Joural of Classical Aalysis Volume 8, Number 06, 3 30 doi:0.753/jca-08- HARMONIC SERIES WITH POLYGAMMA FUNCTIONS OVIDIU FURDUI Abstract. The paper is about evaluatig i closed form the followig classes of series ivolvig the product of the th harmoic umber ad the polygamma fuctios S H ζ! H ψ, 3, T H ζ! H ψ,, ad R H ζ! H ψ, 3, where is a iteger.. Itroductio ad the mai results The celebrated Riema zeta fuctio ζ is a fuctio of a complex variable see [0, p. 65] defied by ζz z z 3 z, Rz >. z Whe z is a iteger, oe has that the Riema zeta fuctio value ζ is defied by the series formula ζ 3. The polygamma fuctios ψ are defied see [7, p. ] by ψ z d d logγz dz dz ψz 0, z / {0,,,...}, or, i terms of the geeralized or Hurwitz zeta fuctio ζ, ψ z! i0 i z!ζ,z, z / {0,,,...}. Mathematics subject classificatio 00: 0G0, 0A05, 33B5, M35. Keywords ad phrases: Abel s summatio formula, harmoic umbers, polygamma fuctio, Riema zeta fuctio. c D l,zagreb Paper JCA-08-3
OVIDIU FURDUI Series ivolvig closed form evaluatio of ζ are collected i [7] ad, more recetly, i [8]. Other series, ivolvig the Riema zeta fuctio ad harmoic umbers, that evaluate to special costats ca be foud i [5]. The th harmoic umber H is defied, for, by H. A famous sum, due to Euler, i which the th harmoic umber is ivolved is give below [], [7, p. 03], [8,p.8]: E q H q q ζq q ζq ζ, q N \{}, where a empty sum is uderstood to be il. For a proof of the reader is referred to [7, pp. 03 05]. We metio that other series of Euler type ca be foud i [9]. I this paper we evaluate three classes of series ivolvig the product of the th harmoic umber ad the tail of ζ. More precisely, we calculate i closed form the ifiite series S H ζ! H ψ, where 3 is a iteger. We also cosider the series T ad R H H ζ! ζ! where is a iteger. The mai results of this paper are the followig theorems. H ψ,, 3 H ψ, 3, THEOREM. Let 3 be a iteger ad let S be the series i. The, S ζ ζ 3 where the last sum is missig whe 3. a i ζ iζi, I particular whe 3or we have the followig result. COROLLARY. The followig equalities hold: H ζ3 3 3 ζ3 ζ;
b HARMONIC SERIES WITH POLYGAMMA FUNCTIONS 5 H ζ 5 ζ ζ3. THEOREM. Let be a iteger ad let T be the series i 3. The, T E E ζ ζ, where E ad E are the series defied i. a b The followig particular cases are worth metioig. COROLLARY. The followig equalities hold: H ζ ζ5 ζ3 5 8 ζ; H ζ5 5 5 7 8 ζ ζ3 3 ζ5 ζζ3. THEOREM 3. A quadratic harmoic sum a The followig equality holds: H ζ3 3 3 3ζ ζ3ζ. b Let 3 be a iteger ad let R be the sum i. The, R H m m E E S ζ, where E ad E are the series defied i. REMARK. We metio that, the quadratic series has bee evaluated m i terms of products of Riema zeta fuctio values, for all eve iteger,i[3]. We eed i our aalysis Abel s summatio formula [, p. 55], [5, p. 58] which states that if a ad b are two sequeces of real umbers ad A a, the a b A b H m A b b. 5 We will also be usig, i our calculatios, the ifiite versio of the precedig formula a b lim A b A b b. 6
6 OVIDIU FURDUI a b. Some lemmas ad the proofs of the mai results Before we prove the mai results of this paper we eed the followig lemmas. LEMMA. Let be a iteger. The followig equalities hold: H H H ; H H. Proof. The lemma ca be proved by iductio or by a applicatio of formula 5. LEMMA. Let 3 be a iteger. The, ζ ζ ζ. Proof. We apply formula 6, with a adb ζ,ad we have ζ lim ζ ad Lemma is proved. ζ ζ, Now we are ready to prove Theorem. Proof. We apply formula 6, with a H ad b ζ,combied to part a of Lemma ad we have, sice b b, that S lim H H H ζ H H H
HARMONIC SERIES WITH POLYGAMMA FUNCTIONS 7 lim H ζ H H m m m m m E ζ ζ ζ 3 lim H i ζ iζi, where the last equality follows based o formula. We also used that ζ ζ 0, H lim lim ad Theorem is proved. Now we prove Theorem. Proof. We apply formula 6, with a H ad b ζ, combied to part b of Lemma adwehavethat T lim H H H ζ lim H H H H H m H H ζ H m H m m ζ ζ ζ ζ
8 OVIDIU FURDUI m H m m ζ ζ E E ζ ζ, ad Theorem is proved. Next we give the proof of Theorem 3. Proof. b We apply formula 6 with a H ad b H ζ combied to part a of Lemma ad we have, sice b b H ζ, that ζ R lim H H H H [ H H H H lim H H H ζ ζ [ H It is a exercise i classical aalysis to show that lim ad this implies, sice 3, that It follows that R lim H H lim H H H Let [ x H ζ ζ, ζ ζ 0. ]. [ H ζ ]. H ] ζ ].
