Journal of Mathematical Analysis and Applications

Transcription

1 J. Math. Aal. Appl ) Cotets lsts avalable at SceceDrect Joural of Mathematcal Aalyss ad Applcatos Asymptotc behavor of termedate pots the dfferetal mea value theorem of dvded dffereces wth repettos Am Xu a,, Feg Cu b,zhcheghu c a Isttute of Mathematcs, Zheag Wal Uversty, gbo 3500, Cha b College of Statstcs ad Mathematcs, Zheag Gogshag Uversty, Hagzhou 3008, Cha c Departmet of Mathematcs, Zheag Uversty, Hagzhou 30027, Cha artcle fo abstract Artcle hstory: Receved 8 Jue 2009 Avalable ole 4 ovember 2009 Submtted by W.L. Wedlad Keywords: Asymptotc behavor Bell polyomal Cycle dex of symmetrc group Dvded dfferece Mea value theorem By the explct represetato for the dvded dffereces wth repettos volvg cycle dex of symmetrc groups, ths paper deals wth the asymptotc behavor of the termedate pot of the mea value theorem of dvded dffereces wth repettos. Our results geeralze some recet terestg results such as the asymptotc expasos of the mmedate pots of the Taylor formula ad the dfferetal mea value theorem of dvded dffereces wth dstct pots Elsever Ic. All rghts reserved.. Itroducto Let f be a fucto whose values are kow at a set of pots x 0, x,...,x. If these pots are dstct, the dvded dffereces of f ca be defed the followg recursve form f [x 0 ] f x 0 ), f [x 0, x,...,x ] f [x 0, x,...,x ] f [x, x 2,...,x ]. x 0 x It s well kow that the dfferetal mea value theorem of dvded dffereces states that for f C, there exsts a termedate pot ξ such that f [x 0, x,...,x ] f ) ξ), ξ m{x 0, x,...,x, max{x 0, x,...,x )..)! Recetly, the asymptotc behavor of the dfferetal mea value of dvded dffereces s also pad more ad more atteto, ad some terestg results have bee obtaed [,2]. Let us cosder the followg Taylor formula f x) 0 f ) a)! x a) f ) ξ ) x a),.2)! where the termedate pot ξ ξ x) s strctly betwee a ad x. I the specal case, t becomes the Lagrage mea value theorem. There was some terest the asymptotc behavor of the termedate pot ξ.2) ad ts specal * Correspodg author. E-mal addresses: xuam009@yahoo.com.c A. Xu), fcu@zgsu.edu.c F. Cu), huzhcheg986@63.com Z. Hu) X/$ see frot matter 2009 Elsever Ic. All rghts reserved. do:0.06/.maa

