A NEW PROOF FOR A CLASSICAL QUADRATIC HARMONIC SERIES. 1. Introduction and the main result

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1 Joural of Classical Aalysis Volume 8, Number (6, 55 6 doi:.753/jca-8-4 A NEW PROOF FOR A CLASSICAL QUADRATIC HARMONIC SERIES CORNEL IOAN VĂLEAN To my parets, Ileaa ad Ioel Abstract. I the followig paper we ited to preset a ew way of calculatig a series similar to the quadratic series of Au-Yeug (see [] H 3 7 ζ(5 ζ(ζ(3, where H deotes the th harmoic umber. We will prove the result by combiig a series of techiques based o the calculatio of two special logarithmic itegrals, the elemetary maipulatios of series ad the the use of the Euler s idetity i (.. Itroductio ad the mai result The followig result H 3 7 ζ(5 ζ(ζ(3, is oe of may such series ivolvig the harmoic umber that Philippe Flajolet ad Bruo Salvy derived i their paper Euler Sums ad Cotour Itegral Represetatios (see [] by meas of cotour itegratio. Like the quadratic series of Au-Yeug that appears i [], the preset series has become a classic i the theory of oliear harmoic series. I this paper we calculate a series similar to the quadratic series of Au-Yeug, but a more advaced oe where the differece is that i deomiator we have ow 3 istead of. Our strategy will ivolve the calculatio of two special logarithmic itegrals, the calculatio of a Euler sum by symmetry reasos ad the use of the Euler s idetity ivolvig harmoic umbers. We state ow a classical quadratic harmoic series summatio result. THEOREM. The followig equality holds: H 3 7 ζ(5 ζ(ζ(3, where H is the th harmoic umber defied, for,byh Mathematics subject classificatio (: 4C, 4A5. Keywords ad phrases: Logarithmic itegrals, harmoic umbers, quadratic series, Euler sums, Riema zeta fuctio. c D l,zagreb Paper JCA

2 56 CORNEL IOAN VĂLEAN Before we prove Theorem we collect some results we eed i our aalysis. Next we prove the lemmas which are used i the proof of Theorem. LEMMA. Let be a iteger. The followig equalities hold: (a I x l( xdx H ; (b J x l ( xdx k H k k H + H( where H ( is the th harmoic umber of order defied, for, byh ( , Proof. (a We have, usig itegratio by parts, that I x l( xdx (x x x log( x x dx +( I ( I x +( I ( I which yields the recurrece relatio i k, ki k (k I k k. Givig values to k from k to ad usig that log( xdx, we obtai that I x log( xdx H, ad the part (a of the lemma is proved. (b We have, usig itegratio by parts as i (a, that J x l ( xdx (x x log x ( x x log( xdx +( J ( J x H +( J ( J, where above we used the part (a of the lemma, I x log( xdx H The, we obtai the recurrece relatio i k, kj k (k J k H k k,.

3 A NEW PROOF FOR A CLASSICAL QUADRATIC HARMONIC SERIES 57 where givig values to k from k to ad usig that log ( xdx, we obtai that J x log ( xdx H k k k H ad the last equality follows immediately from the fact that, whece we get that H i j ij i i i j H i i + H(, ( ij H (, H i ( H + H ( i i ad the part (b of the lemma is proved. We state ow the result of a special Euler sum. LEMMA 3. The followig equality holds: 3 ( ζ(ζ(3 9 ζ(5. Proof. We start with a slightly differet series, S (ζ( 3 where based upo symmetry reasos we have that S Summig both sides, we have that S k k k k 3 ( + k + k k 3 3 ( + k k (k + 3 3k(k + k 3 3 ( + k k 3 + ζ(ζ(3 3 k k k k k 3 ( + k k 3 3 k ( + k k 3 ( + k, 3 (+k k ( + k. k k 3 (+k

4 58 CORNEL IOAN VĂLEAN whece S ζ(ζ(3 3 k k ( + k. Now, we have that k ( + k ( k, k( + k ad multiplyig both sides by, we obtai k ( + k ( 3 k k( + k ( 3 k ( k Therefore, we have that S ζ(ζ(3 3 ζ(ζ(3 3 ζ(ζ(3 3 ζ(ζ(3+3 k 3 4 k k 3 k ( + k ( k 3 4 k H 4 k k ( k + k 9 ζ(5 ζ(ζ(3 where we have used Euler s idetity (see [3, p. 8] H k k Hece, we obtai that. ( k 4 k + k ( k + k k ( + ζ( + ζ( kζ(k +, N,. ( 3 k (ζ( 9 ζ(5 ζ(ζ(3. As a cosequece of the result above, sice ζ( ζ(ζ(3, we obtai that 3 ( ζ(ζ(3 9 ζ(5. 3

5 A NEW PROOF FOR A CLASSICAL QUADRATIC HARMONIC SERIES 59 Now we are ready to prove Theorem. Proof. We have, based o part (b of Lemma,that x ad it follows that x l ( xdx l ( xdx H 3 + H( 3, ( H 3 + H( 3. The, we obtai that ( H 3 + H( 3 H 3 + x H ( 3 log ( xdx. x log ( xdx ( Now, recall the geeratig fuctio of the harmoic umbers is log( t t t H, t <. If itegratig both sides from t tot x, we get that x log ( x Usig (3 i (, we get that t H dt x t H dt x + + H. (3 k x k k log ( xdx k k k k x k+ k ( + H dx x k+ k ( + H dx H k ( + (k + + H k ( + (k + +. (4 Sice we have, by partial fractio decompositio, that ( k (k + + k( + k k + + k ( + ( + ( k k + +,

6 6 CORNEL IOAN VĂLEAN the k k (k + + k Usig (5 i (4, we obtai k H ( k ( + ( + ζ( + H + ( +. H ( ζ( + H + ( + ( k k + + k ( + (k H ζ( ( + H H + ( + 3 H + + ζ( ( + ζ( H+ ζ( (H + + H + ( + 3 H + ( + ζ( ( + 3 H + ( ( + 4 H ζ( 3 (5 H 3 + H 4 H 6ζ(5 3, (6 whereweusedthat H ζ(3 ad H 3ζ(5 ζ(ζ(3 that are both obtaied from Euler s idetity i (. 4 Thus, combiig (6, (4 ad (, we get that H 3 ζ(5 3 Combiig (7 ad Lemma 3, we obtai that H 3 7 ζ(5 ζ(ζ(3, H ( 3. (7 ad the theorem is proved. Ackowledgemet. The author thaks the referee for carefully readig the paper ad makig valuable commets throughout the cotet that led to the preset versio of the paper. Also the author thaks the JCA joural team ivolved i the phases the paper passed through.

7 A NEW PROOF FOR A CLASSICAL QUADRATIC HARMONIC SERIES 6 REFERENCES [] C. I. VĂLEAN AND O. FURDUI, Revivig the quadratic series of Au Yeug, JCA6, (5, o., 3 8. [] P. FLAJOLET AND B. SALVY, Euler sums ad cotour itegral represetatios, Experimet. Math,. 7 (998, [3] H. M. SRIVASTAVA, J. CHOI, Zeta ad q-zeta Fuctios Ad Associated Series Ad Itegrals, Elsevier, Amsterdam (. (Received November 3, 5 Corel Ioa Vălea Teremia Mare, Nr. 63, Timis, 3745, Romaia corel ro@yahoo.com Joural of Classical Aalysis jca@ele-math.com

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