On a general q-identity

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1 O a geeral -idetity Aimi Xu Istitute of Mathematics Zheiag Wali Uiversity Nigbo 3500, Chia xuaimi009@hotmailcom; xuaimi@zwueduc Submitted: Dec 2, 203; Accepted: Apr 24, 204; Published: May 9, 204 Mathematics Subect Classificatios: 05A9, B65 Abstract I this paper, by meas of the -Rice formula we obtai a geeral -idetity which is a uified geeralizatio of three ids of idetities Some ow results are special cases of ours Meawhile, some idetities o -geeralized harmoic umbers are also derived Keywords: -Rice formula; -idetity; -geeralized harmoic umber; Cauchy s itegral formula; Faà di Bruo s formula Itroductio Three ids of idetities will be itroduced i this paper I the paper [2], Va Hamme gave the followig idetity = + 2 = = Oe of the geeralizatios of was give by Dilcher [6]: = 2+λ = λ α α λ = α α 2 Supported by the Natioal Natural Sciece Foudatio of Chia grat 20430, ad the Nigbo Natural Sciece Foudatio the electroic oural of combiatorics , #P228

2 Prodiger [6] gave aother geeralizatio of : =0, m [ ] + 2 = m m m+ 2 =0, m, 3 m where 0 m May wors have bee devoted to the study of the geeralizatios of these idetities See for example [8, 9, 7, 23] Recetly, Guo ad Zhag [2] made use of the Lagrage iterpolatio formula to give a geeralizatio of Prodiger s idetity 3 They also gave a geeralizatio of Dilcher s idetity 2 See Theorems ad 2 i [2], respectively Ismail ad Stato used the theory of basic hypergeometric fuctios to geeralize Dilcher s idetity See Theorem 22 i [3] I the paper[5], Díaz-Barrero et al obtaied two idetities ivolvig ratioal sums: x + = = α β α= x 2 + α + βx + αβ = { x + x + α α<β<γ α β x + αx + βx + γ x + 3, x + αx + β2x + α + β } = x + 4 Recetly, Prodiger [8] made use of partial fractio decompositio ad iverse pairs to preset a more geeral formula: x + = c +2c 2 + =λ s c, c! =, 4 c x + λ+ where s, = α= x + α Almost at the same time, Chu ad Ya [2] preseted a geeralizatio with multiple λ-fold sum: x + =0 0 α α λ = x + α = x 5 x + λ+ A direct proof of 5 ca be foud i Chu [] More recetly, Masour et al [5] established a -aalog for the ratioal sum idetity 4: 2 x + = c +2c 2 + =λ s, c c! c = λ [], 6 [x + ] λ+ where s, = α= α [x + α] I particular, they gave a very ice biective proof for the case λ = the electroic oural of combiatorics , #P228 2

3 I the recet paper [9], Prodiger established a iterestig idetity ivolvig harmoic umbers: + m λ where ad H r =0, m = m m + m c +2c 2 + =λ c!c 2! H = H m+ 2H m + H m, are the geeralized harmoic umbers defied by H r 0 = 0, H r = = = r for, r =, 2, H c, 7 Masour [4] obtaied a geeral ratioal sum to geeralize this idetity He also obtaied a -aalog of this result ivolvig -harmoic umbers Motivated by these iterestig wor, by meas of the -Rice formula used i [6, 7], we will establish a geeral -idetity which is a commo geeralizatio of those three ids of idetities itroduced before Theorem Let λ be ay positive iteger For 0 m ad 0 l + λ, there holds λ +m m /z; z l ; +λ z =0, m x m λ+ = ; z l ; l zx m ; l+λ x m ; m x; m where = c!c 2! c λ!, c = c + 2c λc λ ad l+λ 2 z u = + zx m =0 =0, m = x m c u, 8 This is a very geeral -series sum idetity ivolvig five parameters λ, l, m, x ad z It cotais several ow idetities by choosig differet parameters, which will be show i the third sectio By meas of our idetity, we will also obtai some idetities o -geeralized harmoic umbers Throughout this paper, we will use the stadard otatio For ay real umber x ad ay iteger m, defie [x] = x, x; = x, x; m = x; x m ; =0 the electroic oural of combiatorics , #P228 3

4 For ay oegative iteger, defie []! = [] [2] [], = []! []![ ]! 2 Proof of Theorem I the very iterestig paper [6], Prodiger itroduced the followig formula 2 f = 2πi = where C ecircles the poles, 2,, ad o other formula [7, 20]: = f = 2πi C C ; t; + ftdt,! tt t ftdt, It is a -aalog of Rice s where C ecircles the poles, 2,, ad o other Ideed, by Cauchy s itegral formula oe is ot hard to fid that for ay iteger m {0,,, } there holds =0, m 2 f = ; + 2 2πi C ftdt =0 t, 2 where C ecircles the poles, {0,,, } {m} ad o other Prodiger first applied the -aalog of Rice s formula to prove may idetities such as the idetities of Va Hamme, Uchimura, Dilcher, Adrews-Crippa-Simo, ad Fu-Lascoux, see [6, 7] ad refereces therei It was show that this formula is a very powerful ad useful tool Now, i this sectio we will use this importat formula ad preset a proof of Theorem Proof of Theorem By simple calculatios we have λ m /z; z l ; +λ z =0, m x m λ+ = 2+λ m =0, m x m λ+ z ; z l ; +λ = 2+λ m z ; z l ; =0, m x m λ+ z; z +λ l ; =z l ; l 2+λ m x m λ+ z ; l+λ =0, m the electroic oural of combiatorics , #P228 4