HARMONIC SERIES WITH POLYGAMMA FUNCTIONS 9 A calculatio shows that x H H H ζ, ad sice H H H H ad we get that Thus, sice x H H H ζ H H H, ζ H ζ y. R x H m Lemma y m m y y 0 m y m H m m H m ζ m ζ m m m H m m E S E ζζ ζ m E S E ζ. a Whe 3 we get, based o part b of the theorem, that H ζ3 3 3 H m m E 3 S 3 E ζ 7 ζ 5 ζ ζ3 ζ ζ3ζ 3ζ ζ3ζ, E 3 5 ζ ζ 5 ζ, E ζ3 ad m 7 ζ.
30 OVIDIU FURDUI 7 m We metio that, the idetity ζ was discovered umerically by Erico Au-Yeug ad proved rigorously by David Borwei ad Joatha Borwei i [] who used Fourier series techiques combied to Parseval s formula for provig it ad a recet proof ivolvig itegrals of polylogarithm fuctios was give i [6]. REFERENCES [] D. D. BONAR AND M. J. KOURY, Real Ifiite Series, MAA, Washigto DC, 006. [] D. BORWEIN AND J. M. BORWEIN, O a itriguig itegral ad some series related to ζ, Proc. Amer. Math. Soc., 3 995, 9 98. [3] D. BORWEIN, J. M. BORWEIN AND R. GIRGENSOHN, Explicit Evaluatio of Euler Sums, Proc. Ediburgh Math. Soc., 38 995, 77 9. [] J. CHOI AND H. M. SRIVASTAVA, Explicit Evaluatios of Euler ad Related Sums, Ramauja J., 0 005, 5 70. [5] O. FURDUI, Limits, Series ad Fractioal Part Itegrals. Problems i Mathematical Aalysis, Spriger, New Yor, 03. [6] O. FURDUI, Series ivolvig products of two harmoic umbers, Math. Mag., 8 0, 37 377. [7] H. M. SRIVASTAVA AND J. CHOI, Series Associated with the Zeta ad Related Fuctios, Kluwer Academic Publishers, Dordrecht, 00. [8] H. M. SRIVASTAVA AND J. CHOI, Zeta ad q-zeta Fuctios Ad Associated Series Ad Itegrals, Elsevier, Amsterdam, 0. [9] A. SOFO AND D. CVIJOVIĆ, Extesios of Euler harmoic sums, Appl. Aal. Discrete Math., 6 0, 37 38. [0] E. T. WHITTAKER AND G. N. WATSON, A Course of Moder Aalysis, Fourth Editio, Cambridge, AT The Uiversity Press, 97. Received July 30, 05 Ovidiu Furdui Departmet of Mathematics Techical Uiversity of Cluj-Napoca Str. Memoradumului Nr. 8, 00, Cluj-Napoca, Romaia e-mail: Ovidiu.Furdui@math.utcluj.ro, ofurdui@yahoo.com Joural of Classical Aalysis www.ele-math.com jca@ele-math.com