NOTE. Warren P. Johnson* The Pennsylvania State University, University Park, Pennsylvania Communicated by the Managing Editors

Transcription

1 joural of combiatorial theory, Series A 76, (1996) article o NOTE A q-aalogue of Faa di Bruo's Formula Warre P. Johso* The Pesylvaia State Uiversity, Uiversity Park, Pesylvaia Commuicated by the Maagig Editors Received December 28, 1995 Some years ago Gessel ([Ge]) itroduced a q-aalogue of fuctioal compositio that was strog eough to support a q-aalogue of the chai rule. I this ote we show that Gessel's q-compositio is eve strog eough to support a q-aalogue of Faa di Bruo's formula for the th derivative of a composite fuctio. q-aalogues of the Bell polyomials arise aturally i this cotext Academic Press, Ic. I. ELEMENTARY q-differential CALCULUS We will use a well-kow q-aalogue of the derivative operator that goes back at least to Jackso [Ja]. Related operators were used earlier by Heie ad by Rogers; see also [A2], [GR]. We defie D q f(x)= f(x)&f(qx) x(1&q) ad we call this the q-derivative of f(x). The otatio f $(x) will mea throughout this paper the q-derivative of f(x). I this sectio we develop some properties of D q that we will require. First we itroduce the requisite otatio. We put []=(1&q )(1&q)=1+q+}}}+q &1 ; thus we may write D q x =[]x &1. Next, we defie the q-factorial! q =[1][2] } } } [], where 0! q =1. The the q-multiomial coefficiet may be defied by _ k 1, k 2,..., k m& =! q (k 1 )! q (k 2 )! q }}}(k m )! q * Curret address: Beloit College, Beloit, Wiscosi johsowsci.beloit. edu Copyright 1996 by Academic Press, Ic. All rights of reproductio i ay form reserved.

2 306 NOTE where k 1 +}}}+k m =. Mostly we will be usig the q-biomial coefficiet, which we will write as [ k ]. The q-umbers have a trivial but very useful subtractio property: []&[k]=q k [&k] (1.1) This immediately implies the well-kow recurrece relatio for the q-biomial coefficiets The alterate form _ +1 k & = _ k&1& _ k& +qk (1.2) _ +1 k & = _ k& _ k&1& +q&k+1 (1.3) follows o replacig k by &k+1. Usig (1.2) ad iductio it is easy to prove Schu tzeberger's ocommutative q-biomial theorem ([Sc]) (x+y) = : k=0_ k& xk y &k where yx=qxy. There is a immediate geeralizatio to m variables (x 1 +}}}+x m ) = : k 1 +}}}+k m =_ k 1,k 2,..,k m& xk (1.4) 1 1 }}}xk m m (1.5) where x j x i =qx i x j if ad oly if j>i. This i tur implies the recurrece relatio for the q-multiomial coefficiets _ +1 k 1,..., k m& = _ k 1 &1, k 2,..., k m& 1_ +qk k 1, k 2 &1,..., k m& +q k 1+k 2_ k 1, k 2, k 3 &1,..., k m& +}}}+q k 1+k 2 +}}}+k m&1_ (1.6) k 1,..., k m &1& where k 1 +}}}+k m =+1. With these facts i had, we tur to some basic properties of the q-derivative. It is easy to prove the q-product rule D q f(x) g(x)=f(x) g$(x)+f$(x) g(qx) (1.7)

3 NOTE 307 This immediately exteds by iductio to m fuctios m D q f 1 (x) f 2 (x)}}}f m (x)= : k&1 k=1\ ` j=1 f j (x) + f$ k(x) \ m ` j=k+1 f j (qx) + (1.8) There is a weak form of the chai rule for the q-derivative, amely that if a is idepedet of x, the D q f(ax)=af $(ax) (1.9) We will obtai a stroger form i the ext sectio. (1.9), i cocert with (1.7) ad (1.2), ca be used to prove the q-leibiz rule D q f(x) g(x)= : k=0_ k& f (k) (x) g (&k) (q k x) (1.10) A commo geeralizatio of (1.8) ad (1.10) ca be proved i the same way, usig (1.6) i place of (1.2): D q f 1(x) f 2 (x)}}}f m (x) = : k 1 +}}}+k m =_ k 1,..., k m& f (k 1) 1 (x) f (k 2) 2 (q k 1 x)}}}f (k) (q k 1+}}}+k m&1 x) II. THE q-chain RULE To have a really adequate chai rule for the q-derivative, as was realized by Gessel ([Ge]), we eed a differet otio of fuctioal compositio. We give a slight modificatio of Gessel's costructio: Cosider a fuctio f(x) of the form f(x)= : =1 f x! q (Note that f(0)=0.) Defie the 0th symbolic power of f by f [0] (x)=1, ad for positive iteger k defie the kth symbolic power of f iductively by D q f [k] (x)=[k] f [k&1] (x) f$(x) (2.1) (2.1) gives f [k] (x) up to a additive costat, which is determied by f [k] (0)=0 for k1. A alterative defiitio, agai iductive, ca be give by usig the q-itegral ([GR]). For k1, we have f [k] (x)=x(1&q k ) : =0 q f [k&1] (xq ) f $(xq ) (2.2)

