SOME COMBINATORIAL SERIES AND RECIPROCAL RELATIONS INVOLVING MULTIFOLD CONVOLUTIONS

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1 #A20 INTEGERS 4 (204) SOME COMBINATORIAL SERIES AND RECIPROCAL RELATIONS INVOLVING MULTIFOLD CONVOLUTIONS Leetsch C. Hsu Istitute of Mathematics, Dalia Uiversity of Techology, Dalia, P. R. Chia i-rog Ma Departmet of Mathematics, Soochow Uiversity, Suzhou, P.R.Chia xrma@suda.edu.c Received: /5/3, Revised: /27/3, Accepted: 3/3/4, Published: 4/28/4 Abstract The preset paper cosiders a id of combiatorial series ad its allied reciprocal relatios which are determied by discrete multi-fold covolutios. Furthermore, their various formal ad aalytic expressios i explicit forms are obtaied. Costructive applicatios to some well-ow sequeces such as the Bell umbers, the Fiboacci umbers, the Stirlig umbers ad some others give by iteger partitio fuctios are also preseted.. Itroductio I this paper, we obtai various explicit costructive results for combiatorial sums or series. These results are closely related to what we will call the discrete multi-fold covolutios. Our paper cosists of two mai parts. The first part, composed of Sectios 2, 3, ad 4, deals with oe id of discrete multi-fold covolutio which is determied by a arbitrary fuctio o the set of positive itegers. Some basic idetities ad delta operator summatio formulae are ivestigated ad illustrated with examples. The secod part, cosistig of Sectios 5 ad 6, discusses aother geeral class of discrete multi-fold covolutios which are formed by fiitely may di eret fuctios with discrete variables. Costructive applicatios to some wellow umber sequeces are expouded i detail. I this paper, we will deote the sets of positive itegers, o-egative itegers, real umbers ad complex umbers by N +, N, R ad C respectively. Also, we will use the followig otatio: () : the set of partitios of 2 N +, usually writte as 2 2 with , i 2 N.

2 INTEGERS: 4 (204) 2 (, ) : the subset of () cosistig of the partitios of with parts, i.e., partitios 2 2 subject to (t) t(t )(t 2) (t + ) : the th fallig factorial of t with (t) 0. : the di erece operator defied by f(t) f(t + ) f(t), ad + ( 2 N + ) with 0 deotig the idetity operator. D : the di eretiatio operator with DD D + ad D 0. E : the shift operator defied by E + ad E x f(t) f(t + x), x 2 R. It should be poited out that both ad D, owadays ow as delta operators, could be geerally applied to formal power series to deduce certai formal results. Throughout the preset paper, we will mae frequet use of formal power series, ad provide suitable covergece coditios so that the formal results actually are exact ad aalytic uder these coditios. 2. A Kid of Multi-Fold Covolutio As usual, we will say that (x, x 2,, x ) is a -compositio of with o-egative parts, if x + x x, where x i 2 N, apple i apple. The set of all such compositios of may be deoted by [,, 0], i.e., [,, 0] : (x, x 2,, x ) x i, x i 0. i Also, we will use the stadard represetatio for the set (, ), i.e., (, ) : (x, x 2,, x ) ix i, x i, x i 0. i i Defiitio 2.. Let f(x) be a real-valued or complex-valued fuctio defied o N with f(0). The the -fold covolutio ad the /-partitio sum associated with f(x) are respectively defied by the followig summatios: S(f) f(x )f(x 2 ) f(x ) (2.) [,,0] T (f) (,) f ()f 2 (2) f ()! 2!! (2.2) where the sums o the right-had sides (i short, RHS) of (2.) ad (2.2) rage over the sets [,, 0] ad (, ) respectively.