5 Thus, by the -Rice formula 2 there holds =0, m λ m /z; z l ; +λ z x m λ+ = z l ; l ; + 2 2πi C zt; l+λ dt t x m λ+ =0, m t, 22 where C positively orieted ecloses the poles, {0,,, } {m} ad o other It is obvious that zt; l+λ dt 2πi C t x m λ+ =0, m t = zt; l+λ dt 2πi C t x m λ+ =0, m t, 23 where C positively orieted ecloses the pole x m By Cauchy s itegral formula, there holds zt; l+λ 2πi C t x m λ+ =0, m t = d λ zt; l+λ λ! dt λ =0, m t 24 t=x m Applyig Faà di Bruo s formula [4] yields d λ zt; l+λ dt λ =0, m t = dλ e l+λ 2 dt λ =0 log zt t=x m = zx m ; l+λ =0, m x m =0, m logt t=x m From 22, 23,24 ad 25, the desired result is obtaied = c u 25 Remar 2 Actually, careful checig the proof of Theorem, oe ca fid that Theorem still holds for λ = 0 if i this case we assume the sum of the right had side of 8 is eual to This implies that for 0 l there holds =0, m m /z; z l ; z = ; z l ; l zx m ; l x m x m ; m x; m 3 Coseueces of Theorem Theorem ca help us to fid some ew idetities or retrieve some well ow idetities Let λ = ad x = 8 reduces to the followig idetity the electroic oural of combiatorics , #P228 5

6 Corollary 3 For 0 m ad 0 l, there holds /z; z l ; z m =0, m = m m 2 m z l ; l z m ; l { l =0 z z + m =0, m } m 3 Guo ad Zhag [2] made use of Lagrage iterpolatio formula to obtai this idetity which geeralizes the idetity 3 due to Prodiger It is obvious that 3 reduces to 3 whe l = 0 ad z 0 Let x =, l = λ ad z = i 8 We have Corollary 32 Let λ be ay oegative iteger For 0 m, there holds =0, m + 2+λ m λ = m m 2 m + m m c H, = where H = + m =0 =0, m m This idetity is a -aalog of Prodiger s idetity 7 A alterative form of this -idetity was preseted i [4] For m = 0, l = 0 ad z 0 i 8, the followig idetity is true Corollary 33 Let λ be ay oegative iteger There holds 2+λ x = ; λ+ x; = Sice It is clear that = x t = d λ dt λ = 0x t = λ 0 x t = dλ t=0 = t λ = = α α λ = dt λ e = log x t, t=0 c 32 x x α 33 the electroic oural of combiatorics , #P228 6

7 we apply Faà di Bruo s formula [4] to obtai d λ dt λ x t = t=0 = Comparig 33 with 34, there holds α α λ = Therefore, 32 ca be rewritte as or = x α = λ! = = 2+λ x = ; λ+ x; = = x = x 2+λ x = ; λ+ x; c 34 c α α λ = b =λ = x α x α, b, 35 where b = b + b b By the theory of basic hypergeometric fuctios Ismail ad Stato [3] foud E 35 which reduces to the Dilcher idetity [6] whe x = I fact, it has bee recetly poited out i [] that the Ismail-Stato result 35 is the i = with m = λ + case of followig formula due to Zeg [23]: i =i i i 2 +m z = i ; i ; m ; i z; h m i z,,, i z where i ad h x,, x is the th homogeeous symmetric polyomial i x, x 2,, x defied by h x,, x = x b x b i i x i x i = b = This more geeral formula ca ot follow from Theorem ad it ca be viewed as a differet geeralizatio of the Ismail-Stato result 35 Sice E 32 ca be rewritte as = [ ] =, 2+λ x = ; λ+ x; = α= α c x α the electroic oural of combiatorics , #P228 7

8 Usig the -iverse pair formula [0] f = 2 g g = = we obtai the iverse of 32 2 = = α= 2 f, = ; x; α x α c = λ x λ+ Replacig x by x, we rediscover a idetity due to Masour et al [5]: Corollary 34 Let λ be ay oegative iteger There holds 2 x + = c α [x + α] = λ [] 36 [x + ] λ+ = α= This idetity is a -aalog for the ratioal sum idetity 4 due to Prodiger If we further replace by + ad x by x i 36, the a -aalog of Chu-Ya s idetity 5 is derived: Corollary 35 Let λ be ay oegative iteger There holds =0 + 2 [ x + ] Let the geeralized -harmoic umbers H r 0 = 0, H r = 0 α α λ = α = λ+ [x] [x + α ] [x + ] λ+ r [] r, Recetly, the -geeralized harmoic umber sums have bee useful i studyig Feyma diagram cotributios a relatios amog special fuctios [3] Taig x = 0 i 36, we have the followig idetities o -geeralized harmoic umbers: Corollary 36 For λ, there holds = λ = = H c = [] λ the electroic oural of combiatorics , #P228 8

