The Inverse of Power Series and the Partial Bell Polynomials

Transcription

1 Joural of Ieger Sequece Vol Aricle 1237 The Ivere of Power Serie ad he Parial Bell Polyomial Miloud Mihoubi 1 ad Rachida Mahdid 1 Faculy of Mahemaic Uiveriy of Sciece ad Techology Houari Boumediee USTHB PB 32 El Alia Algier Algeria miloudmihoubi@gmailcom mmihoubi@uhbdz rachida-mahdid@homailfr Abrac Uig he Bell polyomial i hi paper we give he explici compoiioal ivere ad/or he reciprocal of ome power erie We illurae he obaied reul by ome example o Sirlig umber 1 Iroducio The applicaio of he parial Bell polyomial have araced he aeio of everal auhor Come [7] udied hee polyomial ad Riorda [17] ued hem i combiaorial aalyi ad Roma [18] i umbral calculu Recely more applicaio of hee polyomial have appeared i differe framewor icludig iegraio [6] ivere relaio [13] cogruece [14] Dyc pah [11] ad Bliard problem [10] all of which moivae u o apply hee polyomial o deermie he explici compoiioal ivere ad/or reciprocal of ome power erie Ideed recall ha he udy of he exiece of he compoiioal ivere of power erie i a well-ow reul of complex aalyi; ee Foryh [9] ad Saley [19 Propoiio 541] Uder ome codiio o a fucio f o fid he compoiioal ivere f 1 of f aroud zero hree mehod are ued o compue he coefficie y for which he erie 1 Thi reearch i uppored by PNR projec 8/u160/3172 1

2 f 1 = y i he compoiioal ivere of he erie f = 0 0 i baed o he oluio o y = y 1 y 2 i he equaio = 0 y j x j! j! j 0 x The fir mehod If we e x = y = 0 if 0 i wa how by Whiaer [20] ha y 1 = 1 y = 1 1 x 1 x1 2 1 de 1 + i j + j 1x 2+i j 2 1 ij 1 Thi ca be reduced o olve he equaio = y j x j =! j! 1 j 1 which i equivale o olvig he yem y B x 1 x 2 =1 y B x 1 x 2 = δ 1 1 =1 where δ i he Kroicer ymbol ie δ 0 = 1 ad δ = 0 if 1 ad he polyomial B x 1 x 2 are he expoeial parial Bell polyomial defied by heir geeraig fucio = B x 1 x 2 = 1! m x m m! The ecod mehod i baed o Lagrage iverio formula [1] for which we have y = 1 ϕ dϕ fϕ ϕ=0 1 The hird mehod i baed o he h eed derivaive of a fucio g defied i [8] by D 0 [g]ϕ := 1 D [g]ϕ := d fϕd 1 [f]ϕ 1 dϕ for which Domiici [8] howed ha if f = 1 dϕ wih gα 0 ± we have α gϕ f 1 = α + gα m=1 D [g]α For our coribuio we how ha he parial Bell polyomial defie wo familie of power erie for which we ca obai explici compoiioal ivere We alo give he reciprocal of power erie coeced o hee familie For he applicaio we give ome example o power erie whoe coefficie are relaed o Sirlig ad r-sirlig umber Ideed le 2