4 308 NOTE Note that whe k=1 we have D q f [1] (x)=f$(x), which implies that f [1] (x)=f(x) sice both quatities have the same q-derivative, ad both are zero whe x=0. Note also that D q x [k] =[k]x [k&1], so by iductio we have x [k] =x k sice x [1] =x. More geerally, however, we have (x m ) [k] = k! q x mk k! q m Furthermore, if t does ot deped o x, the (tf(x)) [k] =t k f [k] (x) (2.3) as oe ca see by observig that it holds if k=0, 1, ad showig by iductio o k that both sides have the same q-derivative (with respect to x). Usig (2.1) ad iductio we obtai yet aother expressio which we could take as a defiitio of f [k] (x), amely f [k] (x)=k! q : =k x : b 1 +}}}+b k = b i 1 f b1 f b2 }}}f bk [b 1 ][b 1 +b 2 ]}}}[b 1 +}}}+b k ](b 1 &1)! q }}}(b k &1)! q (2.4) If g(x)= =0 g (x! q ), the the q-compositio of g with f is defied by g[ f ]:=: =0 g f []! q (2.5) The q-compositio of g(x) with the idetity fuctio is just g(x) sice x [] =x. Moreover, we have D q g[ f ]= : g =1 which is Gessel's q-aalogue of the chai rule. f [&1] (&1)! q f$=g$[ f ] f $ (2.6) III. PARTITIONS We will require a few facts about partitios, both of umbers ad of sets. A partitio of a positive iteger is a uordered sum of positive itegers equal to. Each summad is called a part. For example, the partitios of 4 are 4, 3+1, 2+2, 2+1+1, , with respectively 1, 2, 2, 3, 4 parts. The we have the Lemma. (1) [ k] is the geeratig fuctio for partitios ito at most k parts ot exceedig &k.

5 NOTE 309 (2) q k [ &1 k ] is the geeratig fuctio for partitios ito exactly k parts ot exceedig &k. These statemets mea that the coefficiet of q m i these expressios couts the umber of partitios of the umber m of the appropriate kid. They may be proved by recurrece usig (1.2); see [A1] for the details. A partitio of a set is a decompositio ito a disjoit uio of oempty subsets, which are called blocks. All the set partitios we cosider will be of the set [1, 2,..., ], which we deote by (), for some positive iteger. For example, [1, 5, 6], [7], [2, 4, 8], [3, 9] is a partitio of (9) with four blocks. We proceed to determie its weight (a cocept itroduced i [Jo]). Begi by crossig out the block with the largest elemet, i this case [3, 9]. Relabel the remaiig elemets with the itegers startig from 1, preservig the order. Here 1 is still 1 ad 2 is 2, but 4 is relabeled dow to 3, 5 to 4, 6 to 5, 7 to 6 ad 8 to 7. The ew partitio is [1, 4, 5], [6], [2, 3, 7] ad the total amout of relabelig was 5 uits. Now repeat the algorithm. [2, 3, 7] is crossed out, ad the remaiig elemets are relabeled as [1, 2, 3], [4], with 6 more uits of relabelig sice 4 is relabeled as 2, 5 as 3 ad 6 as 4. The algorithm is repeated util every block is crossed out, ad the total amout of relabelig is the weight of the partitio. I this case o further relabelig will be ecessary ad the weight is 5+6=11. A combiatorial iterpretatio of Gessel's q-aalogue of the expoetial formula may be based o this idea [Jo]. I fact, this follows from the above lemma; see [Ge] for the origial iterpretatio. IV. q-analogue OF FAA DI BRUNO'S FORMULA Sice Gessel's q-compositio gives us a q-aalogue of the chai rule, amely (2.6), oe ca hope for a q-aalogue of the iterated chai rule, that is, Faa di Bruo's formula for the th derivative of a composite fuctio ([Ch], [Ri], [Co]). I this sectio we obtai such a formula. For orietatio, let us look at D 3 q g[ f(x)]. There are five terms: g$[ f(x)] f$$$(x)+g"[ f(x)] f$(x) f "(qx)+qg"[ f(x)] f$(x) f "(qx) +g"[ f(x)] f"(x) f $(q 2 x)+g$$$[ f (x)] f$(x) f $(qx) f $(q 2 x) These terms correspod to the five partitios of (3), i the order [1, 2, 3]; [1], [2, 3]; [2], [1, 3]; [1, 2], [3]; [1], [2], [3]. The umber of blocks