3 INTEGERS: 4 (204) 3 Note that [,, 0] ad (, ) have o meaig for 0. For coveiece, we defie S 0 (f) T 0 (f) 0, ad thereby have sequeces {S (f)}, 0 ad {T (f)}, 0. Also it is obvious that S (f) f(), S (f) f() ad T (f) f(), T (f) f(); T (f) 0 for >. Moreover, it is easy to see that the RHS of (2.2) with replacemet f()! t ( 2 N + ) is i agreemet with the icomplete Bell polyomial i t i ( apple i apple ), up to a costat factor! (cf.[2]). I what follows we assume that G(t) g()t (2.3) is a formal power series with complex coe ciets. As usual, G () (t) D G(t) deotes the th formal derivative of G(t). Theorem 2.2. For m, 2 N +, the followig idetities hold: m S m (f) (m) T (f) (2.4) T (f) S t! (f) t0 (2.5) g()s(f)t G () (t)t (f)t, (2.6) where the left-had side (i short, LHS) of (2.6) is a formal power series. Proof. To justify (2.4), accordig to Defiitio 2., we oly eed to compute the LHS of (2.4) i this way: for apple apple m, cosider first the fiite sum [,m,0] f(x )f(x 2 ) f(x m ), (2.7) where [, m, 0] deotes the subset of [, m, 0], beig composed of all compositios (x, x 2,, x m ) of with just compoets x i. I other words, there are m compoets x i 0 i (x, x 2,, x m ). Recall that f(0) ad such factors will m tae m ordered places i m m di eret ways. Meawhile, the umber of all possible permutatios of the factors i the product over the set dow to f ()f 2 (2) f () (, ) is eumerated by!/(! 2!!). Thus the sum (2.7) boils m (,)!! 2!! f ()f 2 (2) f () (m) T (f).

4 INTEGERS: 4 (204) 4 Summig o, apple apple m, we therefore obtai (2.4). With (2.4) i had we are able to reformulate S(f) t i the form S(f) t t j! T j (f), j S(f) 0 0. j The it follows that S(f) t t0 j! t j t0 T (f) j j 0 j! T j (f) j! T (f). j Thus (2.5) is proved. Oce agai, by use of (2.4) we may compute the LHS of (2.6) formally as follows: g(m)s m (f)t m m g(m)t m m m (m) T (f) (m) g(m)t T m (f) m D m G () (t)t (f)t. g(m)t m T (f)t The last equality follows from the fact that T (f) 0 for >. This completes the proof of (2.6). 2 Put G(t) /( t) for t < ad e t for t 2 R i (2.6) i successio. The we may get the followig Corollary 2.3. For 2 N, the sequece {S(f)} 0 has the ordiary ad expoetial geeratig fuctios, respectively, as follows: S(f)t S (f) t!! t t t T (f) (2.8) e t T (f)t. (2.9) It should be metioed that as a extesio of a basic result cotaied i [5] by Savits ad Costatie, formula (2.6) was first give by Hsu i his short ote

5 INTEGERS: 4 (204) 5 [0]. Afterwards, it was recovered by Costatie [3] via a di eret approach. I fact, formula (2.6) is actually a cosequece implied by the geeral trasformatio formula of series (cf.[8]) f() () (0) t! f(0) () (t) t!. To see this, it su ces to substitute f(t) with S t (f) ad (t) with G(t), ad the simplify the resultig idetity by (2.5). 3. A Covergece Theorem I this sectio, we will provide a simple ad verifiable covergece coditio for (2.6) i order to mae it a available exact formula. Theorem 3.. If G(t) P 0 g()t is absolutely coverget for t < r, the so is the ifiite series o the LHS of (2.6), thereby (2.6) is a exact formula for t < r. Proof. Comparig the LHS of (2.6) with G(t) ad usig Cauchy s root test for the covergece of ifiite series, we oly eed to show that for every 2 N +, lim! S (f) / apple. To this ed, assume max 0applexapple f(x). By the defiitio of S (f), it is easily foud that every product f(x )f(x 2 ) f(x ) restricted by x + x x (3.) cotais at most factors f(x i ) with x i. Owig to the facts that f(0) ad f(x) < for x 6 0, it follows directly that f(x )f(x 2 ) f(x ) apple. Meawhile, + the total umber of such terms is, a fact comig from the umber of solutios i oegative itegers for the diophatie equatio (3.). Thus we have + S(f) apple, that leads to lim! S (f) / + apple lim! /. Hece the theorem is proved. 2