3 rd be ieger wih d 1 ad x = x 1 x 2 be a equece of real umber wih x 1 = 1 For r > max 2 we coider H 1 = 1 d 1! r ad for r > max d + 1 we coider H 2 = 1 r r+1 d 1 rd+1 B rd+1 r x B r+1 r x r+1 r 1 2 We give below he explici compoiioal ivere 1 of H 1 Theorem 3 ad 2 of H 2 Theorem 16 Alo we pree he explici reciprocal power erie of H 1 H 1 The mahemaical ool ued are baed o he coecio bewee he parial Bell polyomial ad he polyomial of biomial ype For a give real umber α ad for ay equece of biomial ype f ϕ i wha follow we le f ϕ;α deoe ay equece of biomial ype uch ha f ϕ;α := ϕ α + ϕ f α + ϕ 3 ee Come [7 pp ] Aiger [2 pp ] ad Propoiio 1 give i [12] For example he equece ϕ ad ϕα + ϕ 1 are equece of biomial ype Below we ue he followig oaio: B x j or B x wih x = x 1 x 2 for he parial Bell polyomial B x 1 x 2 Dz=0fz for d f 0 2 ad D dz z=0 fz for df 0 dz [ ] [ ad ] for he uiged Sirlig ad r-sirlig umber of he fir id repecively r } { ad } for he Sirlig ad r-sirlig umber of he ecod id repecively { r 2 The fir family of power erie ad heir ivere We give i hi ecio he reciprocal ad/or he compoiioal ivere of a power erie give from a large family of power erie which have coefficie ca be expreed i erm of parial Bell polyomial Propoiio 1 Le x = x 1 x 2 be a equece of real umber wih x 1 = 1 The for r ieger uch ha r > max 2 he reciprocal ad compoiioal ivere of he power erie H = 1 r B r+1 r x r+1 4 3

4 are give by = 1 + r + + H = 1 + r + B r++1+r++ x r B r+1+r+ x 6 r+1+ Proof We coider oly he cae 0 for < 0 we ca proceed imilarly Le f ϕ be a equece of biomial-ype polyomial uch ha f 1 = x +1 / + 1 wih x 2 0 o eure ha Df 1 ϕ 0 The by Propoiio 1 give i [12] we ge 1 + f = B +x 7 I [13 Theorem 1] we proved ha H = 1 r f r ha ivere = 1 + r + + f r + + ad becaue for ay biomial-ype equece of polyomial f ϕ we have f = 1 + f he by replacig i he la ideiy f by f ;r defied by 3 we ge H = 1 + r + f r + The replace f r ad f r++ by heir expreio obaied from he ideiy 7 by f r = B r+1 r x ad f r + + = B r++1+r++x r+1 The propoiio remai rue for he cae x 2 = 0 by coiuiy r++1+ Lemma 2 Le x = x 1 x 2 be a equece of real umber wih x 1 = 1 x = 0 if d 1 ad y = y 1 y 2 wih y j = j!x dj 1+1 /dj 1 + 1! The we have B + x +! = B +y +! ad B ++l x = 0 if 1 l d 1 4

5 Proof Seig z = z 1 z 2 wih z j = y j+1 /j + 1 From he defiiio of parial Bell polyomial we have ie B x = 1 + d j z j =! j!! j 1 B x =! l B l z l=0 l l=0 =! 0 Now from he ideiy [3l] give i Come [7 pp 136] we have mi l=0 ad he above expaio become B x = 1! 0 + l d j l z j l! j! j 1 mi 1 + l B l z = B +y l=0 l B l z 1 + B +y = +! +! B + +y +! Thi give B + x = +! +! B +y ad B ++l x = 0 if 1 l d 1 Theorem 3 Le y = y 1 y 2 be a equece of real umber wih y 1 = 1 rd be ieger uch ha d 1 r > max 2 ad le U r;d = d 1! r + r+1+ d 1 rd+1+ 0 B rd+1+r+ y 1 U 0 r;d = 1 8 The for H 1 = 1 + U r ;d 9 we have 1 = 1 + U r + ;d 10 H 1 = 1 + U r;d 11 Proof I Propoiio 1 choice x = x 1 x 2 uch ha x = 0 if d 1 ad x +1 = + 1!y +1 / + 1! afer ha apply Lemma 2 5

