Remarks on Faber Polynomials

Transcription

1 Iteratioal Mathematical Forum, 3, 008, o. 9, Remarks o Faber Polyomials Helee Airault LAMFA, UMR CNRS 640, Uiversité de Picardie Jules Vere INSSET, 48 rue Rasail, 000 Sait-Queti (Aise), Frace hairault@isset.u-icardie.fr Abstract With the differetial calculus o the Faber olyomials, we calculate the Faber olyomials for owers of iverse fuctios. We aly the same methods to obtai majoratio of the derivatives of the Faber olyomials of a uivalet fuctio of the class Σ. Mathematics Subject Classificatio: 30C50 Keywords: Faber olyomials, uivalet fuctios Itroductio Let g() = + b + b + + b +, the Faber olyomials ψ (t) of g() are give by g () g() t =+ ψ m (t) m () m= Let F k (b,b,.., b k )=ψ k (0), k, the ψ (t) =F (b t, b,.., b ). Let g() t =+ + = G (b t, b,,b ) () If h() =g( ), the h() = 0 G (b,b,,b ). We have G (b t, b,,b )= + ψ +(t) = F k+ (b t, b,,b k ) (3) k + b h() = 0 K (b,b,,b ) real (4) (b + b + + b + ) m = s 0 D m m+s (b,b, ) s+m m N (5)

2 450 H. Airault The D 0 =0if>0 ad D0 0 =,G = K K lim 0 = F ad K 0 =0if>0. We have (6) We deote K for K (b,b,.., b ) ad (G,G,..) for (G (b ),G (b,b ),..). With h() h() =h() +, we see that K satisfies the recurrece K + b K + b K + + b K + b = K + (7) We have, see [3], F (G,G,,G )= F (b,b,,b ), ad see [9], [8] ad [], K (G,G,,G )=K (b,b,,b ) (8) F (b,b,,b )= m= ( ) m m Dm (b,b,,b m+ ) (9) The derivatives of F (b,b,,b ) with resect to ay of the variables b j are give i [3]. With resect to b, k b k F (b,b,,b )=( ) k (k )! K k k (b,,b k ) (0) =( ) k (k )! K k k (G,,G k )=( ) k (k )! D k (,G,,G k ) Deote C m ( ) ( m +) =, the for,, q N, m! ( ) q C q = q C k k= (q )! (q k)!(k )! () The D m (b t, b,.., b m+ )= m Cm k ( )k D m k k (b,b,.., b m+ )t k for m<ad D = b. The Faber olyomials of the iverse fuctio Let g() as i () ad deote g () the iverse fuctio, (gog = Idetity), the, see [, (I..7)-(I..8)] ad [3, roositio 6.], g () = b K + ()

3 Remarks o Faber olyomials 45 [g (u)] = u + + K+ (b,b,,b ) (3) u + This exasio is valid for ay real umber with the covetio that whe + = 0, we relace the coefficiet + K+ by F, this is i accordace with (6). This shows that for ay R, K ( b, K, K 3,, K )= + K+ (b,b, ) (4) where i the left had side, K meas K (b,b,,b ). Makig 0asi (6), we deduce from (4), F ( b, K, K 3,, K )=K (b,b,,b ) (5) The Faber olyomials φ (t) i terms of the (b ), for the iverse fuctio g are well kow, they are the Taylor art i the Lauret exasio of g(), see for examle the roof of Lemma.3 i [8]. We give a differet roof of this fact: I (5), we have the value φ (0) at t = 0. We do a Taylor exasio of φ (t) =F ( b t, K (b,b ), K 3 (b,b,b 3 ), ) (6) Usig (4) at the last ste, we obtai φ (k) (0) = ( )k k! k! k F ( b b k, K(b,b ), K 3(b,b,b 3 ), ) (7) = k K k k ( b, K (b,b ), K 3 (b,b,b 3 ), )=K k (b,b, ) We obtai φ (t) =K (b,b,,b )+ K k(b,b,,b k )t k (8) k= which is the Taylor art of g(t). With the same method, Theorem. The Faber olyomials φ (t) of + [g ()] are give i terms of the (b ) by φ (t) = k( + ) K k(+) k (b,b,,b k ) t k (9) with the covetio ( + ) ( +)k K (+)k k = F k if =( +)k.

