A new approach to Bernoulli polynomials

Transcription

1 Rendiconti di Matematica, Serie VII Volume 26, Roma 2006), -2 A new approach to Bernoulli polynomials F COSTABILE F DELL ACCIO M I GUALTIERI Dedicated to Professor Laura Gori on her 70th birthday Abstract: Si approaches to the theory of Bernoulli polynomials are nown; these are associated with the names of J Bernoulli [2], L Euler [4], E Lucas [8], P E Appell [], AHürwitz [6] and D H Lehmer [7] Inthis note we deal with a new determinantal definition for Bernoulli polynomials recently proposed by F Costabile [3]; in particular, we emphasize some consequent procedures for automatic calculation and recover the better nown properties of these polynomials from this new definition Finally, after we have observed the equivalence of all considered approaches, we conclude with a circular theorem that emphasizes the direct equivalence of three of previous approaches Short review of classical approaches Bernoulli polynomials play an important role in various epansions and approimation formulas which are useful both in analytic theory of numbers and in classical and numerical analysis These polynomials can be defined by various methods depending on the applications In particular, si approaches to the theory of Bernoulli polynomials are nown; these are associated with the names of J Bernoulli [2], 690), L Euler [4], 738), PE Appell [], 882), A Hürwitz [7], 890), E Lucas [8], 89) and DH Lehmer [7], 988) The term Bernoulli polynomials was used first in 85 by Raabe [0] in connection with Key Words and Phrases: Bernoulli polynomials Determinant Hessemberg matri AMS Classification: B68 65F40

2 2 F COSTABILE F DELL ACCIO M I GUALTIERI [2] the following multiplication theorem ) m B n + ) = m n B n m) m m In effect Jacob Bernoulli introduced the polynomials B n m) already in 690 his wor [2] was posthumously published in 73) in parallel with the discovery of m numbers B n related to the calculation of the sum S n m) = n of powers of the first natural numbers: introducing the remarable formula S n m) = B! n + )! mn+ he set S n m) = n + B n+ m) B n+ 0)) After the Jacob Bernoulli s discovery, Leonard Euler [4] proposed an approach to Bernoulli polynomials based on functional series epansion; in order to define the Bernoulli polynomials by this approach, nown as the generating function approach, let us consider the function e t t F, t) = e t if t 0, if t =0, with a fied comple number The function F, t) is, in particular, comple analytic in the dis { t < 2π}, therefore it can be epanded in a convergent power series of t centered at the origin, with coefficients that depend on comple number : e t t e t = B n ) t n n=0 After a number of calculations we can obtain from previous equation the relation B n ) = ) n B n = ) n B n showing that B n ) isapolynomial of degree n A more general approach to Bernoulli polynomials can be obtained by using the so called Appell sequences [], defined as follows: a sequence of polynomials P 0 ),P ),

3 [3] A new approach to Bernoulli polynomials 3 is said to form an Appell sequence if deg P n ) =n for each n =0,, 2 P n ) =np n ) for each n =0,, Usually such a sequence is normalized by setting P 0 ) = Note that an Appell sequence can be obtained constructively producing P n )byanindefinite integration of P n ): P n ) =c n + n 0 P n t) dt n =, 2, and then choosing the constant of integration c n = P n 0) in an appropriate way; in fact, an Appell sequence is completely determined by the numbers P n 0) So, for eample, if we set c n = 0 for each n =, 2, we obtain P n ) = n, and, for this reason, the polynomials forming an Appell sequence are also called generalized monomials; the sequence of Bernoulli polynomials can be obtained by setting c n = B n In 890 A Hürwitz gave the Fourier series epansions for B n ) B n ) = + 2πi) n n e 2πi 0 << = and used the Fourier series approach to Bernoulli polynomials in his lectures, as Lehmer report in [7] In 88 Lucas [8] derived the Bernoulli polynomial sequence using the umbral calculus: inthe identity B n ) =B + ) n he claims that at eponent of B in the power epansion of the right member of previous equation is replaced by the inde of the Bernoulli number B to obtain B + ) n = ) n B n = ) n B n Recently, Lehmer [7] proposed a new approach to Bernoulli polynomials based on the Raabe multiplication theorem ) and derived from this approach the other definitions In particular, Lehmer proved the following assertion: for a given integer n there eists only one monic polynomial of degree n in satisfying the functional equation m f m + m ) = m n f m) for each m>; 2 for each n let us denote the solution of previous equation by B n ); then the sequence {B n )} is an Appell sequence

4 4 F COSTABILE F DELL ACCIO M I GUALTIERI [4] 2 Adeterminantal approach More recently, a new definition for Bernoulli polynomials using a determinantal approach has been proposed by F Costabile [3], 999) This definition requires only the nowledge of basic linear algebra Definition The Bernoulli polynomial of degree n = 0,, 2,, it is denoted by B n ) and is defined by B 0 ) = and 2 3 n n n n+ 0 2) B n ) = )n n n n )! ) n ) n 2 2) n ) n ) n 2 n 2 for each n =, 2, Remar ) 2 If we set in 2) )! := then the entry i, j) isequal to j for each i =2,, n +,j =,, n +,i j i 3 Despite previous definition of B n ) involves the calculation of a n + )- order determinant, its particular form, nown as upper-hessember, allows us to simplify the computational procedure In fact, it is nown that the algorithm of Gaussian elimination without pivoting for computing the determinant of an upper Hessemberg matri is stable [5, p27]; then a stable algorithm for numerical calculation of B n ) can be obtained simply by applying the algorithm of Gaussian elimination without pivoting for computing the determinant 2) The following procedure allows us to recover a well-nown formula for symbolic computation of Bernoulli polynomials Lemma 3 For the determinant H n of an upper Hessemberg matri of order n, with entries h i,j, h i,j =0, i j 2 h, h,2 h,3 h,n h 2, h 2,2 h 2,3 h2,n 0 h 3,2 h 3,3 h3,n 3) H n = 0 h 4,3 h 4,4 0 0 h n,n h n,n

