Quadratic Harmonic Number Sums

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1 Applied Matheatics E-Notes, (), -7 c ISSN 67-5 Available fee at io sites of http// Quadatic Haonic Nube Sus Anthony Sofo y and Mehdi Hassani z Received July Abstact In this pape, we obtain soe identities fo the seies P P H n=(n(n+)) and H n= n n+, whee = P n j= j and is a positive intege. Then we obtain soe seies epesentations fo the Apéy s constant, (). Intoduction The Rieann zeta function is de ned fo s C with <(s) > by (s) = P j= j s. Fo intege n > we let n (s) = P n j= j s. We de ne the n-th haonic nube by H n = n (), and the genealized n-th haonic nube by H n () = n (), fo any eal nube. Moeove, we set H () =. Identities fo sus involving haonic nubes, genealized haonic nubes, and thei powes ae ae in nube in the liteatue. A classical exaple is due to L. Eule [], whee fo integes q > he poved that X q H n n q = (q + ) (q + ) X = ( + ) (q ) Soe ecently obtained identities ae P H n=n = 7(4)=4 due to D. Bowein and J. M. Bowein [], the following one due to A. Sofo [7] which is valid fo integes > X n+ = () +! ( ) H () ; and P n j= H j =j H j=j = H n n () due to M. Hassani [5]. In this pape, we obtain soe identities fo the seies S() = X n(n + ) ; and B() = X n n+ whee and ae positive integes. Moe pecisely, we show the following. Matheatics Subject Classi cations 5A, B65, M6, B5, D6, C. y Victoia Univesity College, Victoia Univesity, P. O. Box 448, Melboune City, VIC 8, Austalia z Depatent of Matheatics, Univesity of Zanjan, Univesity Blvd., P. O. Box , Zanjan, Ian ;

2 A. Sofo and M. Hassani THEOREM. Assue that > is an intege, and let Also, fo j 6= let F() = H () + H H () H () + H A(; j) = H j j + Hj + H () j j( j) Then, we have S() = () + F() X j= A(; j) () THEOREM. Assue that > is an intege, and let G() = ()H + H H () + H() + H 6 Also, fo j 6= let C() = H H + H () X j= H j () j j + H j + H j( j)! Then, we have B() = () + G() + X ( ) + C() () = Then we obtain soe new seies epesentations fo (), which is nown as the Apéy s constant (see [4], pp 4 5). Moe pecisely, applying () with = 5, and () with = and = 4, espectively, we get the following. COROLLARY. We have X n(n + 5) = () + 5() COROLLARY. We have X n(n + ) = () and X n n+4 4 = () () + 6

3 Quadatic Haonic Nube Sus Auxiliay Leas In this section we intoduce two auxiliay leas, which ae the base of poofs of ou esults. In what follows below, we will use both of notations H n and H n () fo the n-th haonic nube, and we will apply the following nown [] integal epesentation H n+ n + = Z x n ln( x)dx () Also, we ecall the polylogaith function de ned by Li n z 7! P j= zj =j n fo integal n > and z in the unit dis. The function Li is nown as dilogaith function. We note that Li n () = (n). The identity (4) and its poof of the following Lea is due to Fudui. LEMMA. Assue that > is an intege, and let Then, we have H() = X H () n H () n+ n(n + ) H() = () + ()H() X j= H () j j + T (); (4) whee Also, we have X T () = ( ) j j H() = () j= + ()H() + (j + ) j + H () H () + + H() PROOF. Fo x (; ) we de ne the function f by + H() H() X j= H () j j (5) f(x) = ln(x) ln( x) Z x ln ( x) + Li (x) + ln(u ) du u Since f (x) = and li x! + f(x) =, we iply f(x) =. Thus, fo x (; ) we obtain Z x ln(u ) ln(x) ln( x) ln ( x) + Li (x) = du (6) u

4 A. Sofo and M. Hassani By using (), we get H() = whee By letting Z Z x ln( x) ln( y) I(x) = Z X (xy) n dydx = xy = t in I(x) we obtain I(x) = ln(x) ln( x) + x ln( y) xy dy Z x Z x ln( ln(x + t) dt t Then, we substitute t = u ( x), and we cobine the esult with (6) to get I(x) = x ln ( x) + Li (x) Thus, we obtain We have and siilaly H() = Z Z Z x ln ( x)dx j= Z X x ln ( x)dx = ( ) j+ j X x ln ( x)dx = ( ) j j j= x)i(x)dx; x ln( x)li (x)dx (7) 6 (j + ) 4 ; (8) (j + ) (9) To evaluate the second pat of the integal in (7) we use integation by pats by setting u(x) = Li (x) and v (x) = x ln( x), fo which we get u ln( x) (x) = x and v(x) = P (x ) ln( x) i= xi i. Hence, by consideing (9) we obtain Z x ln( x)li (x)dx = Z x ln ( x)dx! = X H i i () ()H () i= Cobining (7), (8) and () copletes the poof of (4). To pove (5) we conside (4). In the inteest of expessing T () in tes of haonic nubes, we note that H () T () = + H() H () + H() H () H () H () H () = 4 + H() H() + H() H() + H()