4 45 H. Airault Proof. We give the roof for =. The φ (t) = k K k with the covetio k K k k = F k if =k. With (3), k tk K( b, 3 K3 (b,b ),, + K+ (b,b,,b )) = + K+ (0) With (6), dividig by ad makig 0, φ (0) = F (b, 3 K3 (b,b ),.., + K+ (b,b,.., b )) = K (b,b,.., b ) The with (0), we aly Taylor formula to, φ (t) =F ( b t, 3 K3 (b,b ),, + K+ (b,b,,b )) Similarly, let f() = + b + ad the iverse f (), (fof = Id), the for ay real, see [, (I..6)-(I..9)], [f ()] = [ + with the covetio that we relace articular f () =+b + K (+) ] () K i () by F if =. I K () Theorem. Let f() = + b +, the Faber olyomials φ (t) of + [f ( )] are give i terms of the (b ) by φ (t) = k( + ) K +k(+) k (b,b,,b k )t k (3) Proof. With (), K( b q,.., + K (+) )= q q + K q (b,b,.., b ) if + q, otherwise we use (6). As i (6), we divide by q ad make q 0, F ( b,, + K (+) )= K (b,b,.., b ). We have φ (t) = F ( b t,.., + K (+) ). We calculate its coefficiets φk (0) with (0) k! For =, φ (t) = K k (b,b,,b k )t k ad similarly to (8), this case ca be obtaied directly sice the iverse of g() = f ( ) is g (u) = f( u A differet roof of (9) ad (3) would cosist i makig a chage of ). variables i Cauchy itegral formula. A successio of maiulatios o f() ad g() as raisig to the ower ad takig the iverse as i Theorems. ad. brigs always series with coefficiets i the class of olyomials K.

5 Remarks o Faber olyomials Majoratio of the derivatives of the Faber olyomials for g() = f( ) Theorem 3. Let f() = + b +. The th Faber olyomial of g() = + [f( )] is φ (t) =F (b,b,.., b )+ k= k K k k (b,b,.., b k )t k. I articular for =, see [9, (30)], let φ (t) be the th Faber olyomial of g() =[f( )], the φ (t) = F (b,b,,b )+ k= t k k Dk (,b,b,,b k ) (4) Proof. We have g() = + b + K + + K + +. Because of (), φ (t) =F (b t, K,,K ) (5) By Taylor s formula, φ (t) = φ (0) + φ (k) (0) k= t k! k. derivative is For k, the k th φ (k) (t) =( )k ( k F b k )(b t, K,.., K )=(k )!K k k (b t, K,.., K k ) Thus φ(k) (0) k! = k K k k (b,b,.., b k ). Whe =, we obtai (4) Our roof uses differetial calculus whereas the roofs i [9, 0] are combiatory. Remark that (4) is also a immediate cosequece of () sice (4) is the Taylor art of [f ( )]. See (8). Whe =, the derivatives φ (k) are give by φ(k) (t) = (k )! K k k (G t, G,,G k )= k s=0 k! s + k Ds+k t s (6) I [9], whe g() = f( ) ad f() = + b + b is uivalet i, the ositivity of the coefficiets of the olyomials D k ad De Brages theorem b + ermit to obtai a majoratio of the first derivative φ (t) for t, the Koebe fuctio is extremal. Because of (6), we ca aly the same method, see Theorem 3.3, to all the derivatives φ (k) (t) ad the Koebe fuctio is agai extremal. Let f() = ( ) be the Koebe fuctio ad g() = f( ) = +. We deote x (t) ad x (t) the two roots of x ( + t) x + = 0 (7)

6 454 H. Airault As i sectio 4, the Faber olyomials of g() are the Tchebicheff olyomials, see for examle []. For, π (t) =x (t) + x (t) = F ( ( + t),, 0,, 0, ) (8) π (t) = C k tk (4+t) k =( +t + 4t + t ) +( +t 4t + t ) π (t) =+t, π (t) =+4t + t = π (v), π 3 (t) = ( + t)( + v), with v =4t + t If x(t) is a root of (7), the x(t) 4 ( + v)x(t) + = 0; for y(v) =x(t),it gives π (v) =y (v) + y (v) = x (t) + x (t) = π (t). Thus π (t) = Ck ( + t)k (4t + t ) k = Ck (4 + v)k v k This roves the validity of the exressio of π (t) followig (8). We also have π (t) = T ( t + ) (9) where T is the th Tchebicheff olyomial, see sectio 4. From (7), 3, π (t) = ( + t)π (t) π (t) (30) Sice the olyomial π (t) has ositive coefficiets, its maximum value o t isatt =. Sice π () = 3, π () = 7, we obtai π () from (30). For the derivatives π (k)(t), we have π (t) =,π (t) = ( + t), π (t) = [(+t + 4t + t 4t + t π (t) = + π + (t) = ) ( +t 4t + t ) ] (3) 4t + t π (t) +t π (t) 4t + t j=0 (3) C + j+ (4t + t ) j ( + t) ( j) (33) ad a similar exressio for π (t). Remark that (3) is a articular case of Theorem 9. i [3]. O the other had, (8)-(33) give iterestig relatios if we ut + t = cos(θ) as i sectio 4, or + t = cosh θ. Theorem 3. max t π (k) (t) = π (k) () for k. Moreover a (k) = () = (k )!K k ( 3,, 0,, 0) are the coefficiets of the series π(k) k k ( 3 + ) k = k + (k )! k+ a (k) (34) Theorem 3.3 Let f() = + b + be uivalet i the disc ad g() = f( as i Theorem 3.. For t, k, we have φ(k) (t) π(k) ) ().

