GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION

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1 J Korea Math Soc 44 (2007), No 2, pp GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION Gi-Sag Cheo ad Moawwad E A El-Miawy Reprited from the Joural of the Korea Mathematical Society Vol 44, No 2, March 2007 c 2007 The Korea Mathematical Society

2 J Korea Math Soc 44 (2007), No 2, pp GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION Gi-Sag Cheo ad Moawwad E A El-Miawy Abstract I this paper, we obtai importat combiatorial idetities of geeralized harmoic umbers usig symmetric polyomials We also obtai the matrix represetatio for the geeralized harmoic umbers whose iverse matrix ca be computed recursively Itroductio ad prelimiaries The ordiary harmoic umbers are deoted by H ad are defied as H 0 0 ad H,, 2, The first few harmoic umbers are, 3 2, 6, 25 2, 37 60, These harmoic umbers were studied i atiquity ad are importat i various braches of umber theory ad combiatorial problems They are closely related to the Riema zeta fuctio defied by ζ(s) s ( p s ), p where the product is over all primes p, ad appear i various expressios for various special fuctios It is well ow that s( +, 2) () H,! where s(, ) deotes the Stirlig umbers of the first id defied by (x) : (x r + ) s(, )x For a otatioal coveiece, we deote c(, ) s(, ), ie, c(, ) is the usiged Stirlig umber of the first id which couts the permutatios of Received Jue, Mathematics Subject Classificatio 05A30 Key words ad phrases harmoic umbers, Riema zeta fuctio, Stirlig umbers, Beroulli umbers, symmetric polyomials c 2007 The Korea Mathematical Society

3 488 GI-SANG CHEON AND MOAWWAD E A EL-MIKKAWY elemets that are the product of disjoit cycles Further, It is ow [9] that ζ( + ) c(, )! I may recet wors (see for example []-[5],[7]-[0]), the harmoic umbers have bee geeralized by several ways ad the related idetities were obtaied Our observatios suggest that the geeralized harmoic umbers ca be viewed combiatorially For istace, the geeralized harmoic umbers H (r) of order r are defied to be partial sums of the Riema zeta fuctio: (2) H (r) 0 0 ad H (r) It is ow (p27 i [6]) that the umbers H (r) c( +, )!, c( +, 2)!H, c( +, 3)! 2 (H2 H (2) ),,, r r c( +, 4)! 6 (H3 3H H (2) + 2H (3) ), ad c(, r) are coected by ad so o I [], Adamchi obtaied the geeral formula for c(, m) i terms of geeralized harmoic umbers H (r) : (3) c(, m) ( )! w(, m ), (m )! where the w-sequece is defied recursively by m (4) w(, 0), w(, m) ( ) (m ) H (+) w(, m ) 0 Ad he showed the w-sequece ca be rewritte through a multiple sum: w(, m) i i 2i + i mi m + m! i i 2 i m I [5], Chu ad Doo defied the geeralized harmoic umbers H (r) by (5) H 0 (r) 0 ad H (r) + r,, ad they obtaied several striig idetities o the ordiary harmoic umbers

4 GENERALIZED HARMONIC 489 For the risig factorial [x] x(x + ) (x + ) ( ), by writig as the sum of p + partial fractios we obtai: [r] p+ (6) [r] p+ p 0 A (p) + r p! p 0 ( ) p ( ) + r, where A (p) / p t0 (t ), t It is easy to show that (6) ca also be obtaied by usig the fact that ( x ) (E ) ( x ) ( )! [x] +, where ad E are the forward ad the shift operators with uit step, respectively The risig factorial [r] p+ satisfies the followig idetity, typically proved by iductio or telescopig sums []: (7) [r] p+ ( +p p ) ), p p(p!) ( +p p Usig (6) ad (7), we ca establish a iterestig idetity for the geeralized harmoic umbers H (r): p ( ) ( +p ) p ( ) r p H (r) r p ( ) +p p r0 Besides, i [2], Bejami et al showed that the geeralized harmoic umbers H <r> defied by (8) H <0> ad H<r> H <r >,, r ca be expressed i terms of r-stirlig umbers I 997, Satmyer [0] defied the geeralized harmoic umbers H,r of ra r by (9) H,r,, r 0 0 r r Note that these geeralized harmoic umbers H (r), H (r), H <r> reduce to the ordiary harmoic umbers H whe r or r 0 I this paper, we defie other geeralized harmoic umbers H(, r) : c( +, r + )! ad H,r which are direct geeralizatio of () The purpose of this paper is to obtai some iterestig idetities ivolvig H(, r) These results are derived from symmetric polyomials Further, we give a matrix represetatio for H(, r)

