REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS

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REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x] is a liear operator that commutes with the differetiatio d : R[x] R[x]. dx CONTENTS. The mai result. Some applicatios 3 Refereces 9 We cosider series of the form. THE MAIN RESULT a T P (x, ( where P R[x], ad T : R[x] R[x] is a liear operator such that T D = DT, where D is the differetiatio operator D = d dx. The coditio ( is equivalet with the traslatio ivariace of T, i.e., T U h = U h T, h R, (I where U h : R[x] R[x] is the traslatio operator R[x] p(x p(x + h R[x]. For simplicity we set U := U. Clearly U h O so a special case of the series ( is the series a P (x + h, h R, ( h a U h P (x = which is typically diverget. We deote by O the R-algebra of traslatio ivariat operators. We have a atural map Q : R[[t]] O, R[[t] t c! c! D. It is ow (see [, Prop. 3.47] that this map is a isomorphism of rigs. We deote by σ the iverse of Q σ : O R[[t]], O T σ T R[[t]]. Mathematics Subject Classificatio. 5A, 5A9, B83, 4A3, 4G5, 4G. Key words ad phrases. traslatio ivariat operators, diverget series, summability. Last modified o July 9, 9. (

LIVIU I. NICOLAESCU For T O we will refer to the formal power series σ T as the symbol of the operator T. More explicitely σ T (t = c (T t, c (T = (T x x= R.! We deote by N the set of oegative itegers, ad by Seq the vector space of real sequeces, i.e., maps a : N R. Let Seq c the vector subspace of Seq cosistig of all coverget sequeces. A geeralized otio of covergece or regularizatio method is a pair µ = ( µ lim, Seq µ, where Seq µ is a vector subspace of Seq cotaiig Seq c ad, µ lim is a liear map µ lim : Seq µ R, Seq µ a µ lim a( R such that for ay a Seq c we have µ lim a = lim a(. The sequeces i Seq µ are called µ-coverget ad µ lim is called the µ-limit. To ay sequece a Seq we associate the sequece S[a] of partial sums S[a]( = σ= a(. (. A series a( is said to by µ-coverget if the sequece S[a] is µ-coverget. We set µ a( := µ lim S[a](. We say that µ a( is the µ-sum of the series. The regularizatio method is said to be shift ivariat if it satisfies the additioal coditio µ a( = a( + µ a(. (. We refer to the classic [3] for a large collectio of regularizatio methods. For x R ad N we set { ( i= (x i, x [x] :=, =,, := [x]!. We ca ow state the mai result of this paper. Theorem.. Let µ be a regularizatio method, T O ad f(t = a t R[[t]]. Set c := c (T = T. Suppose that f is µ-regular at t = c, i.e., We deote by f ( (c µ its µ-sum for every N the series a [] c is µ-coverget. (µ f ( (c µ := µ a [] c. The for every P R[x] the series a (T P (x is µ-coverget ad its µ-sum is µ a (T P (x = f(t µ P (x, Hardy refers to such a otio of covergece as covergece i some Picwicia sese.

where f(t µ O is the operator REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS 3 f(t µ := f ( (c µ (T c. (.3! Proof. Set R := T c ad let P R[x]. The so that R = c (T D! I particular this shows that f(t µ is well defied. We have a T P = a (c + R P = a R P =, > deg P. (.4 = ( deg P c R P = = At the last step we used (.4 ad the fact that ( =, if >. ( c R P. This shows that the formal series a (T P (x ca be writte as a fiite liear combiatio of formal series deg P a (T R P (x P (x = a [] c.! From the liearity of the µ-summatio operator we deduce deg P a (T R P (x P (x = µ a [] c! µ = ( deg P = = = f ( (c µ R P (x = f(t µ P (x!. SOME APPLICATIONS To describe some cosequeces of Theorem. we eed to first describe some classical facts about regularizatio methods. For ay sequece a Seq we deote by G a (t R[[t]] its geeratig series. We regard the partial sum costructio S i (. as a liear operator S : Seq Seq. Observe that G S[a] (t = t G a(t. We say that a regularizatio method µ = ( µ lim, Seq µ is stroger tha the regularizatio method µ = ( µ lim, Seq µ, ad we write this µ µ, if Seq µ Seq µ ad µ lim a( = µ lim a(, a Seq µ.