7 Remarks o Faber olyomials Faber olyomials as symmetric fuctios of the roots ad the Tchebicheff olyomials The Faber olyomials ψ (t) itroduced i [5] for exasios of aalytic fuctios ad studied by P. Motel [7] ca be obtaied by Schiffer s elimiatio rocedure [6]. A recet oit of view, see [4], [8], is to cosider the ψ (t) as symmetric fuctios of the roots of a algebraic equatio. Let g() as i () ad m Q m (ξ,t)=ξ m +(b t) ξ m + + b m = (ξ x k (t)) (35) where x (t), x (t),, x m (t) are the roots of Q m (ξ,t) ad let π j (t) =x (t) j + x (t) j + +x j m(t) be the symmetric olyomial of the roots of Q m, it satisfies π m +(b t)π m + + b m π + mb m = 0 (36) From (), the relatio (36) is also valid for ψ m. Comare (36) with (6)-(7). Theorem 4. We have ψ (t) =π (t) Usig De Moivre formula, we iterret Theorem 4. with the Tchebicheff olyomials T (x) ad U (x) where si( +)θ T (cos θ) = cos(θ) ad U (cos θ) = si θ It is well kow that (x )U (x) =( +)T +(x) xu (x), T (x) =U (x) xu (x) ad ( x )U (x) =xt (x) T + (x) k= T (x) =U (x) (37) Theorem 4. Let g() = + b + ad (ψ m(t)) m the Faber olyomials associated to g(). The ψ m (t) =T m ( t b ) = cos(mθ) with cos(θ) =t b (38) Proof. Let Q m (ξ,t)=ξ m +(b t)ξ m +ξ m. It has m ero roots ad the two others o ero roots are x (t) =e iθ ad x (t) =e iθ with cos θ = t b. From Theorem 4., ψ m (t) =x (t) m + x (t) m = e imθ + e imθ =T m (cos θ) Theorem 4.3 Let g() = + b + ad G (t) =G (b t,, 0,, 0) defied by g() t =+ + m= G m (t) m, the G m (t) = si((m +)θ) si θ with cos θ = t b (39)

8 456 H. Airault Proof. From (38), G m (t) = m + ψ m+ (t) = + T m+ (t b ). Usig (37), we get G m (t) =U m ( t b ), thus (39). ACKNOWLEDGEMENTS. The author thaks Emil Michev for the latex versio used i the rearatio of the mauscrit. Refereces [] H. Airault, Symmetric sums associated to the factoriatio of Grusky coefficiets, Coferece, Grous ad symmetries, Aril 7-9 (007) Motreal. [] H. Airault, J. Re, A algebra of differetial oerators ad geeratig fuctios o the set of uivalet fuctios, Bull. Sci. Math., 6 (00), [3] H. Airault, A. Bouali, Differetial calculus o the Faber olyomials, Bull. Sci. Math., 30 (006), [4] A. Bouali, Faber olyomials, Cayley-Hamilto equatio ad Newto symmetric fuctios, Bull. Sci. Math., 30 (006), [5] G. Faber, Uber olyomische Etwickluge, Math. Aale, 57 (903), [6] M. Schiffer, Faber olyomials i the theory of uivalet fuctios, Bull. Amer. Soc., 54 (948), [7] P. Motel, Leços sur les séries de olyômes á ue variable comlexe, Coll. moograhies sur la théorie des foctios, Gauthier-Villars (90). [8] H. Airault, Yu. A. Nereti, O the actio of Virasoro algebra o the sace of uivalet fuctios, Bull. Sci. Math., 3 (008), [9] P. G. Todorov, O the Faber olyomials of the uivalet fuctios of the class Σ, Joural of Math. Aal. ad Al., 6 (99) [0] P. G. Todorov, Exlicit formulas for the coefficiets of Faber olyomials with resect to uivalet fuctios of the class Σ, Proceedigs of the America Mathematical Soc., 8 (98) [] P. K. Sueti, Series of Faber olyomials, Aalytical Methods ad Secial Fuctios, trasl. from Russia (984), Gordo ad Breach Sciece Publ. Received: Setember, 007