2 A. Xu et al. / J. Math. Aal. Appl ) cases whe the legth of the terval volved shrks to zero. A.G. Azpeta [3] proved that, f the dervatve f p) exsts o I ad s cotuous at a wth f ) a) 0 < p) ad f p) a) 0, the lm x a ξ a)/x a) ) p /p. Ths result was geeralzed by U. Abel [4] to ξ a k c kx a) k /k!. It s easy to observe that the Lagrage mea value theorem s a specal case of the Taylor formula ad the mea value theorem of dvded dffereces ca be vewed as aother type of geeralzato of the Lagrage mea value theorem. More geerally, for sutably dfferetable f we allow some of the pots to coalesce, whch case certa dervatves are volved. By extedg the defto gve by de Boor [5] for f [x 0, x,...,x ] the case of dstct argumets, we have a smlar formula for x 0 x x as follows f [x 0, x,...,x ] f [x 0,x,...,x ] f [x,x 2,...,x ] x 0 x f x x 0, f ) x 0 )! f x x 0. Let t 0, t,...,t be dstct ad {t 0,...,t {{ 0, t,...,t {{ p 0 p,...,t,...,t {{ be a set of pots wth repettos. Deote by f [ ] p the dvded dfferece of f at the pots t 0, t,...,t whe repettos are permtted the argumets. By [5], the dfferetal mea value theorem also holds, amely, f ) η),! where η s strctly betwee m{t 0, t,...,t ad max{t 0, t,...,t, ad p 0 p p. Ths mea value theorem cotas Eqs..) ad.2) as specal cases. For other types of mea value theorems, the reader s referred to [6]. Ths paper deals wth the asymptotc behavor of the termedate pot of the mea value theorem of dvded dffereces wth repettos. Our results cota all of the prevous results as specal cases. 2. Ma results We beg ths secto wth some otatos. Let e ν t) t ν, ν 0, ad Ω t) k0, t t k ) p k, S l t) k0, p k t k t) l wth l, 0. Recall that the cycle dex of symmetrc group Z x k ) Z x, x 2,...,x ) a!) a a 2!2) a 2 a!) a xa xa 2 2 xa 2.) a 2a 2 a s oe of the essetal tools eumeratve combatorcs [7]. Usg the cycle dex of symmetrc groups, Wag [8] gave a explct formula for the dvded dffereces of f wth repeated pots as follows..3) Lemma 2.. If f s smooth eough, the 0 p Ω t ) 0 Z p where Z p S ḷ t )) Z p S t ), S 2 t ),...,S p t )). Sḷ t ) ) f ) t ), 2.2)! I the specal case, let t 0 a, t x, p 0, p, p 2 p 0, the t follows from 2.2) that f ) a) f [a,...,a, x] {{ x a) f x) x a) ).! 0 Thus, by.3), wehave f [a,...,a, x] f {{ ) ξ )/!, whch s accorded wth.2). Ifweletp 0 p p, the.3) reduces to.). Let I [a, b] be a terval ad t 0, t,...,t [a, b]. Assume that t a m h, 0,,...,, h b a. Wthoutlossof geeralty, we assume that p 0 p p because of the explct expresso of the dvded dfferece wth repettos. Thus, whe h teds to zeros, we have the followg theorem.

3 360 A. Xu et al. / J. Math. Aal. Appl ) Theorem 2.. Let p, q be tegers ad p, q 0. Assume that f s a fucto admttg a eghborhood of the pot a of R a dervatve of order p q ad that f ) s cotuous at a. If f ) a) f p ) a) 0,ad f p) a) 0, the q c η k ηh) a k! hk o h q) h 0). k Iterpret that p 0, ad the coeffcets c k are gve by the recurrece formula ) [ /p p ) p Ω m ) Z p Sḷ m ) ) ] /p m p, k0 p k c k R k c,...,c k ) k,...,q) wth R k c,...,c k ) k )c k ) /p!b k, [ f ν ] k )B k, c,...,c k ) 2.3) ) /p!b, [ f ν ], 2.4) where B k, [x ν ]B k, x, x 2,...,x k ) deote the expoetal partal Bell polyomals the varables x, x 2,...ad ) p f p ) pk p ) a) k0 p f k Ω m ) Z p S ḷ m ))m p f p) ) a) Ω m ) Z p S ḷ m ))m p, ) p f p f ) a) p f p) a) k0 pk p k p Proof. Sce t a m h, 0,,...,, thefrom2.2) t follows that 0 p Ω m ) 0 0,,...,q). 2.5) h Z p By vrtue of the Taylor formula, we have Sḷ m ) ) f ) a m h). 2.6)! f ) f ν) a) a m h) ν )! m h) ν o h ) h 0). 2.7) ν Substtutg 2.7) to the rght-had sde of Eq. 2.6) yelds k0 p k Ω m ) Z p S ḷ m ))! For coveece, let A,,ν Ω m ) Z p S ḷ m ))m ν,the p k f ν) ) a) ν ν!h ν A,,ν k0 p k ν { pk k0 ν νp k p k νp k Rearragg the terms leads to p r r0 νp r f ν) a) ν!h ν f ν) a) ν!h ν νp [ r k0 p k k0 p k p k ν f ν) a) ν )! m h) ν h ν o h pq) h 0). f ν) ) a) ν ν!h ν A,,ν o h pq) h 0). ) ν A,,ν ν p r ) ] ν A,,ν r ) ν A,,ν o h pq) h 0).

