Baxter Algebras ad the Ubral Calculus arxv:ath/0407159v1 [ath.ra] 9 Jul 2004 L Guo Departet of Matheatcs ad Coputer Scece Rutgers Uversty at Newar Abstract We apply Baxter algebras to the study of the ubral calculus. We gve a characterzato of the ubral calculus ters of Baxter algebra. Ths characterzato leads to a atural geeralzato of the ubral calculus that clude the classcal ubral calculus a faly of -ubral calcul paraeterzed by the base rg. 1 Itroducto Baxter algebra ad the ubral calculus are two areas atheatcs that have terested Rota throughout hs lfe te ad whch he has ade proet cotrbutos. We wll show ths paper that these two areas are tately related. The ubral calculus s the study ad applcato of sequeces of polyoals of boal type ad other related sequeces. More precsely, a sequece {p x N} of polyoals C[x] s called a sequece of boal type f p x + y p xp y, y C, N. 0 Such sequeces have fascated atheatcas sce the 19th cetury ad clude ay of the ost well-ow sequeces, such as the oes aed after Abel, Beroull, Euler, Herte ad Mttag-Leffler. Eve though polyoals of boal type proved to be useful several areas of atheatcs, the foudato of ubral calculus was ot frly establshed for over a cetury sce ts frst troducto. Ths stuato chaged copletely 1964 whe G.-C. Rota[Ro1] dcated that the theory ca be rgorously forulated ters of the algebra of fuctoals defed o the polyoals, later ow as the ubral algebra. Rota s poeer wor was copleted the ext MSC: 16W99, 05A50 1
decade by Rota ad hs collaborators [RKO, RR, Ro]. Sce the, there have bee a uber of geeralzatos of the ubral calculus [Ro, Ch, Lo, Me, Ve]. Durg the sae perod of te whch Rota ebared o layg dow the foudato of the ubral calculus, he also started the algebrac study of Baxter algebra whch was frst troduced by Baxter coecto wth fluctuato theory probablty [Ba]. Fudaetal to the study of Baxter algebra are the portat wors of Rota [Ro2] ad Carter [Ca] that gave costructos of free Baxter algebras. By usg a geeralzato of shuffle product topology ad geoetry, the preset author ad W. Kegher gave aother costructo of free Baxter algebras [G-K1, G-K2]. Ths costructo s appled to the further study of free Baxter algebras [Gu1, Gu2, AGKO]. Our frst purpose ths paper s to gve a characterzato of ubral calculus ters of free Baxter algebras. We show that the ubral algebra s the free Baxter algebra of weght zero o the epty set. We also characterze the polyoal sequeces studed the ubral calculus ters of operatos free Baxter algebras. The secod purpose of ths paper s to use the free Baxter algebra forulato of the ubral calculus we have obtaed to gve a geeralzato of the ubral calculus, the -ubral calculus, for each costat the base rg C. The ubral calculus of Rota s the specal case whe 0. For splcty, we oly cosder sequeces of boal type ths paper. The study of other sequeces the ubral calculus, such as Sheffer sequeces ad cross sequeces, ca be slarly geeralzed to our settg. We hope to explore possble roles played by Baxter algebras other geeralzatos of the ubral calculus. We also pla to gve a forulato of the ubral calculus ters of coalgebras wth operators by cobg the approach ths paper ad the coalgebra approach [RR, NS]. These proects wll be carred out subsequet papers. The layout of ths paper s as follows. I secto 2, we revew the ubral calculus ad gve a characterzato of ubral calculus ters of Baxter algebra. I secto 3, we defe sequeces of -boal type ad forulate the basc theory of the -ubral calculus that geeralzes the classc theory of Rota. I secto 4, uder the assupto that C s a Q-algebra, we gve a explct costructo of sequeces of -boal type. We also study the relato betwee the -boal sequeces ad the classc boal sequeces. 2 The ubral calculus ad ubral algebra 2.1 Bacgroud o the ubral calculus We frst recall soe bacgroud o the ubral calculus. See [RKO, Ro] for ore detals. Let N be the set of o-egatve tegers. Let C[x] be the C-algebra of polyoals wth coeffcets C. The a obects to study the classcal ubral calculus 2