6 Example 4 For y = 1! 2!q + 1! 0 i Theorem 3 he ideiy B y = 12! q ee [3] give H 1 = 1 1 = 1 + H 1 = 1 + r+1 d 1! d 1 r r rd+1 q r++1+ d 1! d 1 r + + r + + r+d+1+ d 1! r r + r + d 1 q where i he coefficie defied by 1 + ϕ + q ϕ2 + + ϕ q = q ϕ 0 q Example 5 For y = 0! 1! i Theorem 16 he ideiy B y = [ ] give H 1 = 1 r+1 [ ] d 1! d 1 rd + 1 r rd+1 r r 1 = 1 + r++1+ [ ] d 1! d 1 r + d r + + r+d+1+ r + + r++ H 1 = 1 + r+1+ [ ] d 1! d 1 rd r + r + rd+1+ r+ Example 6 For y = 1 1 i Theorem 16 he ideiy B y = { } give H 1 = 1 r+1 { } d 1! d 1 rd + 1 r rd+1 r r 1 = 1 + r++1+ { } d 1! d 1 r + d r + + r+d+1+ r + + r++ H 1 = 1 + r+1+ { } d 1! d 1 rd r + r + rd+1+ r+ Propoiio 7 Le rd be ieger wih 0 d 1 ad f ϕ be biomial-ype 6

7 polyomial The for r > max 2 we ge H 1 = 1 ϕ r + 1 f α ϕ 1! d d 1 α ϕ H 1 1 = 1 + ϕ r ! d d 1 H 1 = 1 + ϕ r f α + ϕ d 1! d 1 α + ϕ f α + + ϕ α + + ϕ Proof For H 1 ae y = f ϕ;α = ϕ f α+ϕ α + ϕ i Theorem 3 ad ue Propoiio 1 give i [12] afer ha replace α by α rd ad ϕ by ϕ Example 8 For f ϕ = ϕ i Propoiio 7 we ge H 1 = 1 ϕ r + 1 1! α ϕ d 1 d 1 H 1 1 = 1 + ϕ r ! d d 1 α + + ϕ 1 H 1 = 1 + ϕ r d 1! α + ϕ 1 d 1 ad for f ϕ = ϕ i Propoiio 7 we ge H 1 = 1 ϕ r + 1 α ϕ 1! d d 1 H 1 1 = 1 + ϕ r α + + ϕ 1! d d 1 H 1 = 1 + ϕ r α + ϕ d 1! d 1 The followig heorem geeralize Theorem 3 Theorem 9 Le x = x 1 x 2 be a equece of real umber wih x 1 = 1 ad ruvd be ieger uch ha d 1 r > max 2 ad u > max v d v The for H 1 = 1 1 d 1!v u + 1 v r + 1 B u+1 vu v x u v u v d 1 we have H 1 1 = 1 + H 1 = 1 + r++1+ d 1!v d 1 u + + v u+d+1+v B u+d+1+vu++v x u++v 1 d 1!v u v r u + v u + v d 1 B u+1+vu+v x 7

8 Proof Le f ϕ be a equece of of biomial ype of polyomial uch ha f 1 = x +1 / + 1 wih x 2 0 o eure ha Df 1 ϕ 0 The f ϕ aifie 7 Se α = u ad ϕ = v i Propoiio 7 afer ha ue he ideiy 7 i he hree power erie of Propoiio 7 repecively for = u v = u + + v ad = u + v The heorem remai rue for he cae x 2 = 0 by coiuiy For u = rd ad v = i Theorem 9 we obai Theorem 3 Example 10 Se x = x 1 x 2 wih x = p+ 2 p 1 p 1 i Theorem 9 The ideiy [16 Example 13] B x = + p ! p 1 implie for he power erie H 1 = 1 d 1!v r + 1 up + 1 vp 1 u v d 1 up vp 1 we have H 1 1 = 1 + d 1!v r u + dp vp 1 u + + v d 1 u + dp + vp 1 H 1 = 1 + d 1!v r up vp 1 u + v d 1 up + vp 1 Example 11 Se x = x 1 x 2 wih x = p j+p 1 1 [ p+q+j 1 ] p 1 p+q Theorem 9 The ideiy [16 Example 13] B x = implie for he power erie we have H 1 = 1 H 1 1 = 1 + H 1 = 1 + v p 1 [ ] + p 1 + p + q 1 p 1 p + q pa [ +p+qa ] r + 1! A p+qa v rd + 1! A +A +pa A p 1A pb [ +p+qb ] r ! B p+qb v r + d + 1 +! B +B +pb B p 1B pc [ +p+qc ] r + 1 +! C p+qc Cq rd C +C C +pc p 1C Aq Bq q p 1 q 0 i q C = u + v A = u v 14 B = u + + v 8