4 A. Xu et al. / J. Math. Aal. Appl ) Let {m 0,...,m {{ 0,m,...,m,...,m {{,...,m.fromeq.2.2), foreach0 s we have {{ p 0 p [ e ν ] s k0 p k p ) ν A,,ν ν p s where p s ν p s. For ν p, there holds e ν [ ] k0 p k ν ) ) ν A,,ν, s { 0 0 ν, A,,ν ν. Thus, wth f ) a) f p ) a) 0 ad f p) a) 0wehave f ) a)! f ν) a) ν!h ν νp k0 p k O the other had, by.3) ad Taylor s formula t follows that ) ν A,,ν o h pq) h 0). f ) a)! f ν) a)!ν )! η a)ν o η a) pq) νp η a). Sce η a < h, we coclude that q f pν) a) p ν)! hpν ν0 q ν0 k0 p k ) p ν A,,pν f pν) a)!p ν)! η a)pν o h pq) h 0). The rest of the proof cocdes wth that of Theorem [4] see also the proof of Theorem [2]) ad we omt t. As metoed Secto, the Taylor formula s a specal case of the mea value theorem of dvded dffereces wth repettos. Therefore, let, m 0 0 ad m Theorem2., the the asymptotc behavor of the mmedate pot of the Taylor formula s gve by the followg corollary see also [4]). Corollary 2.. Uder the assumptos of Theorem 2., wehave q c k ξ ξh) a k! hk o h q) h 0). 2.8) k The coeffcets c k are gve by the recurrece formula ) /p p, c k R k c,...,c k ) k,...,q), 2.9) where R k c,...,c k ) s defed as Theorem 2.,ad ) p f p ) a) f f p) a), f p p ) f p ) a) f p) a) 0,,...,q). Aother specal case of the mea value theorem of dvded dffereces wth repettos s for p 0 p p. I ths case, all pots t are dstct as well as all m are dstct. The Theorem 2. gves the asymptotc behavor of the mmedate pot of the dfferetal mea value theorem see also [2]): Corollary 2.2. Uder the assumptos of Theorem 2., wehave q c k ξ ξh) a k! hk o h q) h 0). 2.0) k

5 362 A. Xu et al. / J. Math. Aal. Appl ) The coeffcets c k are gve by the recurrece formula ) /p p e p[m 0,m,...,m ], c k R k c,...,c k ) k,...,q), 2.) where R k c,...,c k ) s defed as Theorem 2., ad ) p f p ) a) e p [m 0,m,...,m ] f f p) a) e p [m 0,m,...,m ], f s gve by 2.5). Ackowledgmet The work s supported by the Educato Departmet of Zheag Provce of Cha Y ). Refereces [] R.C. Powers, T. Redel, P.K. Sahoo, Lmt propertes of dfferetal mea values, J. Math. Aal. Appl ) [2] U. Abel, M. Iva, The dfferetal mea value of dvded dffereces, J. Math. Aal. Appl ) [3] A.G. Azpeta, O the Lagrage remader of the Taylor formula, Amer. Math. Mothly ) [4] U. Abel, O the Lagrage remader of the Taylor formula, Amer. Math. Mothly 0 7) 2003) [5] C. de Boor, Dvded dffereces, Surv. Approx. Theory 2005) [6] T. Trf, Asymptotc behavor of termedate pots certa mea value theorems, J. Math. Iequal ) 5 6. [7] L. Comtet, Advaced Combatorcs, D. Redel Publshg Co., Dordrecht, 974. [8] X. Wag, O the Hermte terpolato, Sc. Cha Ser. A 50 ) 2007)