6 INTEGERS: 4 (204) 6 Corollary 3.2. Idetity (2.8) is aalytic for t < ad so is (2.9) for t < +. Example 3.3. Evidetly, the followig power series 2 G(t) p t 4t coverges absolutely for t < /4. I this case, we have G () (t) (2)! ( 4t) /2, g()! 2. As a cosequece of (2.6), we get the followig exact formula 2 S (f)t (2)! t! ( 4t) T +/2 (f) (3.2) for t < /4. Example 3.4. It is well ow that the Fiboacci umber sequece {F } 0 is geerated by the geeratig fuctio G(t) t t 2 F t, which is coverget absolutely for t < ( p 5 )/2. To ease otatios, we let a ( + p p 5)/2 ad b ( 5)/2. We therefore have ad g() F (a + G(t) ( at)( bt) p (a + b + )t 5 G () (t) p 5 D a at that reduces (2.6) to F S(f)t b + )/ p 5. Simple computatio gives b! apple a p bt 5 at apple + +! a b p 5 at bt which is a exact formula for t < ( p 5 )/2. + b bt +, T (f)t, (3.3) Example 3.5. Settig G(t) ( + t) for 2 R ad t <, we have g() ad G () (t) ( ) ( + t). A direct substitutio of these facts ito (2.6) yields S (f)t t ( ) ( + t) T (f). (3.4)

7 INTEGERS: 4 (204) 7 The further substitutio! leads us to + S (f)( t) ( t) ( + ) ( + t) + + T (f). (3.5) Obviously, both (3.4) ad (3.5) are aalytic for t <. It is also clear that (2.8) ca be deduced from (3.4) via the substitutios! ad! t. As idicated above, formula (2.6) ca be used to costruct various special exact formulae or idetities via the choices of G(t) ad f(t), provided that they are subject to the covergece coditios give by Theorem Some Operator Summatio Formulae It is ow that both D ad are delta operators, so that by use of Mulli-Rota s substitutio rule (cf.[9]), we may deduce some special operator summatio formulae from (2.9), (3.2), (3.4) ad (3.5) respectively. For ay fuctio (t) over R, it is easy to chec that ( + ) (t) E (t) (t + ). Thus, uder the substitutio t!, we see that (3.4) ad (3.5) together yields a pair of -type operator summatio formulae. The results are as follows: S (f) (0) + S (f)( ) (0) ( ) T (f) ( ) (4.) ( + ) T (f)( ) ( ). (4.2) Alteratively, the substitutio t! 4 reduces (3.2) to a -type summatio formula of the form ( ) S (f) ( ) (2)! (0) 2 2 T (f) ( /2). (4.3)! As above, substitutig t by D i (2.9) ad simplifyig the result by the relatios that e D E +, we come up with a D-type summatio formula for (t) 2 C (the set of ifiitely di eretiable real fuctios over R) evaluated at t 0:! S (f)d (0) T (f)d (). (4.4)

8 INTEGERS: 4 (204) 8 I accordace with Example 3.5 i Sectio 3, it is clear that (4.) ad (4.2) are exact formulae uder the coditio lim! (0) / <. (4.5) Actually coditio (4.5) also esures the validity of formula (4.3) iasmuch as it is just equivalet to the coditio (see Example 3.3) lim! / 4 (0) < 4. Moreover, Corollary 3.2 states that (4.4) is a exact formula uder the coditio / lim! D (0) < +. (4.6) It may happe that lim! (0) /. I this case, the covergece coditios for (4.), (4.2), ad (4.3) should be ivestigated separately. As oe might expect, all formulae from (4.) to (4.4) ca be employed to produce various special formulae ad idetities, because f(x) ad (t) are free to choose. I what follows we will detail how to fid cocrete idetities by cosiderig some iterestig examples. Example 4.. Substitute (t) with (t) t+ m ad 2(t) /(t + ) i tur, m 2 N + ad > 0. The we have (t) t + m, 2 (t) ( ) t + +. t + t+ + From ow o, we write briefly for / t+ +. Uder these two choices, it is easy to deduce the followig six formulae respectively from (4.), (4.2), ad (4.3): m + S m (f) ( ) T (f) m m + ( ) m m ( ) 2 4 m S (f) ( ) ( + ) S (f) ( ) (2)! 4! (4.7) T (f) m (4.8) /2 T (f) m (4.9)