5 4 Quadatic Haonic Nube Sus Hence, we get H() = () + () + H() + H () + () H H () H () + and consequently, we obtain (5). This copletes the poof. X j= + H() H () j j + H() H() Ou next lea, gives identities fo P H n=(n(n+)(n+)), whee and ae positive integes. We distinguish two cases 6= and =, which the last case esults in identities involving (). Duing the poof, we need a continuous vesion of haonic nubes (to di eentiate). Such continuous vesions ae available by consideing the elation of haonic nubes with digaa (psi) function and polygaa functions of ode, which ae de ned by (x) = d(log (x))=dx, and () (x) = d (x)=dx, espectively. Note that (x) = R e t t x dt is the Eule gaa function. Since (x + ) = x (x), we have H n = () + (n + ). On the othe hand, it is nown that () = (see [], page 58), whee is the Eule Mascheoni constant (see [4], pp 8 4). Thus H n = + (n + ) () Siila elation fo genealized haonic nubes assets that H (+) n = ( + ) + ( )! whee > is an intege (see [], page 6). LEMMA. Assue that and ae positive integes, and let Then, we have J (; ) = J (; ) = () () Also, fo 6= we have J (; ) = () X + (H ) H () ( ) H () n n(n + )(n + ) () H () + H() () (n + ); () + H H () H () ( ) + H () ( ) + H() PROOF. To pove () we stat fo the nown (see [6]) identity X + H() () ( ) (4) (n) n(n + ) = ( + ) + () () ( + ) = `(); (5) ;

6 A. Sofo and M. Hassani 5 say. Di eentiating both sides of (5) with espect to gives us X (n) n(n + ) = `() `() ; whee `() = d`() d = ( + ) () ( + ) () ( + ) By using (), and consideing the nown popety (x + ) = (x) + =x (see []), we obtain J (; ) = X n(n + ) + X n (n + ) + `() Following the ethod used in the pape by A. Sofo [8], we obtain X n(n + ) = X = n(n + ) (n + ) `() ( + ) + + () () ( + ) ; (6) and X n (n + ) = X n n(n + ) + (n + ) = () + () ( + ) ( + ) (7) Cobining (), (6), (7) and H (p) = H (p) + =p fo p = ; ;, we get (). To pove (4), we assue that 6=, and we apply () and (5) in J (; ) = X H n () n ( ) n + This gives the identity (4) and copletes the poof. n + Finally, by using the esults and techniques fo Wang [9] the following can be shown. LEMMA. Assue that > is an intege. Then, we have X ( ) H () = H() ; (8) = and X ( ) = H () = (H ) 6 + H() H() + H() (9)

7 6 Quadatic Haonic Nube Sus Poof of Theoes We now pove ou Theoes. PROOF of Theoe. We conside the following identity X H n () H n () + P j= n+j X H() = = S() + J (; j) n(n + ) j= Thus S() = H() obtain (). P j= J (; j). By using (5), () and (4) in this identity, we whee PROOF of Theoe. We conside the following expansion Thus, we obtain B() = X A = li! H () n Q n (n + ) = n! B() = = X! H () n n (n + ) C Q A = ( )+! (n + ) = X ( ) + S() = X = A n + ; Now, we use (), and then we apply (8) and (9) to get (). This copletes the poof. Acnowledgent. We than O. Fudui fo the stateent of elation (4) and its poof. Refeences [] M. Abaowitz and I. A. Stegun, Handboo of Matheatical Functions with Foulas, Gaphs, and Matheatical Tables, Dove Publications, 97. [] D. Bowein and J. M. Bowein, On an intiguing integal and soe seies elated to (4), Poc. Ae. Math. Soc., (995) [] L. Eule, Opea Onia, Se., Vol XV, Teubne, Belin, 97. [4] S.R. Finch, Matheatical constants, Encyclopedia of Matheatics and its Applications, Vol. 94, Cabidge Univesity Pess, Cabidge,.

8 A. Sofo and M. Hassani 7 [5] M. Hassani, Identities by L-Suing Method (II), Intenational J. Math. Cobin., (8) 78 86, [6] A. P. Pudniov, Yu. A. Bychov and O. I. Maichev, Integals and Seies Eleentay functions, Vol., Godon and Beach Sci. Publ., New Yo, London, Toyo, 986. [7] A. Sofo. Haonic sus and integal epesentations, J. Appl. Analysis., 6 (), [8] A. Sofo. Tiple integal identities and Zeta functions, Appl. Anal. Discete Math., 4(), [9] W. Wang. Riodan aays and haonic nube identities, Cop. Math. Appl., 6(),