1 X i =0 ( − i (cid:18) k + 1 i (cid:19) (cid:16) i + rm (cid:17) k +1 − t = 0 for all t = 1 , . . . , k + 1 . Hence, the upper limit k + 1 of the first summation in (3) can in fact be lowered to k and thusthe coefficient in (3) can definitely be written as c m,rk,t = m k k + 1 (cid:18) k + 1 t (cid:19) k X j =0 j X i =0 ( − i j + 1 (cid:18) ji (cid:19) (cid:16) i + rm (cid:17) k +1 − t , t = 1 , . . . , k + 1 , (4)consisting of a double sum involving only ordinary binomial coefficients and binomial powers.It is to be noted that (4) can equally be expressed in terms of the r -Whitney numbers of thesecond kind as c m,rk,t = m t − k + 1 (cid:18) k + 1 t (cid:19) k X j =0 ( − j m j j ! j + 1 W m,r ( k + 1 − t, j ) , t = 1 , . . . , k + 1 . (5)This formula can be considered simpler than those in (1) and (2) because each of the numbers S ( j + 1 , t ), S ( i, j ), and S ( k, i −
1) carries at least one extra summation of its own. Indeed,it can be argued (see next section) that (4), or its equivalent form (5), is as simple a formulafor c m,rk,t as one is ever likely to find.Moreover, using (5), one can readily obtain the first few coefficients of highest degree, namely, c m,rk,k +1 = m k k + 1 , k ≥ ,c m,rk,k = m k − (cid:16) r − m (cid:17) , k ≥ ,c m,rk,k − = 112 km k − (cid:0) m − mr + 6 r (cid:1) , k ≥ ,c m,rk,k − = 112 k ( k − m k − r (cid:0) m − mr + 2 r (cid:1) , k ≥ ,c m,rk,k − = 1720 k ( k − k − m k − (cid:0) m − m r + 60 mr − r (cid:1) , k ≥ , etc. 3. Concluding remarks S m,rk ( n ) can alternatively be expressed in terms of the Bernoulli polynomials B k ( z ) in theform [1, Equation 11] S m,rk ( n ) = m k k + 1 k +1 X t =1 (cid:18) k + 1 t (cid:19) B k +1 − t (cid:16) rm (cid:17) n t , k ≥ . (6) NOTE ON THE COEFFICIENTS OF POWER SUMS OF ARITHMETIC PROGRESSIONS 3
Thus, comparing (4) and (6), it follows that B k +1 − t (cid:16) rm (cid:17) = k X j =0 j X i =0 ( − i j + 1 (cid:18) ji (cid:19) (cid:16) i + rm (cid:17) k +1 − t , t = 1 , . . . , k + 1 . In particular, letting t = 1 in the above equation gives us B k (cid:16) rm (cid:17) = k X j =0 j X i =0 ( − i j + 1 (cid:18) ji (cid:19) (cid:16) i + rm (cid:17) k , k ≥ , or, equivalently, B k ( z ) = k X j =0 j X i =0 ( − i j + 1 (cid:18) ji (cid:19)(cid:0) i + z (cid:1) k , k ≥ . (7)Furthermore, by setting z = 0 in (7), we obtain the following well-known explicit formula forthe Bernoulli numbers [9, Equation 1]: B k = k X j =0 j X i =0 ( − i j + 1 (cid:18) ji (cid:19) i k , k ≥ . Regarding the structure of the formulas for Bernoulli numbers, it is worth mentioning Gould’sassertion appearing at the end of his paper [9]: “ the writer has seen no formula for B k whichdoes not require at least two actual summations ,” hinting at the recognition that, in fact, theredoes not exist any elementary formula for B k involving just one (finite) summation. Inspiredby Gould’s conjecture, we can make the ansatz that the above formula for B k , or its equivalentform, B k = k X j =0 ( − j j ! j + 1 S ( k, j ) , k ≥ , (8)as well as the above formula for B k ( z ), constitutes, in a suitably defined sense, the simplestexplicit formula for the Bernoulli numbers and the Bernoulli polynomials, respectively. Con-sequently, with this proviso, and in view of (6) and (7), we may conclude that the expressionin (4), or its equivalent form (5), indeed constitutes the simplest formula that can probably befound for the coefficients of the polynomial S m,rk ( n ).We end this note with the following two additional remarks. Remark 1.
For t = 1 equation (2) becomes c m,rk, = m k k +1 X j =1 k +1 X i = j ji (cid:16) rm (cid:17) j − S ( i, j ) S ( k, i − . By equating this expression to m k B k (cid:0) rm (cid:1) , and replacing the rational rm by the continuousvariable z , we arrive at the following explicit formula for the Bernoulli polynomials involvingthe Stirling numbers of both kinds B k ( z ) = k X j =0 k X i = j j + 1 i + 1 S ( i + 1 , j + 1) S ( k, i ) z j , k ≥ , (9) J.L. CERECEDA which reduces to (8) when z = 0. Let us also observe that, using (9) and the well-knownproperty B k (1) = ( − k B k , yields the identity B k = ( − k k X j =0 k X i = j j + 1 i + 1 S ( i + 1 , j + 1) S ( k, i ) , k ≥ . Remark 2.
For t = 1 equation (1), as well as equation (5), becomes c m,rk, = k X j =0 ( − j m j j ! j + 1 W m,r ( k, j ) , and then B k (cid:16) rm (cid:17) = k X j =0 ( − j j ! j + 1 W m,r ( k, j ) m k − j , k ≥ . This last formula was obtained in [10] as a consequence of [10, Theorem 1].
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