9 INTEGERS: 4 (204) 9 + ( ) S (f) + + S (f) S (f) ( ) ( ) + + ( + ) (2)! 4!( /2) T (f) (4.0) T (f) (4.) /2 T (f). (4.2) We remar that all ifiite series ivolved i (4.0), (4.), ad (4.2) are assumed to be coverget uder suitable coditios for, ad S(f). Example 4.2. The most simple case of S (f) is whe with f(). I such a case, it is easy to chec that S (f), T (f), T (f) 0 for >. Cosequetly, each idetity from (4.7) to (4.2) yields correspodigly a special idetity for. The results are stated as follows: m + (V adermode) (4.3) m m m ( ) 2 ( + ) (4.4) m m m ( ) 2 4 3/2 (4.5) m 2 m + ( ) (4.6) ( + )( + ) + + ( + ) (4.7) ( )( 2) ( /2)( 3/2). (4.8) Note that for > 0 ad > 0 we have the estimates apple b c + 2 O(/), /4 p O(/ p ) ad + + b c O( b c ) (! ), [ ] where is the absolute value of, bxc deotes the largest iteger ot greater tha x, the big O otatio taes the usual meaig of asymptotic. Thus all ifiite

10 INTEGERS: 4 (204) 0 series appearig i (4.6), (4.7), ad (4.8) are absolutely coverget uder their respective coditios, i.e., (4.6) : > 0, 2; (4.7) : > 0, + 3; (4.8) : 2. Both (4.6) ad (4.7) are comparable with the series ( ) + + (4.9) for > 0,. It is of iterest to ote that the special case of (4.9) whe is a iteger is recorded as a theorem i Wilf [7, p.34,theorem], beig employed as a example of the well-ow WZ method. Thus we believe that (4.6), (4.7), ad (4.8) may also be verified by meas of the WZ method. Example 4.3. By the multivariate Vadermode covolutio formula it is easily foud that for 2 R ad r 2 N +, x x 2 x [,,0] x x2 x +. r r r r + [,,0] Now, replace f(x) with x S x T ad r+x r i tur. For both cases, we therefore obtai, T 2 2 ; (4.20) x! 2!! (,) r + x (r + ) + S r r + x r+ r+2 2 r r r+ r. (4.2) r! 2!! (,) Accordigly, (2.6) leads us to a pair of combiatorial series as follows: g() t (r + ) + g() t G () (t)t T x (4.22) r + x G () (t)t T. r (4.23) As is expected, a variety of special idetities ca be deduced from (4.7) ad (4.2) with S (f) ad T (f) beig replaced by those of (4.20) ad (4.2). For istace,

11 INTEGERS: 4 (204) by virtue of (4.) ad (4.20), we may fid a combiatorial series + + ( + ) T. x (4.24) A bit of aalysis shows that this series is absolutely coverget uder the coditios, > 0, Other idetities are left to the iterested reader to wor out. 5. Reciprocal Relatios Let us begi with a basic result origially due to Hsu ad his coauthors of []. Lemma 5.. ([, Corollary ]) Let { } ad { } be two sequeces such that exp t + t (5.) or, equivaletly, log + The the system of relatios t t. (5.2) () 2 2! 2!! (5.3) holds if ad oly if so does the system 2 2 ( )! 2!!, (5.4) () where the sum o the right rages over () { 2 2 }, It is worth metioig that i the cotext of combiatorial aalysis, such a pair of equivalet relatios is said to be a reciprocal relatio. We refer the reader to [4] ad [, Sectios ad 2] for its precise defiitio ad applicatios. Meawhile, the factor ( ) is just the Möbius fuctio of the partitio lattice [6, (30)]. For >, it ca be writte explicitly as ( ) ( ) ( )!.