9 The first few cases are listed as follows H =, =0 [] H 2 + H 2 = 2, =0 [] H 3 + 3H H 2 + 2H3 = =0 = H 2 H 3 = 6, [] 3 H H 2 H H H H 3 + 6H4 = 24, [] 4 H H 3 H 2 + 5H + 20H2 H3 + 30H H H5 H 2 2 = 20 [] 5 These idetities are -aalogs of geeralized harmoic umber idetities which were preseted i [22]: H =, 37 =0 H 2 + H 2 = 2, 38 2 =0 =0 =0 =0 H 3 + 3H H 2 + 2H 3 = 6, 39 3 H 4 + 6HH H 2 H 5 + 0H 3 H 2 + 5H + 20HH H 2 It is worth oticig that startig from = H H 3 + 6H 4 = 24, H 2 H3 + 30H H H 5 = ; = x x; x the electroic oural of combiatorics , #P228 9

10 ad taig the th derivative of both sides at x =, we ca also arrive at Corollary 36 Wag ad Jia [22] applied the Newto-Adrews method to some well ow idetities ad foud may iterestig idetities o harmoic umbers which iclude the idetities from 37 to 3 Acowledgemets We tha the aoymous referee for his/her careful readig of our mauscript ad very helpful commets Refereces [] W Chu Summatio formulae ivolvig harmoic umbers Filomat, 26:43 52, 202 [2] W Chu ad QL Ya Combiatorial idetities o biomial coefficiets ad harmoic umbers Utilitas Mathematica, 75:5 66, 2008 [3] MW Coffey O a three-dimesioal symmetric Isig tetrahedro, ad cotribitios to the theory of the dilogarithm ad Clause fuctios J Math Phys, 49: , 2008 [4] L Comtet Advaced combiatorics, the art of fiite ad ifiite expasios D Reidel Publishig Co, Dordrecht, 974 [5] JL Díaz-Barrero, J Gibergas-Báguea, ad PG Popescu Some idetities ivolvig ratioal sums Appl Aal Discrete Math, : , 2007 [6] K Dilcher Some -series idetities related to divisor fuctio Discrete Math, 45-3:83 93, 995 [7] P Flaolet ad R Sedgewic Melli trasforms ad asymptotics: Fiite differeces ad Rice s itegrals Theoretical Computer Sci, 44:0 24, 995 [8] AM Fu ad A Lascoux -Idetities from Lagrage ad Newto iterpolatio Adv Appl Math, 3:527 53, 2003 [9] AM Fu ad A Lascoux -Idetities related to overpartitios ad divisor fuctios Electro J Combi, 2:#R38, 2005 [0] IP Goulde ad DM Jacso Combiatorial eumeratio A Wiley-Itersciece Publicatio, Joh Wiley & Sos Ic, New Yor, 983 With a foreword by Gia- Carlo Rota [] VJW Guo ad J Zeg Further p, -idetities related to divisor fuctios arxiv: [2] VJW Guo ad C Zhag Some further -series idetities related to divisor fuctios Ramaua J, 253: , 20 [3] MEH Ismail ad D Stato Some combiatorial ad aalytical idetities A Comb, 6:755 77, 202 the electroic oural of combiatorics , #P228 0

11 [4] T Masour Idetities o harmoic ad -harmoic umber sums Afr Mat, 23:35 43, 202 [5] T Masour, M Shattuc, ad C Sog A -aalog of a geeral ratioal sum idetity Afr Mat, 24: , 203 [6] H Prodiger Some applicatios of the -Rice formula Radom Struct Alg, 9: , 200 [7] H Prodiger -Idetities of Fu ad Lascoux proved by the -Rice formula Quaest Math, 27:39 395, 2004 [8] H Prodiger Idetities ivolvig ratioal sums by iversio ad partial fractio decompositio Appl Aal Discrete Math, 2:65 68, 2008 [9] H Prodiger Idetities ivolvig harmoic umbers that are of iterest for physicists Utilitas Mathematica, 83:29 300, 200 [20] W Szpaowsi Average case aalysis of algorithms o seueces Joh Wiley, New Yor, 200 [2] L Va Hamme Advaced Problem 6407 Amer Math Mothly, 40: , 982 [22] W Wag ad C Jia Harmoic umber idetities via the Newto-Adrews method Ramaua J, i press [23] J Zeg O some -idetities related to divisor fuctios Adv Appl Math, 34:33 35, 2005 the electroic oural of combiatorics , #P228