9 Example 12 Se x = x 1 x 2 wih x = p j+p 1 1 { p+q+j 1 } p 1 q 0 i p 1 p+q q Theorem 9 The ideiy [16 Example 13] 1 { } p + p 1 + p + q 1 B x = 15 p 1 p + q implie for he power erie pa { +p+qa } r + 1! A p+qa v rd + 1! A +A +pa A p 1A we have H 1 = 1 H 1 1 = 1 + H 1 = 1 + v pb { +p+qb } r ! B p+qb v r + d + 1 +! B +B +pb B p 1B pc { +p+qc } r + 1 +! C p+qc Cq rd C +C C +pc p 1C Aq Bq q C = u + v A = u v B = u + + v 3 The ecod family of power erie ad heir ivere We give i hi ecio he compoiioal ivere of a power erie give from a ecod family of power erie which have coefficie ca be expreed i erm of parial Bell polyomial Lemma 13 Le x = x 0 x 1 wih x 0 = 1 ad y = y 1 y 2 be equece of real umber wih y j = jx j 1 The he compoiioal ivere of i give by H = 1 + x 16 = ! B 2+1+1y 17 Proof Le f ϕ be a equece of polyomial uch ha f 1 = x ad aume ha x 1 0 o eure ha Df 1 ϕ 0 The we ge f = + 1B+ y I [13 Theorem 1] we proved ha H = 1 + f 1 = f 1 1 H 1 = f 1 The from he la ideiy f 1 ca be expreed by parial Bell polyomial a f 1 = B 2 1y The lemma hold for he cae x 1 = 0 by coiuiy 9

10 Propoiio 14 Le x = x 1 x 2 y = y 1 y 2 be equece of real umber wih x 1 = 1y j = jx j 1 ad d be a poiive ieger The we have he pair of compoiioal ivere power erie H = 1 + x 18 = 1 +! d ! B d+1+1+1y 19 Proof For d 2 choice i Lemma 13 x = 0 if d O uig he oaio of Lemma 2 we ge y = x 1 = 0 if d 1 ad 1 + x = 1 + x! Le z = z 1 z 2 wih z j = j!y dj 1+1 /dj 1 + 1! The o uig Lemma 2 we obai: If d 1 he B y = 0 ad if d 1 we ge! 2 + 1! B 2+1+1y =! d ! B d+1+1+1z Therefore he pair of he power erie give i Lemma 13 ca be wrie a H = 1 + x /! H 1 = 1 +! d ! B d+1+1+1z To fiih hi proof replace x /! by x Example 15 For x = i Propoiio 14 we ge d H = d+1 e d 1 H 1 = for x = 1! 2! 3! i Propoiio 14 we ge ad for x = 1! 2 2! H = 1 d = 0 3! 3 4 i Propoiio 14 we ge { d d + 1 H = d+1 l1 d H 1 = 1 d [ d } ] 10

11 Theorem 16 Le x = x 1 x 2 be a equece of real umber wih x 1 = 1 rd be ieger uch ha r > max d + 1 d 1 ad le V r = r + B r+1+ r+ x 1 V 0 r = 1 20 r+1+ r+ The we have he pair of compoiioal ivere H 2 = 1 + V r 2 = 1 + V r + d Proof Le f ϕ be a equece of biomial ype uch ha f 1 = x +1 Necearily f +1 ϕ aifie 7 The becaue ad 1 r B r+1+r+x = f ;r r + r + H = 1 + r + B r+1+ r+ x r+1+ r+ 1 = f r we ca ae o uig Propoiio 14 ad Propoiio 1 give [12] ha = 1 + r + + f r + + ad by he ideiy 7 we obai = 1 + r + + B r+d+1+ r++ x r+d+1+ r++ I uffice o remar ha from Propoiio 1 we have 1 + B r+1+ r+ x 1 r + = 1 B r+1 r x r+1+ r+ r r+1 r Example 17 Wih x = 1! 2!q + 1! 0 i Theorem 16 he ideiy 12 give H 2 = 1 r r q 2 = 1 + r + + r + + q 11