12 INTEGERS: 4 (204) 2 Keepig Lemma 5. i mid, we ow cosider a m-fold covolutio of the form S m (f) f (x )f 2 (x 2 ) f m (x m ), (5.5) [,m,0] where f j (x) are m ow fuctios with f j (0), the sum rages over all the m- compositios (x, x 2,, x m ) of with x + x x m, x j 0, apple j apple m. We set log + f j () t j() t (5.6) or, equivaletly exp j() t + f j () t, (5.7) where each j () is defied o N. All that we are iterested i is a reciprocal relatio betwee S m (f) ad P m j j(). The relevat coclusio ca be give as follows. Theorem 5.2. Let S m (f) ad relatio holds: m j S m (f) () j() () m () be give as above. The followig reciprocal Y r P m j ( ) Y r j(r) r r! (5.8) Sr m (f) r. (5.9) r! I particular, set f j (x) f(x) with f(0), apple j apple m. The j (x) (x). Therefore, (5.8) ad (5.9) together yield a reciprocal relatio as below. Corollary 5.3. We have S m (f) () m () (2) 2 ()! 2!! (5.0) () S m (f) S2 m (f) 2 S m (f) ( ). (5.) m! 2!! () Proof. Observe first that (5.8) ad (5.9) are special cosequeces of (5.3) ad (5.4) uder the choices that m j(), S m (f). j

13 INTEGERS: 4 (204) 3 Thus, by Lemma 5. it su ces to verify (5.) for such choices. To this ed, oe oly eeds to compute m my my exp t j() exp j() t f j () t j j 0 [,m,0] S m (f) t. 0 j 0 f (x )f 2 (x 2 ) f m (x m ) t Note that S m 0 (f) f (0)f 2 (0) f m (0). Hece idetity (5.) is cofirmed. By Lemma 5. agai, (5.7) ad (5.8) follow directly from (5.3) ad (5.4), respectively. Moreover, whe j (x) (x), the product i (5.8) becomes Y Y (m (r)) r m r r (r) r m Y r (r) r. Hece (5.0) ad (5.) are deduced from (5.7) ad (5.8) correspodigly. Thus the theorem ad the corollary are proved. 2 As already show i [], may classical special fuctios such as the Geebauer- Humbert polyomials ad the She er polyomials ca be represeted by the cycle idicator of the form C (t, t 2,, t ) 2! t t2 t. (5.2)! 2!! 2 () Such a represetatio has a advatage that the sequece {C (t, t 2,, t )} 0 satisfies a id of recurrece relatio. We refer the reader to [] for further details. Comparig (5.0) with (5.2) we may rewrite S m (f)! C m (), 2m (2),, m () (5.3) with each (r) beig give by the coe ciet of t r i the power series expasio of log( + P f()t ), i.e., (r) [t r ] log + f() t. (5.4) As a beefit of doig so, {S m (f)} m, may be computed via the recurrece relatios of C (t, t 2,, t ). It may be of iterest to compare the expressio (5.0) with the basic idetity (2.4). Nevertheless, there does ot seem to exist ay available recurrece relatio for the sum o the RHS of (2.4).