5 490 GI-SANG CHEON AND MOAWWAD E A EL-MIKKAWY 2 Symmetric polyomials I this sectio, we are goig to cosider the so-called symmetric polyomials which are ot oly importat tool i mathematics but also play cosiderable roles i computer sciece, physics ad statistics A polyomial P (x, x 2,, x ) i the variables x, x 2,, x is called a symmetric polyomial or symmetric fuctio if it is ivariat uder all possible permutatios of the variables x, x 2,, x Especially, importat symmetric polyomials that will be cosidered i the curret paper are the elemetary symmetric polyomial σ () (), the complete symmetric polyomial τ, ad the power sum symmetric polyomial S () o the variables x, x 2,, x These polyomials for itegers, with 0 are defied by: (0) () (2) σ () (x, x 2,, x ) τ () (x, x 2,, x ) S () (x, x 2,, x ) r < <r r r x r, where σ () 0 (x, x 2,, x ) ad τ () 0 (x, x 2,, x ) x r x r2 x r, x r x r2 x r, Lemma (Newto-Girard idetity [3]) For positive itegers m, such that m, the followig holds: (3) m σ () m (x,, x ) m ( )r+ S r () (x,, x )σ () m r(x,, x ) It is worthy to metio that two polyomials σ () ad τ () ca be writte through multiple sums as follow: r σ () (x r 2 (4), x 2,, x ) x r x r2 x r, (5) τ () (x, x 2,, x ) r r r r x r x r2 x r r 2r r r It is ow that two geeratig fuctios E(t) ad F (t) for σ () give respectively by (6) E(t) ( + x i t) σ () r (x, x 2,, x )t r, (7) F (t) i i r0 x i t τ () (x, x 2,, x )t r r0 ad τ () are

6 GENERALIZED HARMONIC 49 These fuctios satisfy E(t)F ( t) F (t)e( t) ad r (8) ( ) m σ (r) m τ (r) r m δ r0 m0 for specific o-egative itegers ad r such that 0 r, where δ is the Kroecer symbol The usiged Stirlig umbers of the first id c(, ) ad the Stirlig umbers of the secod id S(, ) defied by x 0 S(, )(x) are related to σ () ad τ () by (9) c(, ) σ ( ) (, 2,, ), (20) S(, ) τ () (, 2,, ) Before we cosider more geeralized harmoic umber idetities it may be useful to give the followig result Lemma 2 For all ozero real umbers x, x 2,, x ad for all oegative iteger i, we have (2) σ () i ( x, x 2,, r0 x ) x x 2 x σ () i (x, x 2,, x ) Proof Replacig each x i by x i i (6) gives: σ r () (,,, )t r x x 2 x x x 2 x Hece i0 σ () i ( x, x 2,, The result follows x )t i x x 2 x i0 (t + x i ) i σ () i (x, x 2,, x )t i As a direct cosequece of Lemma 2, we see that for specific positive iteger, the special case x i i for each i, 2,, yields: (22) σ () i (, 2,, )! σ() i (, 2,, ) Lemma 3 For itegers, m with m 2, we have (23) c(, m) ( )! σ ( ) m (, 2,, ) Proof With the help of (22) ad (9) we obtai c(, m) ( )! ( )! σ( ) m (, 2,, ) σ ( ) m (, 2,, ), which proves (23)

