RECIPROCAL POWER SUMS. Anthony Sofo Victoria University, Melbourne City, Australia.

Transcription

1 #A39 INTEGERS () RECIPROCAL POWER SUMS Athoy Sofo Victoia Uivesity, Melboue City, Austalia. Received: /8/, Acceted: 6//, Published: 6/5/ Abstact I this ae we give a alteative oof ad a itegal eesetatio fo a eciocal owe sum give by Bhataga, amely the esult to qq (q ) (, ) Q (q) (, ), whee Q (q) (, ) dq d q, ad the geealie.. Itoductio ad Pelimiaies Bhataga [] states the followig telescoig theoem. Theoem. ([]) Let u k, v k ad w k be thee sequeces, such that u k v k w k. The we have k u u w k u w k w v u k v v () ovided oe of the deomiatos i () ae eo. Bhataga the idicates that the idetity k k (k )... (k m) m m! ( )... ( m) () ca be oved by the telescoig theoem. I this ae we shall give a itegal eesetatio of () ad the geealie the esult to give idetities fo fiite sums fo oducts of geealied hamoic umbes ad biomial coefficiets. Fist we give some defiitios which will be useful thoughout this ae.

2 INTEGERS: () The hamoic umbes i owe α ae defied as H (α) ad the th Hamoic umbe, fo α, H () H α. t t dt γ ψ ( ), whee γ deotes the Eule-Mascheoi costat, defied by γ lim log ψ () Let C be the set of comlex umbes, R the set of eal umbes, N : N {} {,,, 3...}, the fo w C\Z, (w) is Pochhamme s symbol defied by (w) Γ (w ) Γ () w(w )...(w ), if N, if, (3) hee Z deotes the set of o ositive iteges ad the Gamma ad Beta fuctios ae defied esectively as ad Γ () B (s, ) B (, s) w e w dw, fo Re () >, w s ( w) dw Γ (s) Γ () Γ (s ) fo Re (s) > ad Re () >. The biomial coefficiet is defied as Γ ( ) w Γ (w ) Γ ( w ) fo ad w o-egative iteges, whee Γ (x) is the Gamma fuctio. olygamma fuctios ψ (k) (), k N ae defied by The ψ (k) () : dk dk log Γ () dk d k Γ () Γ () [log(t)] k t t ad ψ () () ψ (), deotes the Psi, o digamma fuctio, defied by ψ () d d log Γ () Γ () Γ (). dt, k N (4)

3 INTEGERS: () 3 We also ecall the elatio, fo m,, 3,... H (m) ζ (m ) ( )m ψ (m) (). (5) m! Thee ae may esults of the tye () fo fiite, ifiite ad alteatig sums. The esult q, fo q q aeas i [5]. Othe esults aea i [, 3, 4, 6, 7, 9,,, 3, 4, 5].. Geealiatios The ext lemma deals with a alteative oof of () fom which moe geealied idetities ca be ascetaied. Lemma. Let R\Z, t R ad N. The ad whe t, t t t ( x) tx ( (tx) ) dx (6),, F t t F t ( x) ( x ) dx (8), whee F t is the Gauss hyegeometic fuctio. Poof. Fist otice that, with t, () may be witte as t t Γ () Γ ( ) t B (, ) Γ ( ) (7) ( x) x (tx) dx t ( x) tx ( (tx) ) dx,

4 INTEGERS: () 4 which is the itegal (6) ad efomig the itegatio, esults i (7). Whe t, we obtai (8). It may be of some iteest to ote that fom (7), with t, ad (8) we have the hyegeometic idetities F, ad F,. Remak 3. It is ossible to massage (8) to oduce idetities of the fom; ( ) fo a ositive itege. H H H The ext lemma deals with the deivatives of biomial coefficiets. Lemma 4. Let, > ad let Q(, ) be a aalytic fuctio of C\ {,, 3,...}. The, Q () (, ) dq d Q(, )P (, ), P (, ) whee fo > Q(, ) [ψ ( ) ψ ( )] H, fo, ad Q (λ) (, ) dλ Q d λ λ λ ρ ρ Q (ρ) (, )P (λ ρ) (, ), fo λ (9) whee P () (, ) P (i) (, ) di P d i, fo N ad Q() (, ) Q(, ). Fo i N, di d i A oof of Lemma 4 is give i [8]. Now we list some aticula cases of Lemma 4. Q () (, ) ( ) i i!, ( ) i.

5 INTEGERS: () 5 ad Q () (, ) Q (4) (, ) Q (3) (, ) s 3 I the secial case whe we may wite ( ) ( ) (s ) 3 (), () 3 6 () 8 () () () 4. Q () (, ) H, () ad Q () (, ) (H ) H (), () Q (3) (, ) (H ) 3 3H H () H (3) () Q (4) (, ) (H ) 4 6 (H ) H () 8H H (3) 3 H () 6H (4). (3) We ca geealie (8) as follows. Lemma 5. Let q be a ositive itege, Q(, ) defied by (9). The, ad Q (q) (, ) be Q (q) (, ) ( x) ( x ) log q ( x) dx fo Z. Poof. The oof follows uo diffeetiatig (8) q times. The followig is a examle fo q.