14 INTEGERS: 4 (204) 4 6. Further Examples Ivolvig Some Classical Number Sequeces I this sectio we will oly focus o some applicatios of Corollary 5.3. As for the reciprocal relatio (5.8) ad (5.9) i Theorem 5.2, we have ot foud ay good example so far. Example 6.. Recall that the Bell umbers $ are defied by the expoetial geeratig fuctio exp(e t t ) exp $() t!!. (6.) Defie the m-fold covolutio S m ($) [,m,0] 0 $(x )$(x 2 ) $(x m ). (6.2) x!x 2! x m! The by taig f() $()/! ad usig (5.4) we fid that () [ t ](e t ) /!. I such a case, a combiatio of the above result with (5.0) ad (5.) fially leads to a reciprocal relatio: S m ($) () m /! /2! 2 /!! 2!! (6.3) m! S m ($) S2 m ($) 2 S m ($) ( ). (6.4)! 2!! () It is of iterest to ote that the case m of (6.3) yields a well-ow idetity $()! () /! /2! 2 /!.! 2!! Example 6.2. Let p() ad M 2 () be the umbers of partitios ad plae partitios of, respectively. Recall further that i the boo [7, p. 30] by Hardy ad Wright, the divisor fuctio () is defied by () d d for, 0. We may write () for (). As is well-ow to us, the Euler formula [7] ad the MacMaho fuctio [4] state that the geeratig fuctios for {p()} 0 ad {M 2 ()} 0 are respectively give by Y Y ( t ) ad ( t ),

15 INTEGERS: 4 (204) 5 yieldig log + p() t log M 2 () t 0 () t 2() t. To proceed further, let us cosider the m-fold covolutios S m (p) p(x )p(x 2 ) p(x m ) ad S m (M 2 ) [,m,0] [,m,0] M 2 (x )M 2 (x 2 ) M 2 (x m ). Taig these ito accout ad maig use of both (5.0) ad (5.), we obtai two pairs of reciprocal relatios as follows: S m (p) () m ()/ (2)/2 2 ()/! 2!! m () S m (p) S2 m (p) 2 S m (p) ( )! 2!! () (6.5) (6.6) ad S m (M 2 ) () m 2()/ 2(2)/2 2 2()/! 2!! (6.7) m 2 () S m (M 2 ) S2 m (M 2 ) 2 S m (M 2 ) ( )! 2!! (). (6.8) I particular, whe m, we recover two ow idetities (cf.[7, (4.6)/(4.7)]): p() () ()/ (2)/2 2 ()/! 2!! (6.9) M 2 () () 2()/ 2(2)/2 2 2()/. (6.0)! 2!! Example 6.3. It is clear from [6, Vol.II, p.76, Exer. 5.3(b)] that for j exp (F j ) x! j x, 0

16 INTEGERS: 4 (204) 6 where F j deotes the free group o j geerators ad (F j ) deotes the umber of subgroups of F j of idex. O accout of this, we thereby obtai the followig reciprocal relatio: m j with the m-fold covolutio S m (F ) () (F j ) () S m (F ) [,m,0] Y r P m j ( ) Y r(f j ) r r! r r (6.) r S m r (F ) r r! (x!) 0 (x 2!) (x m!) m. (6.2) Both (6.) ad (6.2) cotai the well-ow Cauchy idetity ad its dual form as the special case whe m (F a cyclic group): ()! 2!! Y r r r (6.3) ( )! 2!!. (6.4) () We ca put the precedig results i a geeral cotext. To do this, replace F j with a free Abelia group G j which is fiitely geerated by j geerators ad let µ (G j ) be the umber of cojugacy classes, i.e., g G j g for all g 2 F j, of subgroups of G j of idex. Accordig to Group theory or [6, Vol. II, Exer.5.3(a)], we have exp (G j ) x Y ( x ) µ(gj). So it is easily foud that (G j ) d µ d (G j ) d, sice (G j ) x log Y ( x ) µ(gj) d x µ d (G j ) d. which meas that every elemet of the group F j ca be expressed as a combiatio (uder the group operatio) of elemets of certai j-subset of F j ad their iverses.