are specal sequeces of polyoals called sequeces of boal type ad Sheffer sequeces. Defto 2.1 1. A sequece {p x N} of polyoals C[x] s called a sequece of boal type f p x + y p xp y, y C, N. 0 2. Gve a sequece of boal type {p x}, a sequece {s x N} of polyoals C[x] s called a Sheffer sequece relatve to {p x} f s x + y p xs y, y C, N. 0 I order to descrbe these sequeces, Rota ad hs collaborators studed the dual of the C-odule C[x] ad edowed the dual wth a C-algebra structure, called the ubral algebra. It ca be defed as follows. Let {t N} be a sequece of sybols. Let F be the C-odule N Ct, where the addto ad scalar ultplcato are defed copoetwse. Defe a ultplcato o F by assgg + t t t +,, N. 1 Ths aes F to a C-algebra, wth t 0 beg the detty. The C-algebra F, together wth the bass {t } s called the ubral algebra. Whe C s a Q-algebra, t follows fro Eq 1 that t t 1!, N ad so F C[[t 1 ]] as a C-algebra. But we wat to ephasze the specal bass {t }. Oe the detfes F wth the dual C-odule of C[x] by tag {t } to be the dual bass of {x }. I other words, t s defed by t : C[x] C, x δ,,, N. Rota ad hs collaborators reoved the ystery of sequeces of boal type ad Sheffer sequeces by showg that they have a sple characterzato ters of the ubral algebra. Let f, 0 be a pseudo-bass of F. Ths eas that {f, 0} s learly depedet ad geerates F as a topology C-odule where the topology o F s defed by the fltrato { } F c t c 0,. 1 3
A pseudo-bass f, 0 s called a dvded power pseudo-bass f + f f f +,, 0. Much of the foudato for the ubral calculus ca be suarzed the followg theore. Theore 2.2 [RR] Let C be a Q-algebra. 1. A polyoal sequece {p x} s of boal type f ad oly f t s the dual bass of a dvded power pseudo-bass of F. 2. Ay dvded power pseudo-bass of C[[t]] s of the for f x f t for soe! f C[[t]] wth deg f 1.e., ft c t, c 1 0. Sheffer sequeces ca be slarly descrbed. 2.2 Baxter algebras We wll gve a characterzato of the ubral calculus ters of Baxter algebras. For ths purpose we recall soe deftos ad basc propertes o Baxter algebras. For further detals, see [G-K1]. 1 Defto 2.3 Let A be a C-algebra ad let be C. 1. A C-lear operator P : A A s called a Baxter operator of weght f PxPy PxPy + PyPx + Pxy, x, y A. 2. The par A, P, where A s a C-algebra ad P s a Baxter operator o A of weght, s called a Baxter C-algebra of weght. We ofte suppress fro the otatos whe there s o dager of cofuso. Defto 2.4 Let A be a C-algebra. A free Baxter algebra o A of weght s a weght Baxter algebra F A, P A together wth a C-algebra orphs A : A F A such that, for ay weght Baxter algebra R, P ad ay C-algebra orphs f : A R, there s a uque weght Baxter algebra orphs f : F A, P A R, P wth f A f. Free Baxter algebras were costructed [G-K1], geeralzg the wor of Carter [Ca] ad Rota [Ro2]. I the specal case whe A C, we have 4
