EVALUATION OF A CUBIC EULER SUM RAMYA DUTTA. H n

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1 Joural of Classical Aalysis Volume 9, Number 6, 5 59 doi:.753/jca-9-5 EVALUATION OF A CUBIC EULER SUM RAMYA DUTTA Abstract. I this paper we calculate the cubic series 3 H ad two related Euler Sums of weight 6 by a techique ivolvig oly the maipulatio of series. We also provide a secod approach of the computatio ivolvig special logarithmic itegrals. Itroductio H The quadratic series of Erico Au-Yeug appeared as Problem 435 i The America Mathematical Mothly [948, pp. 43] proposed by H. F. Sadham ad a solutio due to Marti Keser ivolvig maipulatio of series ad the use of logarithmic itegral appeared i AMM [95, pp. 67]. A recet paper [6] gaveaew proof based o the calculatio of a quadratic logarithmic itegral combied with Abel s summatio formula. I this paper we provide two approaches of evaluatig the cubic Euler sum 3 H.Thefirst is by a techique ivolvig oly the maipulatio of series ad the secod is similar to the techique ivolvig the logarithmic itegral i [6]. The closed form of liear Euler sums: Eq, : H q + q ζq + q j ζ j + ζq j, for q is well ow. We cosider o-liear Euler sums ivolvig the geeralized Harmoic umbers H m m of the form H m H m m,wherem,...,m ad m are positive itegers. W m + +m +m is ow as the weight of the o-liear Euler sum. I this paper we restrict our attetio to o-liear Euler sums of weight 6 that are ivolved i the evaluatio of the cubic Euler sum. We metio that our results are ot ew ad they eist i the mathematical literature. The first Euler sum i Theorem.. ca be evaluated by results proved i [3] ad the secod Euler sum Theorem.3. ca be evaluated by the methods i [] ad[]. Mathematics subject classificatio : 33B5, 33B3, 4B, 4C5. Keywords ad phrases: Logarithmic itegrals, harmoic umbers, Euler sums, Riema zeta fuctio. c D l,zagreb Paper JCA-9-5 5

2 5 R. DUTTA. The two mai lemmas LEMMA.. The followig equality holds: Proof. We have H H + + j j j+ j j H + H + ζ H j + j j j + + j j + j + j j + j + + j + j j + + j + j H j j + j H ζ H. j + j j + j The eplaatios i the previous calculatios are as follows:. Order of summatio iterchaged..3 The chage i variable + j was made..4 Used the symmetry of the summatio with respect to ad j, j j j + j + + j j + j + + j j + j + + j + j + j + + j j j + + j..5 We used the idetity H m m m.6 j j + j j j j j + j j m + j j ζ H jm + j.. H

3 EVALUATION OF A CUBIC EULER SUM 53 Agai, the first series i.6 ca be writte as follows H + + H H H H H + j j j H+ H + j + j H + H..7 Thus, combiig.6 ad.7 the lemma is proved. LEMMA.. The followig equality holds: where it is uderstood that the sum is il whe. H + H 3, Proof. Straightforward from partial fractio decompositio.. Two Euler sum of weight 6 LEMMA.. The followig equality holds: H ζ6 H 5 ζ 3 3 ζ6. Proof. Divide both sides of the idetity i Lemma.. by ad tae the summatio over ad we have, 4 H + H 3 m m + m, where i the later step we made the chage of variable m + i the double summatio. The double summatio ca be evaluated i the followig maer:

4 54 R. DUTTA m 3 m 3 m 3 3 m + m m m m + 3 m m 3 + m 3 3 m m 3 3 m m m> 3 + m m ζ6. >m m m 3 3 m 3 + m 3 These types of double summatios are ow i literature. It is a special case of the Torheim double summatios appearig i [4], [5]. Thus, H 4 + H ζ6. Rearragig the terms ad substitutig the value of E5, we get that H ζ6 H 5 ζ 3 3 ζ6. This completes the proof. THEOREM.. The first Euler sum. The followig equality holds: H 4 ζζ4 ζ3 H + ζ H 3 4 H ζ ζ6 ζ 3. Proof. Dividig both sides of the idetity i Lemma.. with 3 ad taig the summatio over wehave, H + H 4 + ζζ4 H 5 H 3 +.