6 310 NOTE i the partitio is the umber of derivatives o g, ad the block sizes determie the umbers of derivatives of f. The powers of q will be explaied presetly, but ote that the weight of [2], [1, 3] is 1 while the weights of the other four partitios of (3) are all 0. Let? be a partitio of () ito the blocks B 1, B 2,..., B k. As with the partitios of (3) above, we put the blocks i icreasig order of the maximal elemets. Suppose there are b i elemets i the block B i, for each i, ad deote the set of partitios of () by 6(). The we have Theorem (q-aalogue of Faa di Bruo's Formula, First Form). D q g[ f(x)]= :? # 6(){ q wt? g (k) [ f(x)] f (b1) (x) f (b2) (q b 1 x) _f (b3) (q b1+b2 x)}}}f (bk) (q x)=. (4.1) b1+b2+}}}+bk&1 The proof is by iductio o. We have verified the case =3 already, ad the lower-dimesioal cases are trivial. Observe that the expoet of q i the factor f (b i) (q b 1+}}}+b i&1 x) couts the umber of elemets cotaied i B 1 _ }}} _B i&1, i.e., i the blocks with a smaller maximal elemet tha that of B i. Thus, for example, the theorem claims that to the partitio of (9) there correspods the term [1, 5, 6], [7], [2, 4, 8], [3, 9] q 11 g (4) [ f(x)] f$$$(x) f $(q 3 x) f $$$(q 4 x) f "(q 7 x) of D 9 q g[ f(x)]. Assumig that (4.1) holds for, we show it holds for +1. Usig (1.8) i (4.1), we have D +1 q g[ f(x)] k = : q wt? g (k+1) [ f(x)] \`? # 6() + :? # 6(){g 1ik i=1 (k) [ f(x)] q wt?+ i&1 i&1 b l=1 l\` f (b i) (q b 1+}}}+b i&1 +1 x) + j=1 _f (bi+1) (q i&1 b l l=1 x) \ k ` j=i+1 j&1 l=1 f (b j) (q b l x) + = (4.2) j&1 f (bj) (q 1+ b l l=1 x) + where we also used (1.9). We may obtai all the partitios of (+1) from the partitios of () by relabelig the elemets 1, 2,..., as 2, 3,..., +1 i

7 NOTE 311 that order, ad the either addig the elemet 1 as a sigleto block, or addig it to oe of the relabeled blocks. Doig the former icreases the umber of blocks by oe, icreases the umber of elemets i the blocks with a smaller maximal elemet tha that of B i by oe for each of the origial blocks, ad does ot chage the weight. Therefore this class of partitios of (+1) correspods to the first summad i (4.2). I the latter case, suppose we have adjoied 1 to the relabeled block B i. For the B j with j>i, the umber of elemets i the blocks of smaller idex has icreased by oe, but this umber has ot chaged for the B j with j<i, ad of course the size of B i has icreased by oe. The effect o the weight is as follows. The blocks B j with j>i are crossed out oe by oe. Each elemet has a label oe larger tha it previously had both before ad after relabelig, so there is o et effect o the relabelig weight. Whe B i is crossed out, however, the remaiig elemets have to be relabeled dow oe more uit tha before. Thereafter, each elemet has the same label it would have had before the elemet 1 was adjoied, so there is o further effect o the weight. Thus the weight icreases by the umber of elemets i the blocks with idex less tha i, that is, by b 1 +}}}+b i&1. Therefore these partitios correspod to the secod summad i (4.2). Faa di Bruo's formula is ot usually stated i the form of a sum over set partitios, although it is most easily proved whe so stated. Oe ca the ask whether our q-aalogue ca be give without referece to set partitios. The followig lemma will allow us to give such a statemet. Lemma. Let? be a partitio of () ito the blocks B 1, B 2,..., B k, listed i icreasig order of their maximal elemets, with B i =b i,1ik.the : q wt?? #(B 1,..., B k ) = _b 1+b 2 +}}}+b k &1 b k &1 &_ b 1+b 2 +}}}+b k&1 &1 where the sum is over all such?. b k&1 &1 & }}} _ b 1+b 2 &1 b 2 &1 & Proof. To determie the weight of?, we begi by crossig out B k ad relabelig the elemets i the other blocks with [1, 2,..., b 1 +b 2 +}}}+ b k&1 ], preservig the order. Before relabelig, the largest elemet i these blocks does ot exceed b 1 +b 2 +}}}+b k &1, so the cotributio to the relabelig weight from ay oe elemet is o larger tha b k &1. For each?, the, the relabelig weight is a partitio ito at most b 1 +b 2 +}}}+b k&1