6 INTEGERS: () 6 Examle 6. Fo q, () 3 (ψ () ψ ( )) ψ () ψ ( ) 3 H H () H H (). H H Theoem 7. Let R\Z ad let, q N. The qq (q ) (, ) Q (q) (, ) Q (q) (, ), (4) ad whe, whee Q () (, ) qq (q ) (, ) ad Q () (, ). Poof. Fom the left had side of (8), let F (, ) Q (q) (, ), (5) Q(, ). Diffeetiatig q times with esect to esults i o qf (q ) (, ) F (q) (, ) Q (q) (, ) qq (q ) (, ) Q (q) (, ) Q(,) so that F (, ) Q (q) (, ), aivig at (4). Fom Lemma 5, we may wite i itegal fom qq (q ) (, ) Q (q) (, ) Q (q) (, ) ( x) ( x ) log q ( x) (q log ( x)) dx.

7 INTEGERS: () 7 Fo ad hece (5). qq (q ) (, ) Q (q) (, ), The followig is a examle. Examle 8. Fo q 3 3Q () (, ) Q (3) (, ) Q (3) (, ), i exlicit fom 3 (H H ) H () H () (H H ) 3 H (3) H (3) 3 H () H () (H H ) (H H ) 3 H (3) H (3) 3 H () H () (H H ) ad whe 3 (H ) H () (H ) 3 H (3) 3H () H. The ext Lemma as stated i [], gives a alteate eesetatio fo Q (q) (, ). Lemma 9. Let q ad be ositive iteges, Q(, ), ad Q (q) (, ) be defied by (9). It was oved i [] that a alteate eesetatio fo Q (q) (, ) is: ( ) q q! ( ) ( ) q Q (q) (, ) (6)

8 INTEGERS: () 8 ( )q q! ( ) q qf q (q) tems,...,,,..., (q) tems. (7) Fom (4) ad (6) we see that qq (q ) (, ) Q (q) (, ) ( ) q q! ( ) ( ) q Q(q) (, ). Fom Examle with q 3, we have 6 ( ) ( ) 4 H (3) H (3) 3 (H H ) H () H () (H H ) 3 3 H () (H H ). H () The ivesio fomula, see [5], states g () ( ) k f (k) f () k k k ad hece ad whe, ( ) k g (k), k ( ) Q (q) (, ) ( ) q q! ( ) q, It ca also be oted that ( ) ( ) Q (q) (, ) ( )q q! q. Q (q) (, ) ( ) q q! ( ) q ( ) q qψ (q ) ( ) ψ (q) ( ) ( ) q q! ζ (q) H (q) ζ (q ) H (q),

9 INTEGERS: () 9 by (5); ad whe, ( ) Q (q) (, ) ( ) q q!ζ (q). Refeeces [] H. Belbachi, M. Rahmai ad B. Suy. Alteatig sums of the eciocals of biomial coefficiets. J. Itege Seq. 5 (), Aticle..8. [] G. Bhataga. I aise of a elemetay idetity of Eule. Elec. J. Combiatoics. 8 () (), P3. [3] J. Choi. Cetai summatio fomulas ivolvig hamoic umbes ad geealied hamoic umbes. Al. Math. Comut. 8 (), [4] J. Choi ad H.M.Sivastava. Some summatio fomulas ivolvig hamoic umbes ad geealied hamoic umbes. Math. Com. Modellig. 54 (), -34. [5] R. L. Gaham, D. E. Kuth, ad O. Patashik. Cocete Mathematics. Addiso-Wesley, Massachusetts, d Ed., 994. [6] A. Sofo. Comutatioal techiques fo the summatio of seies. Kluwe Academic/Pleum Publishes, 3. [7] A. Sofo. Summatio fomula ivolvig hamoic umbes. Aalysis Mathematica. 37 (), [8] A. Sofo. Itegal foms of sums associated with hamoic umbes. Al. Math. Comut. 7 (9), [9] A. Sofo. Sums of deivatives of biomial coefficiets. Advaces Al. Math. 4 (9), [] A. Sofo. New classes of hamoic umbe idetities. submitted. [] A. Sofo ad H. M. Sivastava. Idetities fo the hamoic umbes ad biomial coefficiets. Ramauja J. 5 (), [] J. Sieß. Some idetities ivolvig hamoic umbes. Mathematics of Comutatio. 55 No.9, (99), [3] R. Sugoli. Sums of eciocals of cetal biomial coefficiets. Iteges. 6 (6), Aticle #A7. [4] W.Y.G. Wa. Idetities ivolvig owes ad ivese of biomial coefficiets. J. Math. Res. Exositio3 (), -9. [5] Z. Zhag ad H. Sog. A geealiatio of a idetity ivolvig the ivese of biomial coefficiets. Tamkag J. Math. 39 (8), 9-6.