Theore 2.5 [G-K1] Let U C be the drect product N Cu of the ra oe free C-odules Cu, N. 1. Wth the product defed by u u 0 U C becoes a C-algebra. 2. The operator + u +,, N, P C : U C U C, u u +1, N s a Baxter operator or weght o the C-algebra U C. Further, the par U C, P C s the free coplete Baxter algebra o C. Proposto 2.6 Fx C. Let {v } be a pseudo-bass of U C. the followg stateets are equvalet. 1. v v 0 + v +,, N. 2. The operator P v : U C U C, v v +1, N s a Baxter operator or weght o the C-algebra U C. Further, the par U C, P v s the free coplete Baxter algebra o C. Proof: The proposto follows edately fro the defto. Defto 2.7 Let {v } be a pseudo-bass of U C. If ay of the equvalet codtos Proposto 2.6 s true, we call {v } a -dvded power pseudo-bass of U C. We ca ow gve a characterzato of the ubral algebra ad polyoal sequeces of boal type. Theore 2.8 C. 1. The ubral algebra s the free Baxter algebra of weght zero o 2. Let {p x} N be a sequece of polyoals C[x]. The {p x} s of boal type f ad oly f t s the dual bass of a dvded pseudo-bass of U C. 5
3 The -ubral calculus We ow develop the theory of the -ubral calculus. We costruct the -ubral algebra ad establsh the relato betwee the -ubral calculus ad -ubral algebra. I the specal case whe 0, we have the theory started by Rota o the ubral calculus. 3.1 Deftos I vew of Theore 2.8, we wll study the followg geeralzato of the ubral calculus. I order to get terestg exaples, we wll wor wth C[[x]] stead of C[x]. Defto 3.1 A sequece {p x N} of power seres C[[x]] s a sequece of -boal type f p x + y p + xp y, y C, N. 0 0 Whe 0, we recover the sequeces of boal type. We also ote the followg syetrc property. Lea 3.2 A sequece {p x N} of power seres C[[x]] s a sequece of -boal type f ad oly f p x + y p xp p+ y, y C, N. Proof: We have 0 0 0 0 0 0 0 p + xp y + 0 p + xp y + 0 p xp y use substtuto +, + + p xp y 0 0 0 + 0 + + 6 p xp y p + xp y
0 l l l p xp +l y usg substtuto l +, + l l p xp +l y 0 l0 l0 l 0 l l p xp +l y. Ths proves the lea. Defto 3.3 Fx a C. The algebra U C Cu, together wth the -dvded power pseudo-bass {u }, s called the -ubral algebra. Fx a pseudo-bass q {q x} of C[[x]] of -boal type see 4 for the exstece of of such sequeces. Let C<q> be the subodule of C[[x]] geerated by the pseudobass q. As the case of 0, we detfy U C wth the dual C-odule of C<q> by tag {u } to be the dual bass of {q x}. Thus each eleet of U C ca be regarded as a fuctoal o C<q>. We deote < > for the resultg parg U C C<q> C. It s characterzed by < u q > δ,, N. 2 Lea 3.4 The parg < > s pseudo-perfect the sese that 1. for a fxed u U C, f < u fx > 0, fx C<q>, the u 0, ad 2. for a fxed fx C<q>, f < u fx > 0, u U C, the fx 0. Proof: It follows fro Eq.2 ad the fact that {u } ad {q } are pseudo-bases. 3.2 Basc propertes The followg eleetary propertes ca be proved the sae way as the classcal case. Proposto 3.5 Let {p x} C<q> be a sequece of -boal type. Let {v } U C be the dual bass of {p x}. 7
1. The Expaso Theore For ay u U C, u < u p x > v. 0 2. The Polyoal Expaso Theore For ay px C<q>, we have px < v px > p x. 0 Note that U C acts o tself o the rght, ag U C a rght U C-odule. Ths U C-odule structure, through the parg < >, aes C<q> to a left U C- odule. More precsely, for u U C ad f C<q>, the eleet uf C<q> s characterzed by < v uf > < vu f >, v U C. 3 Proposto 3.6 Wth the otato above, we have u q x 0 Proof: For each N, we have q + x,, N. < u u s > < u u s > < + u + s > 0 0 0 0 0 + + + + δ +, δ, + δ, + + δ, + > 0 the defto of δ, + + + < u s + > Eq.2 0 + 8