8 312 NOTE parts ot exceedig b k &1, ad we get all such partitios from summig over all?. By the lemma of sectio III, we therefore have :? #(B 1,..., B k ) 1+b q wt? 2 +}}}+b k &1 = _b : q b k &1 & wt?? #(B 1,..., B k&1 ) ad the lemma follows upo iteratio sice the weight of a sigle block is 0. Observe that we may rewrite the expressio i the lemma as (&1)! q [b 1 ][b 1 +b 2 ]}}}[b 1 +}}}+b k&1 ](b 1 &1)! q (b 2 &1)! q }}}(b k &1)! q ad that this expressio has (more or less) arise already i (2.4). From this poit of view, it therefore gives a atural q-aalogue of the Bell polyomial ([Ri], [Co]). That is, we may put B, k, q ( f 1, f 2,..., f &k+1 ) = : b 1 +}}}+b k = b i 1 ad rewrite (2.4) as (&1)! q f b1 f b2 }}} f bk [b 1 ][b 1 +b 2 ]}}}[b 1 +}}}+b k&1 ](b 1 &1)! q }}}(b k &1)! q (4.3) f [k] (x)=k! q : =k x! q B,k,q ( f 1, f 2,..., f &k+1 ) (4.4) Makig the abbreviatio f i, j := f (i) (q j x), we may ow state our q-faa di Bruo formula i the followig form: Theorem (q-aalogue of Faa di Bruo Formula, Secod Form). D q g[ f(x)]= : b 1 +}}}+b k = b i 1 g (k) [f(x)] B,k,q ( f b1,0,f b2,b 1,f b3,b 1 +b 2,...) We coclude this ote with a few properties of these q-bell polyomials. It follows from (4.4) ad results i [Jo] that { =B, k, q (1,1,...,1) k=q c q (,k)=b,k,q (0! q,1! q,..., (&k)! q )

9 NOTE 313 where the latter are the (sigless) q-stirlig umbers of the first kid that occur i the literature most ofte, e.g., i [Sa], ad the former are some q-stirlig umbers of the secod kid that were discussed i [Jo]. Recall the q-expoetial fuctio (4.4) ad (2.3) imply that e q [tf(x)]= : k0 e q (x)= : =0 x! q B,k,q ( f 1,..., f &k+1 ) x! q t k (4.5) If we take the q-derivative of (4.4) or of (4.5) ad equate coefficiets, we fid the recurrece B +1, k+1, q ( f 1,..., f &k+1 )= : l=k_ k& B l,k,q( f 1,..., f l&k+1 ) f &l+1 The welter of other recurrece relatios satisfied by the Bell polyomials (see, e.g., [Co]) do ot seem to go through i this settig, maily because e q [ f ] is ot really a expoetial fuctio. It is a easy cosequece of (1.4), however, that if yx=qxy ad all other parameters commute, the e q (x) is a hoest expoetial fuctio i the sese that e q ((x+y)t)=e q (xt) e q ( yt) If oe is willig to use power series with ocommutig coefficiets, therefore, oe ca defie some other q-bell polyomials with a richer recursive structure. But we shall ot take this topic up here. ACKNOWLEDGMENTS This work was begu i the author's Ph.D. thesis at the Uiversity of Wiscosi, ad cotiued at Pe State. I wish to thak both istitutios, ad especially Professors R. Askey ad G. Adrews, for their support ad kidess. REFERENCES [A1] G. E. Adrews, ``The Theory of Partitios,'' Ecyclopedia of Mathematics ad Its Applicatios, Vol. 2, AddisoWesley, Readig, MA, [A2] G. E. Adrews, O the foudatios of combiatorial theory. V. Euleria differetial operators, Studies i Appl. Math. 50 (1971), [Co] L. Comtet, ``Advaced Combiatorics,'' Reidel, Dordrecht, 1974.

10 314 NOTE [Ge] I. M. Gessel, A q-aalog of the expoetial formula, Discrete Math. 40 (1982), [GR] G. Gasper ad M. Rahma, ``Basic Hypergeometric Series,'' Ecyclopedia of Mathematics ad Its Applicatios, Vol. 35, Cambridge Uiv. Press, Cambridge, [Ja] F. H. Jackso, O q-fuctios ad a certai differece operator, Tras. Roy. Soc. Ediburgh 46 (1908), [Jo] W. P. Johso, Some applicatios of the q-expoetial formula, Discrete Math. 157 (1996), [Ri] J. Riorda, ``A Itroductio to Combiatorial Aalysis,'' Wiley, New York, [Sa] B. E. Saga, Cogruece properties of q-aalogs, Adv. i Math. 95 (1992), [Sc] M.-P. Schu tzeberger, Ue iterpre tatio de certaies solutios de l'e quatio foctioelle: F(x+y)=F(x) F(y), C. R. Acad. Sci. Paris 236 (1953),