7 492 GI-SANG CHEON AND MOAWWAD E A EL-MIKKAWY From Lemma 3 ad (3), we immediately obtai a simple represetatio for the w-sequece: (24) w(, m) m!σ ( ) m (, 2,, ) Further, usig the recurrece relatio (25) eables us to compute the umbers w(, w) recursively as follows: w(i, j) w(i, j) + j w(i, j ), i 2, j i so that w(i, i) 0 ad w(i, 0) for i 0, ad w(i, ) H i for i 3 Geeralized harmoic umbers H(, r) We begi with the recurrece relatio for the usiged Stirlig umbers of the first id c(, r) i [6]: (25) c(, r) c(, r ) + ( )c(, r),, r with c(, 0) δ 0, c(, ), c(, ) ( )! for The relatio (25) also gives c( +, 2) c(, ) + c(, 2) Dividig both sides by! taig ito accout the fact that c(, ) ( )!, we obtai (26) g() + g( ),, where g() c( +, 2)/! Sice H + H,, from (26) we obtai g() H which proves () More geerally, we may rewrite the recurrece relatio (25) i the form: (27) c( +, r + )! c(, r) ( )! c(, r + ) + ( )! Let us defie the geeralized harmoic umbers H(, r) by (28) H(, 0) ad H(, r) c( +, r + )!, r First ote that from (27) the umbers H(, r) satisfy the recurrece relatio: (29) H(, r) H(, r) + H(, r ) Theorem 4 The geeralized harmoic umbers H(, r) satisfy (30) H(, r) 2 r Proof From Lemma 3, we have < < r H(, r) σ () r (, 2,, )

8 GENERALIZED HARMONIC 493 Hece usig (0) yields This completes the proof H(, r) < < r 2 r The formula (30) ca be used to fid a formula for coefficiets of powers of m i the Stirlig umbers S(m +, m) of the secod id (see [8]) At this stage it is coveiet to obtai a geeratig fuctio for the geeralized harmoic umbers H(, r) From (6), we obtai (3) r0 σ () r(x, x 2,, x )t r (t + x i ) i Settig x m m for each m, 2,, i (3) gives σ () r(, 2,, )t r (t + i) r0 Thus from (9) we have (32) c( +, r + )t r r0 i (t + i) Usig (28) together with (x)! ( x ) ad (32) proves that the geeratig fuctio we are looig for is give by ( ) + t (33) H(, r)t r r0 Sice H(, r) r! w( +, r), the geeratig fuctio for w(, ) i (3) ca be easily obtaied The Beroulli polyomials B (x) defied by ( ) B (x) B x 0 are importat i obtaiig closed form expressios for sum of powers of itegers [] such as (34) r + (B +( + ) B + ), 0, r0 i where B B (0) is the -th Beroulli umber Theorem 5 The geeralized harmoic umbers H(, r) satisfy the followig idetities: (i) H(, r), (ii) ( )r+ H(, r),

9 494 GI-SANG CHEON AND MOAWWAD E A EL-MIKKAWY (iii) jr j H(j, r ) H(, r), (iv) r0 ( )r H(, r)(b r+ ( + ) B r+ ) r+, (v) r0 H(, r)(b r+( + ) B r+ ) r+ ( ) 2+, (vi) rh(, r) ( + )(H + ), (vii) r0 (2r )H(, r) ( ) + 2 Proof Puttig t i (33) ad taig ito accout the fact that H(, 0) gives (i) Similarly puttig t i (33) yields the idetity (ii) To prove (iii), sice applyig (4) gives H(, r) σ () r (, 2,, ) H(, r) r r r r r r r 2 r r r r 2r 2 + r + + r < < r 2 r 2 r r r r 2 r 2 r r r r 2 r 2 2 r r, 2 r + 2 r H(r, r ) + H(r, r ) + r r + + H(, r ) H(j, r ), j jr as required To prove (iv), puttig m ad x for each, 2,, i (3) gives Hece we get σ () (, 2,, ) ( ) r+ S r () (, 2,, )σ () r(, 2,, ) ( ) r+ S r () (, 2,, )σ () r(, 2,, )!

10 GENERALIZED HARMONIC 495 Cosequetly, by usig (34) ad (3) we obtai (35) ( ) r+ H(, r)(b r+ ( + ) B r+ ) r + The idetity (35) ca also be writte i the form: ( ) r H(, r)(b r+ ( + ) B r+ ) r + r0 To prove (v), from (33) we have ( ) + t H(, r)t r Hece (36) t0 r0 r0 H(, r)t r H(, r)( t r ) r0 t0 ( + t)! ( ) ( ) + t 2 + Usig (34) the (36) yields ( ) 2 + H(, r)(b r+ ( + ) B r+ ) r +, r0 which is the required result To prove (vi), usig (33) we obtai (37) H(, r)t r (t + ) (t + r)!! r0 Differetiatig both sides of (37) with respect to t gives ( ) + t ( ) (38) rh(, r)t r + t t + r (H +t H t ) t0 Puttig t i (38) yields rh(, r) ( + )(H + ), as required To prove (vii), by usig the forward operator with uit step, the (37) yields ( H(, r)t r ) ( (t + ) )! Hece r0 H(, r) (t r )! ((t + ) ) r0