0 0 < u 0 < u s + > + < u s + > > s + >. Now the proposto follows fro Lea 3.4. + 0 Proposto 3.7 Let q {q x} be the fxed sequece of -boal type. For ay u, v U C, we have < uv q x > 0 0 Proof: By Proposto 3.5, we have ad The we have uv 0 0 0 0 u v < u q + x > < v q x >. 4 < u q x > u, 0 < v q x > u. 0 < u q x > < v q x > u u < u q x > < v q x > 0 Proposto 2.6 < u q x > < v q x > 0 0 replacg by < u q x > < v q x > 0 0 exchagg the secod ad the thrd su u u u 9
Sce 0 < u q x > < v q x > exchagg the frst ad the secod su. 0 for > + ad 0 for >, we have 2 < u q x > < v q x > t+ t t0 t t < u q x > < v q x > < u q x > < v q t+ x > replacg by t + t < u q x > < v q t+ x > t0 0 < t t t 0 ad > 0. u Thus uv 0 t0 0 t t < u q x > < v q t+ x > u. Applyg ths to q x ad usg Eq. 2, we have < uv q x > t < u q x > < v q +t x >. t0 0 Exchagg u ad v the equato, we have t < uv q x > < vu q x > t < v q x > < u q +t x > t0 t0 0 t 0 Ths proves the proposto. t t < u q +t x > < v q x >. 10
Proposto 3.8 Let {p x} C<q> be a sequece of -boal type ad let u ad v be U C. The < uv p x > 0 0 < u p + x > < v p x >. Proof: We follow the case whe 0 [RR]. Let C[[x, y]] be the C-odule of power seres the varables x ad y. Sce C[[x, y]] C[[x]] C C[[y]] ad q x s a pseudo-bass of C[[x]], eleets of the for q xq y,, N for a pseudo-bass of C[[x, y]]. Thus ay eleet px, y of C[[x, y]] ca be expressed uquely the for px, y, c, q xq y, c, C. For u U C, defe u x px, y, c, < u q x > q y ad u y px, y, c, q x< u q y >. The by Proposto 3.7 ad the fact that q x s a -boal sequece, we have < uv q x > 0 0 u x v y q x + y. < u q + x > < v q x > Sce q x s a bass of C<q>, by the C-learty of the aps < uv >, u x ad v y, we have < uv px > u x v y px + y for ay px C<q>. Sce p x s of -boal type, we obta < uv p x > u x v y p x + y u x v y 0 0 0 0 p + xp y < u p + x > < v p x >. 11
Let c C. Defe the shft operator E c o C<q> by E c fx fx + c. A operator L o C<q> s called shft varat f E c L LE c, c C. Proposto 3.9 The eleets U C, regarded as operators o C<q>, are shft varat. Proof: We oly eed to chec that u are shft varat whe appled to q x. The sce q x s a bass of C<q>, u s shft varat o C<q>. Sce u s a pseudo-bass of U C, we see further that every eleet U C s shft varat o C<q>, as s eeded. Fx, 0. We have Also E c u q x s s0 0 s0 s s0 s s+l l0 + w 0 w0 s +s l0 w s0 lettg w l + s. u E c q x u q x + c u l l0 0 l0 l s0 0 0 l l+s l0 l s q +s + c s +s +s +s l q +s+l q c q +l xq c s s0 l +l +l l w s s s q +l+s q c q +w xq c q +l +s q c q +l +s q c 12
+ w 0 w0 w s0 lettg w l + s + w w 0 w0 s0 +w s w s w s replacg s by w s. +s s w+s s s q +w q c q +w q c Thus we oly eed to prove w s0 s +s w s w s0 w s +s for,,, w 0. Wth the help of the Zelberger algorth[pwz], we ca verfy that both sdes of the equato satsfy the sae recursve relato + w + w 1 s + 2 + 3 w + 2 w + 2 3 w +2 w 2 4 + 2 4 + 5 w + 3 W w + 2 + w 2 W 2 F,,, w 0 for,, 0. Here W s the shft operator WF,,, w F,,, w + 1. we ca also easly verfy Eq.5 drectly for w 0, 1. Therefore Eq. 5 s verfed. Ths copletes the proof of Proposto 3.9. s 5 3.3 Sequeces of -boal type ad Baxter bases Now we are ready to state our a theore the theory of the -ubral calculus. Theore 3.10 Let {p x} N be a bass of C<q>. The followg stateets are equvalet. 1. The sequece {p x} s a sequece of -boal type. 2. The sequece {p x} s the dual bass of a -dvded power pseudo-bass of U C. 3. For all u ad v U C, < uv p x > 0 0 < u p + x > < v p x >. Rear 3.1 As the case whe 0, the thrd stateet the theore has the followg terpretato ters of coalgebra. 13