5 Usig the partial fractio EVALUATION OF A CUBIC EULER SUM 55 H we have: + ζ3 H H 3 + ζ3 ζ Substitutig the value from i ad rearragig, + H 3 H ζ H 3 + H 4. H 4 ζζ4 ζ3 H + ζ H 3 H 5 + H 4 ad usig the result from Lemma.. alog with the values of E,,E3, ad E5, we get the desired closed form. THEOREM.3. The secod Euler sum. The followig equality holds: H H 3 6 ζ3 4ζζ4+4ζ3 97 ζ6+7 4 ζ4ζ+ 5 ζ3 + 3 ζ3. H 4ζ H H ζ6 Proof. Multiplyig both sides of Lemma.. with H ad taig the summatio over, 3 H H + H 3H H m m j m j m j 3 3 H m+ m m + m jm + + j m + + j m j m + + j The eplaatios i the previous calculatios are as follows:. The chage of variable m + was made. m j 3 ζ3....3

6 56 R. DUTTA. Used the idetity H..3 Used the symmetry of the summatio with respect to m, ad j. H H Thus, 3 6 ζ E5, ad the first Euler sum from Theorem.. completes the calculatio. H H 3. Proof of the mai theorem We are ow ready to prove our mai theorem. ad substitutig the value of THEOREM 3.. The cubic Euler sum. The followig equality holds: 3 H 93 6 ζ6 5 ζ3. Proof. Multiplyig both sides of the idetity i Lemma.. by H ad taig the summatio over, 3 H3 + H H + ζh H H H +. Due to the symmetry with respect to, i the double summatio i the right had side we have H H + H H + + H H + H H H. Thus, rearragig the terms we have 3 H H H H 3 ζ H 3 + H 4, ad substitutig the values of the first Euler sum from Theorem.. ad the secod Euler sum from Theorem.3. alog with the value of E,,E3, we get the desired closed form. 4. A secod approach LEMMA 4.. A logarithmic itegral For itegers the followig equality holds: log 3 d H3 + 3H H + H 3.

7 EVALUATION OF A CUBIC EULER SUM 57 Proof. We cosider the Beta Fuctio give by Ba,b a b d ΓaΓb defied over positive reals a,b >. Thus, whe a is a positive Γa + b iteger we calculate the third order partial derivative with respect to b of B,b at b. Usig differetiatio uder the itegratio sig we have, [ ] 3 b 3 B,b b [ 3 b 3 b d ] b log 3 d. O the other had, B,b ΓΓb Γ + b! bb + b +. Taig three successive logarithmic derivatives of B,b with respect to b, b B,b B,b, b + j j j b B,bB,b b + j +, b + j 3 b 3 B,b B,b j j b + j b + j j j b + j + j Whe b wehave [ ] 3 b 3 B,b log 3 d H3 + 3H H + H 3. b The secod proof of Theorem 3.. b + j Proof. Dividig both sides of the idetity i Lemma 4.. by ad taig the summatio over, 3 H H H + H log 3 d log 3 d where Li 3. Li log 3 d, 3 for,,, is the Polylogarithmic fuctio. Iterchage of itegratio ad summatio ca be justified by Toelli s theorem. The Dilogarithm fuctio admits the followig reflectio formula: Li +Li ζ loglog. 4

8 58 R. DUTTA I Usig 4 i the itegral i 3 we have Li log 3 d Li log 3 I + ζi I 3. d+ ζ log 3 d The first itegral: Usig Li H we have I Li log 3 H log 3 d 6 d Li log 3 d H H 4 loglog 4 d + 6ζ6. Thus, substitutig the value from Lemma.. we get, I 6ζ3 + 8ζ6. The secod itegral: I log 3 log 3 d d log 3 d 6 4 6ζ4. The third itegral: Usig log H we have, I 3 loglog 4 log log 4 d d H log 4 H d Substitutig the value of E5, we have I 3 8ζ6+ζ3. Thus, H 5 + 4ζ6. I ζ6 6ζ3 6ζζ4 ζ6 6ζ3. Substitutig the value of the secod Euler sum from Theorem.3. ad H 3 3 ζ6+ ζ3 alog with the value of the itegral I i 3 we get the desired result.

9 EVALUATION OF A CUBIC EULER SUM 59 The sum H 3 3 ζ6+ ζ3 is a special case of the summatio formula a a a + a, where we too a 3. Acowledgemet. The author thas the referee for carefully readig the paper ad maig valuable commets throughout the cotet that led to the preset versio of the paper. Also the author thas the JCA joural team ivolved i the phases the paper passed through. REFERENCES [] J. M. BORWEIN, R. GIRGENSOHN, Evaluatio of triple Euler sums, Electro. J. Combi. 996, 7. [] JUNESANG CHOI, H. M. SRIVASTAVA, Eplicit evaluatio of Euler ad related sums, Ramauja Joural 5, 5 7. [3] C. MARKETT, Triple sums ad the Riema zeta fuctio, J. Number Theory , 3 3. [4] L. TORNHEIM, Harmoic double series, Amer. J. Math. 7 95, [5] J. G. HUARD, K. S. WILLIAMS, AND N.-Y. ZHANG, O Torheim s double series, Acta Arithmetica , 5 7. [6] C. I. VALEAN, O. FURDUI, Revivig the quadratic series of Au-Yeug, J. Classical Aal. 6 5, o., 3 8. Received Jue 3, 6 Ramya Dutta Udergraduate studet Departmet of Mathematics Cheai Mathematical Istitute H, SIPCOT IT Par, Siruseri, Kelambaam 633, Idia dattaramya.ramya@gmail.com Joural of Classical Aalysis jca@ele-math.com