11 496 GI-SANG CHEON AND MOAWWAD E A EL-MIKKAWY Cosequetly, we have (39) H(, r)((t + ) r t r )! (t + ) (t + ) ( )! r0 Settig t i (39), we obtai (2 r )H(, r) r0 This completes the proof of the theorem ( ) + ( + 2 ) ( ) + t We list more ew idetities for the geeralized harmoic umbers H(, r) without proofs (i) H(, r) ( ) r r H(, ), (ii) r H(, r)s(r +, + ) ( ) + (+)!, (iii) 0 ( )+ H(, )B , (iv) r+ rh(, r) H(,r ) 4 Matrix represetatio For the geeralized harmoic umbers H(, ) defied by (28) we defie the matrix H [h ij ] i,j as follows: { H(i, j) if i j, h ij 0 if i < j For example, the 5 5 matrix H is give by H Here the elemets of the matrix H are computed recursively usig (29) Usig (8) we see that the iverse matrix Q [q ij ] i,j of H is give by { ( ) q ij i+j j!τ (j+) i j (, 2,, j + ) if i j, 0 if i < j Usig (2) yields q ij 8 20 { ( ) i+j j!s(i +, j + ) if i j, 0 if i < j

12 GENERALIZED HARMONIC 497 For example, we obtai the 5 5 matrix Q as follows: Q It is worthy to metio that the elemets of the iverse matrix Q ca also be computed recursively usig the recurrece relatio with q ij jq i,j (j + )q i,j, i 3, 4,, ; j 2, 3,, i, q i ( ) i+ (2 i ) for i ; q ii i! for i Due to the idetity (i) of Theorem 5, the product DH is a stochastic matrix, where D diag(, 2,, ) We coclude this paper by describig that the matrix represetatio for the geeralized harmoic umbers may be used to get more idetities ad some combiatorial coectios with other combiatorial umbers Acowledgmet The authors would lie to tha the referee for valuable commets ad suggestios Refereces [] V Adamchi, O Stirlig umbers ad Euler sums, J Comput ad Appl Math 79 (997), o, 9 30 [2] A T Bejami, D Gaebler, ad R Gaebler, A combiatorial approach to hyperharmoic umbers, Itegers (Elec J Combi Number Theory) 3 (2003), A5, 9 [3] S Chaturvedi ad V Gupta, Idetities ivolvig elemetary symmetric fuctios, J Phys A 33 (2000), o 29, L25 L255 [4] W Chu, Harmoic umber idetities ad Hermite-Padé approximatios to the logarithm fuctio, J Approx Theory 37 (2005), o, [5] W Chu ad L De Doo, Hypergeometric series ad harmoic umber idetities, Adv i Appl Math 34 (2005), o, [6] L Comtet, Advaced Combiatorics, D Reidel Pub Compay, 974 [7] A Gertsch, Geeralized harmoic umbers, C R Acad Sci Paris, Sér I Math 324 (997), o, 7 0 [8] I M Gessel, O Mii s idetity for Beroulli umbers, J Number Theory 0 (2005), o, [9] T M Rassias ad H M Srivastava, Some classes of ifiite series associated with the Riema Zeta ad Polygamma fuctios ad geeralized harmoic umbers, Appl Math Comput 3 (2002), o 2-3, [0] J M Satmyer, A Stirlig lie sequece of ratioal umbers, Discrete Math 7 (997), o -3, [] F Scheid, Numerical Aalysis, McGraw-Hill Boo Compay, 968 [2] A J Sommese, J Verscheide, ad C W Wampler, Symmetric fuctios applied to decomposig solutio sets of polyomial systems, SIAM J Numerical Aalysis 40 (2002), o 6, [3]

13 498 GI-SANG CHEON AND MOAWWAD E A EL-MIKKAWY Gi-Sag Cheo Departmet of Mathematics Sugyuwa Uiversity Suwo , Korea address: gscheo@suedu Moawwad E A El-Miawy Departmet of Mathematics Faculty of Sciece of Masoura Uiversity Masoura 3556, Egypt address: miawy@yahoocom