SHIFTED HARMONIC SUMS OF ORDER TWO

Transcription

1 Commu Koea Math Soc 9 0, No, pp SHIFTED HARMONIC SUMS OF ORDER TWO Athoy Sofo Abstact We develop a set of idetities fo Eule type sums I paticula we ivestigate poducts of shifted hamoic umbes of ode two ad ecipocal biomial coefficiets Itoductio ad pelimiaies I this pape we study the summatio of the poduct shifted hamoic sums of ode two ad ecipocal biomial coefficiets, of the fom, H ad its fiite coutepat, i the pocess we shall obtai some ew two paamete hypegeometic fuctio idetities Aalogous esults of Eule type fo ifiite seies have bee developed by may authos, see fo example, ad efeeces theei Let, H γ ψ j jj t 0 t dt be the th hamoic umbe, γ deotes the Eule Mascheoi costat, ψz : dlogγz/dz is the digamma fuctio ad Γz is the well ow gamma fuctio Let R ad C deote, espectively the sets of eal ad complex umbes ad N : {,,3,} the set of atual umbes A geealizedbiomial coefficiet w z may be defied by w : z Γw Γz Γw z ; w, z C Received Septembe 30, Mathematics Subject Classificatio Pimay 05A0, 05A9, B65; Secoday B83, M06, 33B5, 33D60, 33C0 Key wods ad phases hamoic umbes, biomial coefficiets ad gamma fuctio, polygamma fuctio, combiatoial seies idetities ad summatio fomulas, patial factio appoach, hypegeometic idetity 39 c 0 The Koea Mathematical Society

2 0 ANTHONY SOFO ad i the special case whe z, N, we have w : ww w w,!! whee w λ : Γwλ Γw, λ 0; w C\{0} ww wλ, w C, λ N, is ow as the Pochhamme symbol Some well ow Eule sums ae,, H 3 3H H H 3 6ζ5, i we have, fo H ζ3 H ζ ad 8, H H 3 p H p H p H p H H H with 0 0 : j j H j j H jp H j j, whee the geealized th hamoic umbe i powe, H, is defied fo positive iteges ad as H : m m May fiite vesios of hamoic umbe sum idetities also exist i the liteatue, fo example i, 0 we have m m H H H H m m m H m m, 0

3 SHIFTED SUMS ad fom, 8 H H H H H H H 0 Futhe wo i the summatio of hamoic umbes ad biomial coefficiets has also bee doe by Sofo 6 The wos of,, 5, 6, 7, 8, 0,, 5, 0,, 3, 7, 9, ad efeeces theei, also ivestigate vaious epesetatios of biomial sums ad zeta fuctios i simple fom by the use of the Beta fuctio ad by meas of cetai summatio theoems fo hypegeometic seies Lemma Let 0, m, p ad be positive iteges The j mjj 3 m H pm H m H p H H H, m p 5 ad 6 j mjj m H H m m Poof Fo 3, j j H m m H H m mjj m mj m j m H pm H m H H m p H s H s s H 3 mj j m H H p m H, 3 m j Similaly fo Fo 5 we fist ote that fo a abitay seuece X,l the followig idetity holds X,l X,l, l l l

4 ANTHONY SOFO hece Fo 6 m H m m m ms s s sm sms H H s H H s H s s sm m m m m m m H H I the case of o itege values of the agumet z, we may wite the geealized hamoic umbes, H z α, i tems of polygamma fuctios 7 H α ζα α ψ α α!, {,, 3,}, whee ζz is the zeta fuctio Whe we ecoute hamoic umbes at possibleatioalvaluesoftheagumet,ofthefomh α they maybe evaluated by a available elatio i tems of the polygamma fuctio ψ α z o, fo atioal agumets z, 7 ad we also defie H γ ψ, ad H α 0 0 The evaluatio of the polygamma fuctio ψ α a at atioal values of the agumet ca be explicitly doe via a fomula as give by Kölbig 9, o Choi ad Cvijovic 3 i tems of the Polylogaithmic o othe special fuctios Some specific values ae give as 8 H 3 3 H π3 7ζ3, H l, 3 π 3l, ad H 7 may othes ae listed i the excellet boo 98 8G 5ζ Hamoic umbe idetities We ow pove the followig two theoems

5 SHIFTED SUMS 3 Theoem Let N ad fo eal umbe > ad < The we have H H H ζ H H Poof Let h H H Now whee h H ζh H H H H h 3 A lim s H s ad coside the followig expasio:! h! h!! h A, Fo a abitay positive seuece X,p the followig idetity holds hece fom whee! h 0p0 A X p, 0p0 h H H X p,p j j j s j,

6 ANTHONY SOFO h j j j j 0 jj ψj ψj Sice we otice that ψj ψj m0 mj the h Fom, 5 ad 6 m0j m0 j mj H H m ζ H m m h H H H H H m ζ H H H H Hm m ζ H H H H s s s H H Now h H H ζ H H ζ H H H H H H H s s s

7 SHIFTED SUMS 5 ad sice, fom the H h H H H hece the idetity follows H H H ζ H H ζ H H H H H H H s s s The two special cases of ± follows i the ext coollay Coollay Ude the assumptios of Theoem, fom let, the H ζ H H j j j ad fo, 5 H ζ3h H ζ3 H H H H j H j j Poof Coside, the 6 H H 3

