Some identities related to reciprocal functions

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1 Discree Mahemaics Some ideiies relaed o reciprocal fucios Xiqiag Zhao a;b;, Tiamig Wag c a Deparme of Aerodyamics, College of Aerospace Egieerig, Najig Uiversiy of Aeroauics ad Asroauics, Najig Jiagsu 2006, People s Republic of Chia b Deparme of Mahemaics, Shadog Isiue of Techology, Zibo Shadog 25502, People s Republic of Chia c Deparme of Applied Mahemaics, Dalia Uiversiy of Techology, Dalia Liaoig 6024, People s Republic of Chia Received 22 November 999; received i revised form March 2002; acceped April 2002 Absrac The cocep of Riorda array is used o reciprocal fucios, ad some ideiies ivolvig biomial umbers, Sirlig umbers ad may oher special umbers are obaied. c 2002 Elsevier Sciece B.V. All righs reserved. MSC: 05A9; 05A0; C20; 05A40 Keywords: Riorda array; Geeraig fucio; Combiaorial ideiy. Iroducio I 99 [4,5,7] Shapiro iroduced he cocep of he Riorda group, which correspods o a se of iie lower-riagular marices. Riorda groups are paricularly impora i sudyig combiaorial ideiies ad combiaorial sums. For example, i 994 [8], Sprugoli sudied Riorda arrays relaed o biomial coecies, coloured wals ad Sirlig umbers. His wor veried ha may combiaorial sums ca be solved by rasformig he geeraig fucios. I 995 [9], Sprugoli paid aeio o he ideiies of Abel ad Gould, respecively. I his paper, we coiue he wors of Shapiro ad Sprugoli o discuss some ew applicaios of Riorda arrays. We also obai may ew ideiies relaed o special umbers, such as Sirlig umbers of boh ids, ad Beroulli umbers. Correspodig auhor. address: zhaodss@yahoo.com.c X. Zhao X/03/$ - see fro maer c 2002 Elsevier Sciece B.V. All righs reserved. S X

2 324 X. Zhao, T. Wag / Discree Mahemaics Riorda arrays ad Lagrage iversio formulas Noaio R se of real umbers R[] a rig of formal power series i some ideermiae N N {0; ; 2;:::} f [ ]f f R[], f [ ]f deoes he coecies of i he expasio of f i f he composiioal iverse fucio of f, i.e., ff f f fg{f } f is he ordiary geeraig fucio of he sequece {f }. fe{ f } f is he expoeial geeraig fucio of he sequece { f }. ordf is he smalles ieger for which f 0, ad is called he order of f I his paper, we resric ourselves o he cocep of Riorda array as i [7]. This may be described as follows: Le g;f R[], g g, f f wih f 0 0 here we assume f 0, ad f as is composiioal iverse. The sequece of fucios {d } N is ieraively deed by d 0 g; d gf ; which also dees a iie lower-riagular marix {d ; ; N; 066}, where d ; [ ]d. The iie lower-riagular marix {d ; } is called a Riorda array i. Ad we deoe D g;f {d ; }. I [9], Sprugoli proved a impora formula [Theorem 3., p. 28], which ca be used o obai may ideiies. Similarly, we give Theorems ad 2. Theorem. Le D g;f be a Riorda array ad f f. The we have d ; f [ ]gg ; 0: 0 Proof. 0 d ; f 0 [ ]gf [y ] fy [ ]g ff[ ]g g. Example. Le D ;. The d ; ad f +. So we have ; 0:

3 X. Zhao, T. Wag / Discree Mahemaics Example 2. Le D ; l. The we have d ; [ ] l! [ ] ;! where [ ] deoes he usiged Sirlig umbers of he rs id, f l ; f e, ad { f [ ] e! ; 0; 0; 0: Therefore, Theorem gives [ ]! ; ; 0; where is he Kroecer dela. Le D 2 l m ; l ad 0. The [ ] d 2 ; [ ] l m l m +!! m + ad by, we have 0 m + [ m + ] [ ] : m If { } deoes he Sirlig umbers of he secod id, he e p { } p! :! p p If we cosider he Riorda arrays D 3 ; e ad D 4 e p ; e, he, by, we have he followig ideiies: {!! } ; ; 0 ad + p! { } { } p! ; 0: + p p Furhermore, all of he above ideiies ca be proved by a direc applicaio of he Riorda array cocep, for example, if 0, he rs ideiy may be obaied

4 326 X. Zhao, T. Wag / Discree Mahemaics as follows:! 0 [ 0 ]!! i i + i0 [ ] [ ]e e [ e [ y ] ] e y y l [ ] [ ] ; : Theorem. The hypoheses are he same as hose i Theorem. The, by usig he Lagrage iversio formula see [], we have f d ; [ ] [ ]g; 0: 2 Example 3. Le D he m m+ ; a+ ; + a d ; see [8; p: 272]; m +a + f a+ ; f [ ] a + a+ [ ] a+ ad [ ]g[ ] m m+ : m

