ON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS A. BAZSÓ Astrct. Depending on the prity of the positive integer n the lternting power sum T k n = k + k + + k...+ 1 n 1 n 1 + k. cn e extended to polynomil in two wys sy s T k+ x nd x. In this note we clssify ll the possile decompositions of T k these polynomils. 1. Introduction We denote y C[x] the ring of polynomils in the vrile x with complex coefficients. A decomposition of polynomil F x C[x] is n equlity of the following form which is nontrivil if F x = G 1 G x G 1 x G x C[x] deg G 1 x > 1 nd deg G x > 1. Two decompositions F x = G 1 G x nd F x = H 1 H x re sid to e equivlent if there exists liner polynomil lx C[x] such tht G 1 x = H 1 lx nd H x = lg x. The polynomil F x is clled decomposle if it hs t lest one nontrivil decomposition; otherwise it is sid to e indecomposle. It is well known see e.g [1] tht the lternting power sum T k n := 1 k + k... + 1 n 1 n 1 k cn e expressed y mens of the clssicl Euler polynomils E k x vi the identity: T k n = E k0 + 1 n 1 E k n 1 010 Mthemtics Suject Clssifiction. 11B68 11B5. Key words nd phrses. Euler polynomils decomposition. 1
A. BAZSÓ where the clssicl Euler polynomils E k x re usully defined y the generting function e xt e t + 1 = n=0 E n x tn n! t < π. The motivtion for studying the decomposition of lternting power sums is the lck of results in this direction s well s the long history of the investigtion of the decomposition of the relted power sum S k n := 1 k + k +... + n 1 k. It strted in the 16th century when Johnn Fulher [6] discovered tht for odd vlues of k S k n cn e written s polynomil of the simple sum N given y N = 1 + + 3 +... + n = nn + 1. Anlogously to 1 there exists reltion etween S k n nd the clssicl Bernoulli polynomils B k x those defined y their generting function te xt e t 1 = B n x tn t < π n! nmely we hve n=0 S k n = 1 k + 1 B k+1n B k+1 with of course the clssicl Bernoulli numers B n given y B n = B n 0. Using this connection one cn extend S k n ppropritely to polynomil S k x = 1 k + 1 B k+1x B k+1 x R In [8] Rkczki proved tht the polynomil S k x is indecomposle for even vlues of k. Further he oserved tht for k = v 1 ll the decompositions of S k x re equivlent to the following decomposition: S k x = S v x 1 where S v x is rtionl polynomil of degree v. His result is consequence of theorem of Bilu et l. [5] which sttes tht the Bernoulli
ON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS 3 polynomil B k x is indecomposle for odd k while if k = m is even then ny nontrivil decomposition of B k x is equivlent to B k x = B m x 1 with rtionl polynomil Bm x of degree m. In recent pper Bzsó et l. [3] considered the more generl power sum S k n := k + + k + + k +... + n 1 + k for positive integers n > 1 0 with gcd = 1. In prticulr S10n k = S k n. Agin y the ove definition of S k n cn e extended to hold true for every rel vlue of x s S k x = B k k+1 x + B k+1. k + 1 Bzsó et l. [3] oserved tht the polynomil S k x is indecomposle for even k. If k = v 1 is odd then ny nontrivil decomposition of S k x is equivlent to the following decomposition S k x = Ŝv x + 1 where Ŝv is rtionl polynomil of degree v. Recently Rkczki nd Kreso [9] proved the following result which will e used in the proof of our min result elow out the decomposition of Euler polynomils: Proposition 1. Euler polynomils E k x re indecomposle over C for ll odd k. If k = m is even then every nontrivil decomposition of E k x over C is equivlent to E k x = Ẽm x 1 where Ẽmx = m n=0 m En n n xm n nd E j = j E j 1/. In prticulr the polynomil Ẽ m x is indecomposle over C for ny m N. Proof. This is Theorem 1 in [9]. For positive integer n > 1 nd for 0 coprime integers let T k n := k + k + + k... + 1 n 1 n 1 + k.