12 Wih x = 1! 2! i Theorem 16 we obai H 2 = 1 r r r 1 2 = 1 + r + + Wih x = 1 1 i Theorem 16 we obai H 2 = 1 2 = 1 + r r + + Wih x = 0! 1! i Theorem 16 we obai H 2 = 1 2 = 1 + r r + d r { } r + 1 r + 1 r r r + d r [ ] r + 1 r + 1 r r r + d r + + r { r + d r [ r + d r + + Oher example ca be derived by uig he ideiie ad 15 } Propoiio 18 Le rd be ieger wih r > max d + 1 d 1 ad f ϕ be a equece of biomial ype The we have he pair of compoiioal ivere H 2 = 1 ϕ α ϕ f α ϕ 2 = 1 + ϕ α + dϕ + ϕ f α + dϕ + ϕ Proof Se x = f 1 ϕ;α i Theorem 16 o ge H 2 = 1 2 = 1 + ad chage ϕ by ϕ a ad α rϕ ad ϕ by ϕ ϕ α + rϕ ϕ f α + rϕ ϕ ϕ α + rϕ + dϕ + ϕ f α + rϕ + dϕ + ϕ ] 12

13 Example 19 Wih f ϕ = ϕ i Propoiio 18 we ge H 2 = 1 ϕ α ϕ 1 2 = 1 + x α + dϕ + x 1 Wih f ϕ = ϕ i Propoiio 18 we ge H 2 = 1 ϕ α ϕ α ϕ 2 = 1 + ϕ α + dϕ + ϕ α + dϕ + x 4 Coequece ad complemeary reul We give i hi ecio ome properie ad complemeary remar o he compoiioal ivere by aig paricular cae of Theorem 3 ad 16 Corollary 20 Le x = x 1 x 2 be a equece of real umber ad rdf be ieger wih d 1 ad f 2 The for r > max 2 he ivere power erie give by 9 ad 10 hold for U r;d = r + 1 +! r + rd B rd+1+ f 1[/f]r+ x rd f 1[/f]! ad for r > max d + 1 he ivere power erie give by 21 ad 22 hold for V r = r + 1! B r+1+ f 1[/f] r+x r f 1[/f]! where [ϕ] i he large ieger ϕ Proof Se y 1 = 1 y 2 = = y f = 0 ad = mf + δ 0 δ f 1 i Theorem 3 o ge B + y j = B mf+δ+y j + = mf+δ mf+δ+ i=0! i! B mf+δi yj+1 j + 1 m+δ mf + δ!! = m + δ! i! B j!yj+f m+δi j + f! i=0 1 mf + δ! m + δ + j!yj+f 1 = B m+δ+ m + δ! j + f 1!! = + f 1[/f]! B + f 1[/f] 13 j!yj+f 1 j + f 1!

14 The for x = y +f 1 / + f 1! we obai 1 d 1! rd r U r;d = B rd+1+r+ y j r + d 1 1 r + 1 +! rd B rd+1+ f 1[/f]r+ x j = r + rd f 1[/f]! V r = B r+1+ r+ y j = r + 1! B r+1+ f 1[/f] r+x j r + r f 1[/f]! r+1+ r+ Propoiio 21 Le r be ieger ax be real umber ad f ϕ be a biomial-ype polyomial The for r > max 2 he ivere power erie give by 9 ad 10 hold for U r;d = d 1! r + r d 1 D r+ z=0 e z f r + ϕ + z;α ad for r > max d + 1 he ivere power erie give by 21 ad 22 hold for V r = r + Dr+ z=0 e z f r + ϕ + z;α Proof For y = D z=0 e z f 1 ϕ + z;α i Theorem 3 ad 16 ad ue he ideiy give i [15 Lemma 1] by B jd z=0 e z f j 1 ϕ + z;α = D z=0 e z f ϕ + z;α Theorem 3 ad 16 remai rue whe oe ue a fiie produc of power erie a follow: Propoiio 22 Le x = x 1 x 2 be a equece of real umber wih x 1 = 1 1 m rm be ieger ad H r be power erie defied by H r = 1 B r+1 r x r > max 0 := + r The we have m H ri i=1 r+1 = H r wih r > max m = m i Proof Le f ϕ be a equece of polyomial of biomial ype uch ha f 1 = x +1 /+ 1 ad le f;r ϕ be he expoeial geeraig fucio of he equece of biomial ype f ;r Aume ha x 2 0 By Propoiio 1 give i [12] we ge m m m i=1 H ri = i=1 ad hi i exacly 0 f i ;r 1 r = i=1f;r i = 0 B r+1 r x m = r+1 The propoiio remai rue for he cae x 2 = 0 by coiuiy 14 i=1 f ;r = f ;r 0 i=1 H ri