17 INTEGERS: 4 (204) 7 Thus we obtai m j S m (G) () (G j ) () Y r P m j ( ) Y r(g j ) r r! r r (6.5) r Sr m (G) r, (6.6) r! where the m-fold covolutio S m (G) µ d (G ) d [,m,0] d x d x 2 µ d (G 2 ) d 2 d x m µ d (G m ) d m. Example 6.4. Recall that i Example 3.4, we write a ( + p 5)/2, b ( p 5)/2. Thus, the Fiboacci umbers F may be writte as F (a + b + )/ p 5. Startig with the geeratig fuctio for {F } 0, amely F t ( at)( bt), 0 it is ot hard to fid that log F t a + b t. 0 O cosiderig the m-fold covolutio S m (F ) [,m,0] F x F x2 F xm, we immediately obtai the followig reciprocal relatio via (5.0) ad (5.): S m (F ) () m (a + b)/ (a 2 + b 2 )/2 2 (a + b )/ (6.7)! 2!! m a + b S m (F ) S2 m (F ) 2 S m (F ) ( ). (6.8)! 2!! () It is worth metioig that the case m of (6.8) is correspodet to the wellow idetity (cf.[7, Example 6]) () ( ) ( )! F F 2 2 F! 2!! p p

18 INTEGERS: 4 (204) 8 Example 6.5. For p 2 N +, as is ow to us, there holds a ratioal geeratig fuctio for the Stirlig umbers of the secod, amely 0 + p p t ( t)( 2t) ( pt), where we have adopted Kuth s otatio for the Stirlig umbers (cf.[2] or [7]). By this we at oce obtai + p log p 0 t p log( jt) j ( p ) t s t p, where s p P p j j is the sum of arithmetic progressio of degree. Give m ad p 2 N +, let us cosider the m-fold covolutio S m (S) [,m,0] x + p p x2 + p p xm + p p I view of (5.0) ad (5.), it is easy to fid the followig reciprocal relatio:. S m (S) () m s p/ s 2 p/2 2 s p /! 2!! (6.9) m s p S m (S) S2 m (S) 2 S m (S) ( )! 2!! () Certaily (6.9) may also be writte as. (6.20) S m (S)! C ms p, ms 2 p,, ms p. (6.2) Evidetly, the case m is just the cycle idicator of the Stirlig umbers + p p! C s p, s 2 p,, s p. 7. Cocludig Remars I our precedig discussio, we have illustrated the applicatios of Theorem 2.2, Theorem 5.2 ad Corollary 5.3 to Aalytic Combiatorics by establishig some

19 INTEGERS: 4 (204) 9 summatio ad trasformatio formulae. All these examples demostrate that the -fold covolutio ad / partitio give by Defiitio 2. are closely related with the geeratig fuctios of admissible combiatorial costructios (cf.[5, Def..5; I.2]), such as the iteger partitio, the rooted or labeled tree, etc. Perhaps most oteworthy is that two summatios ofte occur i the rig of symmetric fuctios [3]. For example, the elemetary symmetric fuctios ad the th power sum i idepedet variables x i are respectively defied by e x i x i2 x i applei <i 2< <i apple s x + x x. I these two expressios, the iteger is purposely suppressed for clarity. Recall that Girard s famous formula states s m m ( ) ( m )! e! 2!! e2 2 e. (7.) (m,) I [4, Sectio I, Chapter I, Sectios 5 ad 6], MacMaho geeralized this formula by establishig m s m t m e t 2e 2 t 2 + 3e 3 t 3 + e t + e 2 t 2 e 3 t 3 +. (7.2) This clearly states that as a reciprocal of (7.), each e ca be expressed i terms of s m e m (m,) ( ) m++2+ +! 2!! s 2 s2 2 s. (7.3) Aother pair of reciprocal relatios for e m ad s m via the use of determiats ca be foud i [3, p.28, Example 8]. Thus, it is iterestig to study such reciprocal relatios as Theorem 5.2 ad Corollary 5.3 i full geerality i the rig of symmetric fuctios. This will be discussed i our forthcomig paper. Acowledgmets. The authors would lie to tha the aoymous referee ad Ruizhog Wei for their helpful suggestios which improved the readability of this paper. The revisio of this paper was doe whe the secod author was visitig the Departmet of Computer Sciece, Leahead Uiversity, which was partially supported by NSERC grat This research was also supported by the Natioal Natural Sciece Foudatio of Chia uder Grat No

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