8 6 ANTHONY SOFO The fist sum o the ight had side of 6 ca be eplaced by, similaly ζ3 H ζ H 3,,, F 3,, ad substitutig ito 6 we obtai Fo H H ζ3 H ζ H j j If we use the same method as i Theoem, by patial factio expasio we ae able to evaluate 7 we ca also evaluate 8 the fom 7 9 H j ζ3h ζ H H H 3 F 3,,,,, H, ζ3h ζ H 3 H H H ζ F 3,,,,,

9 SHIFTED SUMS 7 Fom ad 9, 5 follows Rewitig the last euality we ae able to wite the oe paamete hypegeometic idetity 0 F 3,,,,, ζ3 H ζ H Fom 8, afte some maipulatios ad simplificatio we ca also obtai some ew idetities, which will be useful i the followig wo: H H H, H p H p p, H H H H Idetity 0 holds, i geeal, fo R\{,, 3,} If ad usig 8, the,,, F 3,, 3lζ 7 ζ3 Example Fom Theoem, let, H fo, l G H 8388π ζ, 5 0 8l G 6π ζ, whee G is the Catala costat Rema Fom Theoem the case ± m, m follows much the same pocedue as i Theoem Some specific cases ae: 6 H H ζ ,

10 8 ANTHONY SOFO ad H ζ Now we coside the followig fiite vesio of Theoem Theoem Let, p N ad fo eal umbe > ad < The we have H H p H H H p H H H p H p H H p H H m H H H p Hmp H m m m Poof To pove we may wite h! h A, whee A is give by 3, ad by a eaagemet of sums! h A j j j j j ψj ψj ψpj ψp

11 j Fom 3, 5 ad 6 h m0 SHIFTED SUMS 9 j m0 mj m0 mp pm ψ p ψ m ψ ψ p m ψmp ψm ψ ψ p 3 h H p m0 p H H mp H m m H p H p H H H H H p H H p m H H p H H H p H p H H mp H m m H H p H H H H p H H H p

12 50 ANTHONY SOFO Sice fom h H H H H p substitutig ito 3 ad afte simplificatio follows, The two special cases of ± follows i the ext coollay ad we will also give two emaable idetities fo hypegeometic fuctios cotaiig two paametes Coollay Ude the assumptios of Theoem, fom let, the ad fo 5 H H H p 3 H p H p H p H Hp H H H p H p H p j H pj H j j p p H H p H3 p H p H p H H p H H p j H jp H j j

13 SHIFTED SUMS 5 Poof Fist let us coside the case Let 6 p p p p ζh 3 F,p,p p3,p,p,p,p F 3 p p p3,p3,p,,, F 3,, Now, by patial factio decompositio we have 7 p p H p H3 p H p H,,, H H p F 3,,3 H H p ad fom 7 ad 0 we ow that p p H H p H3 p H p H Hp H H p

14 5 ANTHONY SOFO The last step is to wite 8 H H, substitutig the ow esult, ad 7 ito 8 we obtai, upo usig, the idetity Fom 6, 7 ad 7 we obtai the ew, two paamete idetity,p,p 3F 9 ad p3,p p p ζ H p H ph p 3F,p,p p3,p pp Fo, let 0 H ad by patial factio expasio Similaly 3 3 Hp 3 F,p,p p3,p H 3 H p 3 H p H H p H H H H p

15 SHIFTED SUMS 53 3 F 3,,,,, p 3,p,p,p F 3 p,p,p p 3 p ad fom, ad we obtai the ew two paamete hypegeometic idetity,p,p,p,p,p F 3 p3,p3,p 3F p3,p 3 p p p p ζ3 H ζ H p H 3 p H p H p Usig the idetities ad ito 0 ad simplifyig we obtai the euied idetity 5 Fom 9 ad 3 we ca also explicitly deive,p,p,p a idetity fo F 3 p3,p3,p Idetities 9 ad 3 hold, i geeal, fo p R\{,, 3,,} If p 3, 5 ad usig 8, the, 3F,, 7 5, 3 3F, 5 7, , ad F 3,,, 5, 5 7 5, 3 0ζ3 5π 3 65, 3F, 5, ζ G

16 5 ANTHONY SOFO Example Some examples follow Usig the idetities,, ad 5, the Also, H H H G 67p8 9pp 6 9 π 75 6 ζ8h p 8 9 H p l H p 3, pp G 63p 5pp 9 5 π 7 ζ 38 5 H p 38 5 H p lh p 5 p 3 H p p H p p 3p3 pp 7p3 9p 7p H 3p p p p H p, H p H p p pp, H 3 p Rema Usig simila piciples as discussed above, it is possible to expess the moe geeal sums, a fothcomig pape H p m, i closed fom This will be doe i Refeeces J Choi, Fiite summatio fomulas ivolvig biomial coefficiets, hamoic umbes ad geealized hamoic umbes, J Ie Appl 9 03, p, Cetai summatio fomulas ivolvig hamoic umbes ad geealized hamoic umbes, Appl Math Comput 8 0, o 3, J Choi ad D Cvijović, Values of the polygamma fuctios at atioal agumets, J Phys A: Math Theo 0 007, o 50, ; Coigedum, ibidem 3 00, o 3, 3980, p J Choi ad H M Sivastava, Some summatio fomulas ivolvig hamoic umbes ad geealized hamoic umbes, Math Comp Modellig 5 0, o 9-0, W Chu, Summatio fomulae ivolvig hamoic umbes, Filomat 6 0, o, 3 5 6, Ifiite seies idetities o hamoic umbes, Results Math 6 0, o 3-, 09

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