3 The C-lear ap q x p x, N, defes a autoorphs of the C- coalgebra C<q>. Here the coproduct s defed by frst assgg q x 0 : C<q> C<q> C<q> 0 ad the exted to C<q> by C-learty. q + x q x, N Proof: 1 3 has bee proved Proposto 3.8. 3 2: Let v be the dual bass of p x U C. The we have < v v p x > 0 0 0 0 0 0 + < v p + x > < v p x > δ,+ δ, δ,+ δ, 0 for δ +, +. O the other had, < 0 + 0 0 + + + v + p x > + δ +, δ +,. 14
So by Lea 3.4, v v 0 + v + ad v s a -dvded power pseudo-bass of U C. 2 1 Let p x be the dual bass of a -dvded pseudo-bass v of U C. Let y def be a varable ad cosder the C-algebra C 1 C[[y]]. Regardg p x as eleets C 1 [[x]], the sequece p x s stll of -boal type. Also, let U C 1 N C 1 u be the C 1 -algebra obtaed fro U C by scalar exteso. The v s also a -dvded power pseudo-bass of U C 1. Further, the C-lear perfect parg 2 exteds to a C 1 -lear perfect parg betwee C 1 < q > ad U C 1. Uder ths parg, {p x} s stll the dual bass of {v }. Note that all prevous results apples whe C s replaced by C 1 sce we dd ot put ay restrctos o C. Wth ths d, we have p x + y < u p x + y > p x Lea 3.5 0 < u E y p x > p x defto of E y 0 < u E y p x > p x Eq.3 0 < E y u p x > p x Proposto 3.9 0 < E y u p x > p x Eq. 3 0 < E y 0 0 0 0 0 0 0 0 p + x > p x Proposto 3.6 p + yp x defto of E y p + yp x p xp + y > 0, > exchagg the sus. 0 15
4 -boal sequeces C[[x]] I ths secto we study -boal sequeces C<q> whe C s a Q-algebra C. We show that U C s soorphc to C[[t]] regardless of. We further show that the parg the -ubral calculus ad the parg the classcal ubral calculus whe 0 are copatble. The stuato s qute dfferet whe C s ot a Q-algebra ad wll be studed a subsequet paper. 4.1 The -ubral algebra ad -dvded powers We frst descrbe the -ubral algebra U C. Defto 4.1 A power seres ft C[[t]] s call delta f deg f 1, that s, ft 1 a t wth a 1 0. Proposto 4.2 1. Let ft be a delta seres. The τ ft def ftft ft 1, 0! for a -dvded power pseudo-bass for C[[t]]. 2. The ap C[[t]] U C, τ t tt t 1! detfes C[[t]] wth the -ubral algebra U C. t, 0 Proof: 1. Sce ft s delta, we have deg τ f. So τ ft for a pseudo-bass for C[[t]]. Thus we oly eed to show that the pseudo-bass s a -dvded power seres. For ths we prove by ducto o N that, for ay N, τ ftτ ft 0 + τ + ft. 6 The equato clearly holds for 0. Assue that equato 6 holds for a N ad all N. The for ay N, we have +1 0 +1 ++1 +1 0 +1 ++1 +1 τ ++1 ft τ ++1 ft 16
+ +1 0 0 + ++1 +1 1 +1 ++1 +1 +1 + +1 0 + 0 + + 0 + 0 ft τ ftτ ft + 1 τ +1 ftτ ft. τ ++1 ft τ ++1 ft +1 τ + ft ft + τ + ft + 1 τ + ft + 1 ft + τ + ft + 1 + 1 Ths copletes the ducto. 2. The stateet s clear sce {τ, t} s a weght -dvded power pseudo-bass. We ext costruct a sequece of -boal type. For a gve C, we use e x ex 1 1 x to deote the seres. Whe 0, we have e x x.! 1 Proposto 4.3 Let C be a Q-algebra. {e x} s a -boal pseudo-bass for C[[x]]. Proof: We prove by ducto the equato e x + c e + xe c, c C, N. 7 0 0 Clearly Eq. 7 holds for 0. It s also easy to verfy the equato e x + c e x + e c + e xe c whch s Eq. 7 whe 1. Usg ths ad the ducto hypothess, we get e +1 x + c ex + c 0 0 e + xe c 17
0 0 +1 + 1 +1 0 + 1 + +1 0 0 0 +1 1 + 1 1 [ e +1+ xe c + e+ [ e +1 x + + +1 + +1 +1 0 +1 +1 0 0 1 e +1 +1 +1 e +1+ 1 xe c + e +2+ e c e +1 xe c 1 + 1 0 xe +1 +1 1 0 1 ] +1 e xe+1 c + +1 xe c + Thus Eq. 7 s proved. +1 1 +1 0 e 2+2 xe c e +1+ xe c. +1 + 1 1 1 c + e+1+ 1 1 1 1 e +1 c ] e +1+ xe c e +1+ xe c e 2+2 xe c e +1+ xe c Thus we ca use the -boal pseudo-bass q {e x} of C[[x]] ad the - tt a t a 1 dvded power pseudo-bass τ t, 0, of C[[t]] to defe! the parg < > : U C C<q> C as s descrbed Eq.2. Defto 4.4 Let ft C[[t]] be a delta seres. The dual bass of {τ ft} C<q> s called the assocated sequece for ft. We show that every -boal sequece s assocated to a delta power seres, as the case whe 0. Theore 4.5 Let C be a Q-algebra. Let {s x} be a bass of C<q>. The {s x} s a sequece of -boal type f ad oly f {s x} s the assocated sequece of a delta seres ft C[[t]]. 18