4 A. BAZSÓ Clerly T10 k n = T k n. Using generting functions Howrd [7] showed tht T k n cn e written s follows y mens of Euler polynomils: T k n = k E k + 1 n 1 E k n +. 3 Thus depending on the power of 1 in 3 we cn extend T k n to polynomil in the following two wys: T k+ k x := T k k x := E k + E k x + E k E k x +. In this work our gol is to determine ll the possile decompositions of the polynomils T k+ x nd Tk x defined ove. We note tht the decomposition properties of polynomil with rtionl coefficients ply n importnt role in the theory of seprle Diophntine equtions of the form fx = gy see [4]. For relted results on such equtions involving the ove mentioned power sums we refer to [5] [8] [] nd [9].. The min result In this section we pply Proposition 1 in order to derive refinement of Fulher s theorem [6] nd n nlogue of the results of Bzsó et l. [3] in the cse of lternting sums of powers of rithmetic progressions. Theorem 1. The polynomils T k+ x nd Tk x re oth indecomposle for ny odd k. If k = m is even then ny nontrivil decomposition of T k+ x or Tk x is equivlent to T k+ m+ x = T respectively where x + 1 T m+ x = m T m x = m with Ẽmx specified in Proposition 1. or T k m x = T E m + Ẽmx E m Ẽmx x + 1
ON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS 5 Proof. We detil the proof for T k+ proof is essentilly the sme. Let k e n odd positive integer nd put x. For the polynomil Tk x the t := x +. Suppose tht there exists polynomils f 1 t f t C[t] such tht nd deg f 1 t > 1 nd deg f t > 1 T k+ k x = E k + E k t = f 1 f t. 4 From the second equlity in 4 we otin E k t = k f 1f t E k Now putting ft := k f 1 t E k 5 implies tht. 5 E k t = ff t. 6 Thus we otined nontrivil decomposition of the kth Euler polynomil which contrdicts Proposition 1. If k N is even then similrly we hve reltion 6 whence y Proposition 1 it follows tht f t = l t 1 = l x + 1 where lx is liner polynomil. This completes the proof of Theorem 1. As n exmple we consider the lternting sum of the squres of the rithmetic progression + +... n 1 + : Tn = 1 n 1 n + n + +. One cn esily otin the following decompositions: { Tn = n + 1 + +4 4 if n is odd 8 n + 1 + if n is even. 8
6 A. BAZSÓ Acknowledgements The uthor is grteful to the referee for her/his helpful remrks. Reserch ws supported y the Hungrin Acdemy of Sciences the OTKA grnts K75566 nd NK10408 nd in prt y the TÁMOP 4..1./B-09/1/KONV-010-0007 project implemented through the New Hungry Developement Pln co-finnced y the Europen Socil Fund nd the Europen Regionl Developement Fund. References 1. M. Armowitz nd I.A. Stegun Hndook of Mthemticl Functions Ntionl Bureu of Stndrds 1964.. A. Bzsó D. Kreso F. Luc nd Á. Pintér On equl vlues of power sums of rithmetic progressions Gls. Mt. Ser. III 47 01 53 63. 3. A. Bzsó Á. Pintér nd H. M. Srivstv On refinement of Fulher s Theorem concerning sums of powers of nturl numers Appl. Mth. Letters 5 01 486 489. 4. Y. F. Bilu nd R. F. Tichy The Diophntine eqution fx = gy Act Arith. 95 000 61 88. 5. Y. F. Bilu B. Brindz P. Kirschenhofer Á. Pintér nd R. F. Tichy Diophntine equtions nd Bernoulli polynomils with n Appendix y A. Schinzel Compositio Mth. 131 00 173 188. 6. J. Fulher Drinnen die mirculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden Johnn Ulrich Schönigs Augspurg 1631. 7. F. T. Howrd Sums of powers of integers vi generting functions Fioncci Qurt. 34 1996 44 56. 8. Cs. Rkczki On the Diophntine eqution S m x = gy Pul. Mth. Derecen 65 004 439 460. 9. Cs. Rkczki nd D. Kreso Diophntine equtions with Euler polynomils sumitted. Institute of Mthemtics MTA-DE Reserch Group Equtions Functions nd Curves Hungrin Acdemy of Sciences nd University of Derecen P.O. Box 1 H-4010 Derecen Hungry E-mil ddress: zso@science.unide.hu