15 Corollary 23 Le y = y 1 y 2 be a equece of real umber wih y 1 = 1 rd be ieger wih d 1 ad le U r;d V r;d be give by Theorem 3 ad Theorem 16 The for r > max 2 we have B U j r ;d + 1 = U r + ;d =1 B U j r + ;d + 1 = U r ;d =1 ad for r > max 1 + d we have B V j r + 1 = V r + d =1 B V j r + d + 1 = V r =1 Proof From Come [7 pp 151] for h = 1 + a we have h 1 = 1 + b wih b = B a 1 a 2 =1 The i uffice o combie wih Theorem 3 by aig a = U r ;d ad b = U r + ;d ad combie wih Theorem 16 by aig a = V r ad b = V r + d Referece [1] M Abramowiz ad I A Segu ed Hadboo of Mahemaical Fucio wih Formula Graph ad Mahemaical Table Dover Publicaio 1992 [2] M Aiger Combiaorial Theory Spriger 1979 [3] H Belbachir S Bouroubi ad A Khelladi Coecio bewee ordiary muliomial geeralized Fiboacci umber parial Bell pariio polyomial ad covoluio power of dicree uiform diribuio A Mah Iform [4] E T Bell Expoeial polyomial A Mah [5] A Z Broder The r-sirlig umber Dicree Mah [6] C B Colli The role of Bell polyomial i iegraio J Compu Appl Mah [7] L Come Advaced Combiaoric D Reidel

16 [8] D Domiici Need derivaive: a imple mehod for compuig erie ex-paio of ivere fucio I J Mah Mah Sci [9] A R Foryh Theory of Fucio of a Complex Variable Vol 1 ad 2 Third ed Dover Publicaio 1965 [10] B Germao ad M R Marielli Bell polyomial ad geeralized Bliard problem Mah Compu Modellig [11] T Maour ad Y Su Bell polyomial ad -geeralized Dyc pah Dicree Appl Mah [12] M Mihoubi Bell polyomial ad biomial ype equece Dicree Mah [13] M Mihoubi Parial Bell polyomial ad ivere relaio J Ieger Seq Aricle 1045 [14] M Mihoubi Some cogruece for he parial Bell polyomial J Ieger Seq Aricle 0941 [15] M Mihoubi The role of biomial ype equece i deermiaio ideiie for Bell polyomial To appear i Ar Combi Prepri available a hp://arxivorg/ab/ v1 [16] M Mihoubi ad M S Maamra Touchard polyomial parial Bell polyomial ad polyomial of biomial ype J Ieger Seq Aricle 1131 [17] J Riorda Combiaorial Ideiie R E Krieger 1979 [18] S Roma The Umbral Calculu Academic Pre 1984 [19] R P Saley Eumeraive Combiaoric Vol 2 Cambridge Uiveriy Pre 1998 [20] E T Whiaer O he reverio of erie Gaz Ma Mahemaic Subjec Claificaio: Primary 11B73; Secodary 30B10 70H03 05A10 Keyword: Parial Bell polyomial; biomial-ype polyomial; reciprocal ad compoiioal ivere; Sirlig umber; biomial coefficie Received April ; revied verio received March Publihed i Joural of Ieger Sequece March Reur o Joural of Ieger Sequece home page 16