4 LIVIU I. NICOLAESCU The Abel regularizatio method A is defied as follows. We say that a sequece a is A coverget if the radius of covergece of the series a t is at least ad the fuctio t ( t a t has a fiite limit as t. Hece A lim a( = lim t ( t a t, ad Seq A cosists of sequece for which the above limit exists ad it is fiite. Usig ( we deduce that a series a( is A-coverget if ad oly if the limit lim a t t exists ad it is fiite. We have the followig immediate result. Propositio.. Suppose that f(z is a holomorphic fuctio defied i a ope eighborhood of the set {} { z } C. If a z is the Taylor series expasio of f at z = the the correspodig formal power series [f] = a t is A-regular at t =, [f] ( ( A = f (, ad the series [f](r A = [f] ( ( A r! coicides with the Taylor expasio of f at z =, ad it coverges to f( + r. Corollary.. Suppose that f(z is a holomorphic fuctio defied i a ope eighborhood of the set {} { z } C ad a z is the Taylor series expasio of f at z =. The for every T i O such that c (T =, ay P R[x], ad ay x R we have A a T P (x = f ( (T P (x.! Let N. A sequece a Seq is said to be C -coverget (or Cesàro coverget of order if the limit S [a]( lim ( + exists ad it is fiite. We deote this limit by C lim a(. A series a( is said to be C - coverget if the sequece of partial sums S[a] is C coverget. Thus the C -sum of this series is C More explicitly, we have (see [3, Eq.(5.4.5] C a( = lim a( = lim ( + S + [a](. ( + ( ( ν + ν= a( ν This was apparetly ow ad used by Euler.

Hece REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS 5 C ( + a( S + [a]( A A!, where a( a b lim b( =, if a(, b(, for. The C covergece is equivalet with the classical covergece ad it is ow (see [3, Thm. 43, 55] that C C A, <. Give this fact, we defie a sequece to be C-coverget (Cesàro coverget if it is C -coverget for some N. Note that C A. Both the C ad A methods are shift ivariat, i.e., they satisfy the coditio (.. We wat to commet a bit about possible methods of establishig C-covergece. To formulate a geeral strategy we eed to itroduce a classical otatio. More precisely, if f(t = a t is a formal power series we let [t ]f(t deote the coefficiet of t i this power series, i.e. [t ]f(t = a. Let f(t = a t. The the series a t C-coverges to A if ad oly if there exists a oegative real umber α such that [t ] ( ( t (α+ f(t A α Γ(α +, where Γ is Euler s Gamma fuctio. For a proof we refer to [3, Thm. 43]. Defiitio.3. We say that a power series f(t = a t is Cesàro coveiet (or C-coveiet at if the followig hold. (i The radius of covergeces of the series is (ii The fuctio f is regular at z = ad has fiitely may sigularities ζ,..., ζ ν o the uit circle { z = }. (iii There exist ε > ad θ (, π such that f admits a cotiuatio to the dimpled dis { } z ε,θ := z C; z < + ε, arg( > θ, j =,..., ν. ζ j (iv For every sigular poit ζ j there exists a positive iteger m j such that f(z = O ( (z ζ j m j as z ζ j, z. The results i [, Chap. VI] implies that the collectio R C of C-coveiet power series is a rig satisfyig f R C df dt R C. Ivoig [, Thm VI.5] we deduce the followig useful cosequece. Corollary.4. Let f R[[t]] be a power series C-coveiet at. The f is C-regular at ad f ( C = f ( ( A = f ( (. Usig [, VII.7] we obtai the followig useful result.

6 LIVIU I. NICOLAESCU Corollary.5. (a The power series ( + t m = ( + m ( t, m, log( + t = ( + t are C-regular at. (b If f(z is a algebraic fuctio defied o the uit dis z < ad regular at z = the the Taylor series of f at z = is C-regular at. Recall that the Cauchy product of two sequeces a, b Seq is the sequece a b, a b( = a( ib(i, N. i= A regularizatio method is said to be multiplicative if ( µ a b( = µ for ay µ-coverget series a( ad b(. The results of [3, Chap.X] show that the C ad A methods are multiplicative. For ay regularizatio method µ ad c R we deote by R[[t]] µ the set of series that are µ-regular at t =. Propositio.6. Let µ be a multiplicative regularizatio method. The R[[t]] µ is a commutative rig with oe ad we have the product rule ( (f g ( ( µ = f ( ( µ g ( ( µ. = a( Moreover, if T O is such that c (T = the the map is a rig morphism. ( µ R[[t]] µ f f(t µ O b( Proof. The product formula follows from the iterated applicatio of the equalities D t (fg = (D t fg + f(d t g, (fg( µ = f( µ g( µ, f ( µ = (D t f( µ, where D t : R[[t]] R[[t]] is the formal differetiatio operator d dt. The last statemet is a immediate applicatio of the above product rule. Remar.7. The iclusio R[[t]] C R[[t]] A is strict. For example, the power series f(z = e /(+z satisfies the assumptio of Propositio. so that the associated formal power series [f] is A-regular at. O the other had, the argumets i [3, 5.] show that [f] is ot C-regular at. Cosider the traslatio operator U h O. From Taylor s formula p(x + h = h! D p(x we deduce that σ U h(t = e th. Set h := U h. Usig Corollary.5 ad Theorem. we deduce the followig result.,

REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS 7 Corollary.8. For ay P R[x] we have C ( P (x + h = ( h P (x. (. Observe that ( + ( h h = so that ( h is the iverse of the operator + h. We thus have C ( P (x + h = ( + h P (x = ( + U h P (x. (. Remar.9. Here is a heuristic explaatio of the equality (. assumig the Cesàro covergece of the series ( P (x + h. Deote by S(x the Cesàro sum of this series. The S(x + h = C ( P ( x + ( + h Hece (. = C ( P (x + h + P (x = S(x + P (x. S(x + h + S(x = P (x, x R. If we ew that S(x is a polyomial we would the deduce S(x = ( + U h P (x. The iverse of + U h ca be explicitly expressed usig Euler umbers ad polyomials, [4, Eq. (4, p.34]. The Euler umbers E are defied by the Taylor expasio cosh t = e t + e t = E! t. Sice cosh t is a eve fuctio we deduce that E = for odd. They satisfy the recurrece relatio ( ( E + E + E 4 + =,. (.3 4 Here are the first few Euler umbers. 4 6 8 4 6 The Hece E 5 6, 385 5, 5, 7, 765 99, 36, 98 9, 39, 5, 45 h + U h = U U h + U h = U e D + e D C ( P (x + h = = U h cosh hd = U h E h! D. E h (! P ( x h. (.4

8 LIVIU I. NICOLAESCU Whe P (x = x m, h =, we have C ( (x + m = ( ( m E x m. (.5 Settig x = ad usig the equality E j+ =, j we coclude that C ( m = ( m m+ ( m E = ( m ( m m+ E. (.6 Usig (.3 we deduce that whe m is eve, m = m, m > we have C ( m =. (.7 For example + + C=, ( + 3 + 4 C= 4, ( + 3 3 3 + 4 3 C= 8, ( 3 5 + 5 3 5 + 4 5 C= 4. ( 5 Whe P (x = ( x m, x =, h = the it is more coveiet to use (. because ( ( x x =,, x. We deduce ( C ( = m m ( ( = ( m. (.8 m m+ = Example.. Cosider the traslatio ivariat operator T : R[x] R[x], P (x e s P (x + sdx. Set R = T. As explaied i [, II.3.B], the operators T ad R are itimately related to the Laguerre polyomials. We have R = DT = T D ad 3 σ T (t = t σ R(t = t = t t. If P R[x] is a polyomial of degree m the T P (x x= = ( + D + + D m P (x x= = e (s +s + +s P (s + + s ds ds. R For t we deote by (t the ( simplex (t := { (s,..., s R ; s + + s = t }., 3 We ca write formally T = R e s U s ds = R e s( D ds = ( + D, so that σ T (t = t.

REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS 9 ad by dv (t the Euclidea volume elemet o (t. Itegratig alog the fibers of the fuctio f : R [,, f(s,..., s = s + + s we deduce ( e (s +s + +s P (s + + s ds ds = f dv (t e t P (tdt R = v e s s P (sds, (t where v is the ( -dimesioal volume of the ( -simplex = (t t=. To compute the volume v we view is a regular -simplex with distiguished base, ad distiguished vertex (,...,, R +. The distace d from the vertex to the base is the distace from the vertex to the ceter of the base. We have d = + +, d =, v = ( + / d v = 3 v. Sice v = we deduce v = ( + /, T P (x x= =! (! e s s P (sds, ad R P (x x= = e s s P ( (sds. (! Usig Theorem. ad Corollary.4 with the C-coveiet series f(t = ( + t we deduce C ( T P (x x= = C ( e s s P (sds (! ( deg P ( = + (! s P ( (s ds. If we let P (s = s m we deduce ad = e s s P (sds = (m +!, ( C m + ( = m e s s P ( (sds = [m] (m! = [m ] m!, m = ( + ( m. (.9 Let us poit out that (.9 ca be obtaied from (.8 usig the shift-ivariace of the Cesàro regularizatio method. REFERENCES [] M. Aiger: Combiatorial Theory, Spriger Verlag, 997. [] P. Flajolet, R. Sedgewic: Aalytic Combiatorics, Cambridge Uiversity Press, 9. [3] G.H. Hardy: Diverget Series, Chelsea Publishig Co. 99. [4] N.E. Nörlud: Mémoire sur les polyomes de Beroulli, Acta Math. 43(9, -94. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556-468. E-mail address: icolaescu.@d.edu URL: http://www.d.edu/ licolae/

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