Proof: If {s x} s the assocated sequece of a delta seres ft, the by Theore 3.10 ad Proposto 4.2, {s x} s a -boal sequece. Coversely, let {s x} be a -boal sequece ad let {f t} be the dual bass of {s x} C[[t]]. It follows fro Theore 3.10 that {f t} s a -dvded power seres. We oly eed to show 1. f 1 t s a delta seres, ad 2. f t τ f 1 t, 0. We frst prove 2 by ducto. Sce {f t} s a -dvded power bass, we have f tf t 0 + f + t. Tag 0, we have f 0 tf 0 t f 0 t. Thus f 0 t 1. Assug f t τ f 1 t, fro f tf 1 t f +1 t + f t we have f +1 t f tf 1 t τ f 1 tf 1 t τ +1 f 1 t. Ths proves 2. The 1 follows sce f f 1 t s ot a delta seres, the {τ ft} caot be a pseudo-bass of C[[t]]. 4.2 Copatblty of -calculus We ow show that, for ay gve C, the -ubral calculus s copatble wth the classcal ubral calculus whe 0. The copatblty s the followg sese. Sce {x } ad q {q } are pseudo-bass of C[[x]], the pargs < > 0 : C[t] C[x] C ad exted to pargs ad < > : C[t] C<q> C [ ] 0 : C[t] C[[x]] C [ ] : C[t] C[[x]] C. Defto 4.6 We say that < > 0 ad < > are copatble f [ ] 0 [ ]. 19
Theore 4.7 The pargs < > 0 : C[t] C[[x]] C ad are copatble. < > : C[t] C[[x]] C Ths theore has the followg edate cosequece. Corollary 4.8 Let C be a Q-algebra. A pseudo-bass {p x} of C[[x]] s of -boal type f ad oly f t s the dual bass of a -dvded power pseudo-bass of C[[t]] uder the parg < > 0. Proof of Theore 4.7: Sce {τ, t} s a bass of C[t] ad {e x} s a pseudobass of C[[x]], we oly eed to prove [τ, t e x] 0 δ,,, 0. 8 We wll prove ths equato by ducto o. Whe 0, τ, t 1. Sce deg e x 1, we have [τ,0 t e x] 0 δ 0,. Assue that Eq. 8 holds for a 0 ad all 0. The we have [τ,+1 t e x] 0 t [ + 1 τ,t + 1 τ,t e x] 0 1 + 1 [tτ,t e x] 0 + 1 [τ,t e x] 0 1 + 1 [tτ,t e x] 0 + 1 δ,. By [Ro, Theore 2.2.5], we have Sce we have [tτ, t e x] 0 [τ, t d dx e x] 0. d dx e x e 1 xe x e x + e 1 x, [τ,+1 t e x] 0 1 + 1 [τ,t e x + e 1 x] 0 + 1 δ, 20
Ths copletes the ducto. + 1 δ, + + 1 δ, 1 + 1 δ, + 1 δ, + + 1 δ, 1 δ +1,. Acowledgeet: The author thas Wlla Kegher for helpful dscussos. Refereces [AGKO] G.E. Adrew, L. Guo, W. Kegher ad K. Oo, Baxter algebras ad Hopf algebras, preprt. [Ba] G. Baxter, A aalytc proble whose soluto follows fro a sple algebrac detty, Pacfc J. Math. 10 1960, 731-742. [Ca] P. Carter, O the structure of free Baxter algebras, Adv. Math. 9 1972, 253-265. [Ch] W. Y. C. Che, The theory of copostoals, Dscrete Math. 122 1993, 59 87. [Gu1] L. Guo, Propertes of free Baxter algebras, Adv. Math., 151 2000, 346-374. [Gu2] [G-K1] [G-K2] [Lo] [Me] [NS] L. Guo, Ascedg cha codtos free Baxter algebras, preprt. L. Guo ad W. Kegher, Baxter algebras ad shuffle products, Adv. Math., 150 2000, 117-149. L. Guo ad W. Kegher, O free Baxter algebras: copletos ad teror costructos, Adv. Math., 151 2000, 101-127. D. E. Loeb, A geeralzato of the boal coeffcets. Dscrete Math. 105 1992, 143 156. M. A. Médez, The ubral calculus of syetrc fuctos. Adv. Math. 124 1996, 207 271. W. Nchols ad M. Sweedler, Hopf algebras ad cobatorcs, : Ubral Calculus ad Hopf Algebras, Coteporary Matheatcs 6, Aer. Math. Soc. 1982, 49-84. [PWZ] M. Petovsě, H. Wlf, ad D. Zelberger, A B, A. K. Peters, 1996. 21
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