5 So we have m 0 X. Zhao, T. Wag / Discree Mahemaics a m +a + a + ; 0: m Theorem 2. Le D g;f be a Riorda array, h be he geeraig fucio of he sequece {h } N ad h f h f. The we have d ; h f [ ]gh: 3 Proof. d ; h f [ ]gh ff[ ]gh. Example 4. Now we cosider he Riorda array D ; e ad h expe. The, by 3, we have!! { }![ ] expe or { } B, where B are he Bell umbers, whose expoeial geeraig fucio is expe 0!B. Example 5. Le B geeraig fucio is e Le 0 D ; l + ; deoe he Beroulli umbers of high order, whose expoeial B! : h l p : + The d ; [ ]l + e [ ] B! ; fe, h f e P, ad h f p!b p. Therefore, we have p p! B B p [ ] l ;! + or + p! B Bp +p p!! [ ] + p : p By he Lagrage iversio formulas of all ids i [], we ca easily obai may formulas. These formulas ca be used i diere cases.

6 328 X. Zhao, T. Wag / Discree Mahemaics Theorem 3. Le D g;f be a Riorda array. The we have q f d ; [ q ] [ q ]g: 4 Proof. See []. Theorem 4. Le D g;f be a Riorda array ad h 0 h. The we have d ; 0 h 0 + Proof. See []. f d ; [ ]h [ ]gh: 5 Whe oly owig he coecies of powers of f wih posiive iegral expoes, we have he followig heorem. Theorem 5. Le D g;f be a Riorda array, q be a posiive ieger ad 6q6. The we have q Proof. See []. Example 6. Le because 2 q q qd ; j j + j f j q [ q+j ]f j j [ q ]g: 6 D p+ l ; q { G H m+ H m } m + m m+ l ; he p + +q d ; H p++q H p+q ;

7 X. Zhao, T. Wag / Discree Mahemaics where H. By6, we have 2 l p + +q l H p++q H p+q l l j j + j l j H p+ l H p qj + l l p + l : p Theorem 6. Le hree fucios Fx;Gx ad Hy of a real variable be give, where Fx ad Gx are of class C i x a, ad Hy is of class C i y b Fa, ad le PxHFx. If we pu g l l! p m m! d l G dx l ; f xa! d m P dx m ; xa d F dx ; h xa! d H dy ; yb f 0 Fa; f 0; g 0 0, h 0 Hbp 0 PaHFa, ad dee he followig formal power series: g l 0 g l l ; f f ; hu 0 h u ; p m 0 p m m : The g;f is a proper Riorda array ad we have d ; h [ ]gp 0 g j p j : 7 j0 Proof. By he deiio of he proper Riorda array ad f 0, we ow ha g; f is a proper Riorda array, ad we have d ; h [ ]ghf. O he oher had, from Theorem B i [, p. 38], we obai formally phf. So we have d ; h [ ]ghf[ ]gp 0 g j p j : j0 Example 7. Le H m ; G, ad F e + 2! ; F0: 3!

8 330 X. Zhao, T. Wag / Discree Mahemaics The f2! + 2 3! + ; gg, ad g;f {d ; } is a proper Riorda array, where d ; [ ] [ ] j0 2! + 2 3! + + +! + 2! + 2 3! + + +! j ; 2 ;:::; i i +! i : Ad h m ; where m is he risig facorial of m of order. So from 7, we have j0 j m ; 2 ;:::; i i +! i j0 B m j j! ; where j deoes se of pariios of jj N, represeed by 2 2 j j wih j j j, i N; i ; 2;:::;j. 3. The expoeial Riorda array For he expoeial geeraig fucio of a sequece, we have Deiio. Le ge{ g }, fe{ f } R[], ordg 0, ordf. For a iie lower riagular array D {d ; ; N; 066}, if for xed, E{d ; } gf! 0, he we wrie D g;f ad say ha g;f is a expoeial Riorda array. Le g;f ad d;h be wo expoeial Riorda arrays. Le d;h g;f dgh;fh. The he se of all expoeial Riorda arrays form a iie group ad ; is is ui eleme. Jus as i [8,9,2], we ca obai may ideiies relaed o he expoeial Riorda arrays. Besides, he expoeial Riorda arrays are direcly relaed o he classical Umbral Calculus. The ieresed readers ca see he relaive papers ad he wors of Roa, Roma ad Kuh [3,6]. Le f f!, g g! R[], fg gf, ad f0 g0 0. We dee umber pair {A ; ;A 2 ; } as follows: d f! A ;! ; d g! A 2 ;! : If fg gf ad d, he {A ; ;A 2 ; } is he geeralized Sirlig umber pair i [2].

9 X. Zhao, T. Wag / Discree Mahemaics Theorem 7. For umber pair {A ; ;A 2 ; } ad 0, we have A ; g A 2 ; f ; : 8 If, i paricular, {A ; ;A 2 ; } is he Sirlig umber pair, we have g A 2 ; A 2 ; j f j ; 9 f 6j66 6j66 A ; A ; j g j : 0 Proof. We have d gf d f g! g A ;! A ; g! : O he oher had, d fg d g! f A 2 ; f! : f A 2 ; The we obai 8 by ideifyig he coecies of! i d fg ad d gf. If d ad fg gf, he {A ; } {A 2 ; } ;f ;g ;, hus A ; A 2 ; I iie ui marix. Ad we obai he iverse relaio: a A ; b ; b A 2 ; a : Le a A ; g i 8. By he above iverse relaio, we have g A 2 ; a. The, by 8, we obai! g A 2 ; j A 2 ; j f j 6j66 A 2 ; A 2 ; j f j :

10 332 X. Zhao, T. Wag / Discree Mahemaics The proof of 0 is similar o ha of 9. Example 8. Le {A ; ;A 2 ; } be he geeralized Sirlig umber pair. If {A ; } g ;f ; {A 2 ; } g 2 ;f 2, he we deoe simply {A ; ;A 2 ; } as { g ;f ; g 2 ;f 2 }. For he geeralized Sirlig umber pair { L ; ;! } { ; ; ; where L ; is he Lah umber, which has he expressio L ;!! by Theorem 7, we ca obai he followig ideiies: +!L ; ; ; 0;! 6j66! j! 6j66 + j!l ; L ; j ; } ; + j ; 0; j 0: Le f R[] ad gff. The fg gf. So we have he followig example. Example 9. Le fe ad gffe e m B m m m!, where B m is he Bell umber. The f, g B. If d, he we have d f!! e { }! ad d g!! m m B m B ; B ;:::;B + m!! where B ; x ;:::;x + is he parial Bell polyomial. Therefore, we obai umber pair {{ } {A ; ;A 2 ; } ;B } ; B ;:::;B + { ; e ; ; e e }:

11 ad X. Zhao, T. Wag / Discree Mahemaics From 8, we have he followig ideiy: { } B B ; B ;:::;B + : If de, he we have d f!! e { } +! d g!! e e e [ B ; B ;:::;B +!! i B i i; B ;:::;B i + ]! : ad Therefore, we have 0; ; { } A ; + ; + + A 2 ; i From 8, we have { } + B + B i i; B ;:::;B i + : i i B i; B ;:::;B i + : Example 0. I [], Xu Lizhi L.C. Hsu ad Yu Hogqua have deed he geeralized Sirlig umber pair {S; ; ; ;S; ; ; } as follows:!! + + S; ; ;! ; S; ; ;! ;

12 334 X. Zhao, T. Wag / Discree Mahemaics i.e. { {S; ; ; ;S; ; ; } ; + ; ; + } : Le f+ ad g+. The f, g, ad we have, by 8 ad 9, respecively, ha S; ; ; ; ; 0; 6j66 S; ; ; S; j; ; j ; 0; 2 while he oher wo ideiies ivolvig S; ; ; may be obaied i a lie way sice ad are symmeric. [ I paricular, aig ad leig 0, we easily d ha S; ; ; 0 + ] { ad S; ; 0; } jus sad for he ordiary Sirlig umbers of he rs ad secod ids, respecively. If aig ad, he S; ; ; + S ; ad S; ; ; S;, where S ; ad S; are called he degeerae Sirlig umbers of he rs ad secod id by Carliz [0] ad deed by! S ;! ;! + S;! ; where. Moreover we have, by ad 2, respecively, he followig ideiies, + S ; S; + ; ; 6j66 S; S; j 0; j +; 0;

13 X. Zhao, T. Wag / Discree Mahemaics S ; S ; j 6j66 j ; 0: Acowledgemes This wor was suppored by he Naioal Naural Sciece Foudaio of Chia The auhor wishes o ha he referee for may useful suggesios ha led o he improveme ad revisio of his oe. Refereces [] L. Come, Advaced Combiaorics, Reidel, Dordrech, 974. [2] L.C. Hsu, Geeralized Sirlig umber pairs associaed wih iverse relaios, The Fiboacci Quar [3] D.E. Kuh, The Ar of Compuer Programmig, Vol. I III, Addiso-Wesley, Readig, MA, [4] D. Merlii, D.G. Rogers, R. Sprugoli, M.C. Verri, O some aleraive characerizaios of Riorda arrays, Caad. J. Mah [5] D.G. Rogers, Pascal riagles, Caala umbers ad reewal arrays, Discree Mah [6] S.M. Roma, G.C. Roa, The Umbral calculus, Adv. i Mah [7] L.W. Shapiro, S. Geu, W.J. Woa, L. Woodso, The Riorda group, Discree Appl. Mah [8] R. Sprugoli, Riorda arrays ad combiaorial sums, Discree Mah [9] R. Sprugoli, Riorda arrays ad he Abel Gould ideiy, Discree Mah [0] Wag Tiamig, Yu Hogqua, Yao Hog, A marix represeaio of combiaorial umbers wih applicaios, J. Dalia Uiv. Techol [] Xu LizhiL.C. Hsu, Yu Hogqua, A uied approach o a class of Sirlig-ype pairs, Appl. Mah. -JCU 2B [2] Yi Dogsheg, Umbral calculus ad Hsu-Riorda array, Ph.D. Thesis, Dalia Uiversiy of Techology, 999.