5 [5] A new approach to Bernoulli polynomials 5 the following recursive relation holds n 4) H n = ) n q n)h +,n H with the following settings n q n n) =, q n) = h j,j, =0,,,n 2 j=+2 or equivalently q n n) =, q n) =h +2,+ q +, =0,,,n 2 Proof A proof of previous result can be accomplished by using the Laplace formula to calculate the determinant H n By applying previous result to determinant 2), we find: 5) B n ) = n n ) n + B ) n + that can be used independently to define the Bernoulli polynomials [] In addition, previous formulas can be used to calculate the coefficients of the polynomial B n ); in fact, by setting for each =0,, 2,, n B ) = b j j j=0 and by substituting previous relations in 5) we obtain j=0 b nj j = n n + n ) n + b j j = n n n n + j=0 =j j=0 n+ ) bj j ; comparing term by term the first member polynomial with the last member we finally obtain the relations b nj = n ) n + b j, j =0,,, n, n + j=0 b nn =

6 6 F COSTABILE F DELL ACCIO M I GUALTIERI [6] 3 Properties of Bernoulli polynomials Some of well-nown properties of Bernoulli polynomials can be easily recovered from the determinantal definition 2) with some calculation and the nowledge of basic notions related to the theory of determinant; let us consider here some eample Property Differentiation):For the differentiation of Bernoulli polynomials can be used the relations B n ) =nb n ), n =, 2, Proof In order to recover this property starting from the determinantal approach, one can differentiate the determinant 2) using the properties of linearity, epand the resulting determinant with respect to the first column and recognize the factor B n ) after multiplication of the i-th row by i 2 i =3,, n and j-th column by /j j =,, n Property 2 Integral means conditions): For each n there is 0 B n ) d =0 Proof The proof of this property consists in a direct calculation In fact, after the definite integration the first two line of determinant coincide 0 B n ) d shall Property 3 Differences): For each n there is B n +) B n ) =n n Proof A proof of previous property based on determinantal definition 2) can be accomplished with some calculation by using the linearity property of a determinant with respect to each row and the following well-nown identity i ) +) i i i =

7 [7] A new approach to Bernoulli polynomials 7 In force of the primary connection between Bernoulli numbers B n Bernoulli polynomials, namely and B n 0) = B n we obtain from determinantal definition of Bernoulli polynomials 2) a determinantal definition for the Bernoulli numbers as well, by the setting 6) B 0 = B n = )n n )! n n n n ) n ) n 2 2) ) n ) n n 2 n 2 n =, 2, Property 4 A series representation in terms of Bernoulli numbers): For Bernoulli polynomials we have B n ) = ) n B n = ) n B n, n =0,, Proof In this case, we can start the proof argumentations by epanding the determinant 2) with respect to the first row; then, woring on the cofactor of the power, =0,, n, after some calculation by using the property of linearity with respect to each row or column one recognize that this cofactor is eactly n ) Bn Property 5 The value at = ): For each n 2 B n ) = B n Proof Even in this case the proof consists in a direct calculation: in fact, evaluating the determinant 2) at = previous equality results by epanding the evaluated determinant with respect to the first column

8 8 F COSTABILE F DELL ACCIO M I GUALTIERI [8] 4 Equivalence of definitions 7) It is possible to prove that all previous approaches lead to at least one of { B n )=nb n ), B n )=nb n ), B 0)= B ), B n 0)=B n,n, B n 0)=B n ),n>, { B n )=nb n ), 0 B n t) dt=0,n, that yield, jointly with condition B 0 ) =, the same sequence of polynomials, ie the sequence of Bernoulli polynomials In this sense previous definitions are equivalent In addition, we prove the following Theorem 4 The following circular diagram holds, were the arrows mean that the pointed approaches can be derived from the previous one as theorems: Appell Sequence Determinant Generating Function Proof Determinantal approach Appell s approach As we saw, from determinantal definition 2) of Bernoulli polynomials easily follows that these polynomials form the following Appell sequence B 0 ) =, B n ) =nb n ), 0 B n t) dt =0, n Appell s approach Euler s approach On the other hand, it is nown that the polynomials forming an Appell sequence {P n )} can be introduced by means of closed form formulas by using the generating function of the sequence, ie the function e t f t) = n=0 P n ) tn

9 [9] A new approach to Bernoulli polynomials 9 with the following setting f t) = n=0 P n 0) tn Euler s approach Determinantal approach Finally a nown algorithm for calculating the quotient of two power series [9] can be used to derive the determinant form 2) for Bernoulli polynomials from the Euler approach In fact, in the equation e t t e t = B n ) t n let us replace functions e t and e t with their Taylor series epansions in t and in t respectively) at the origin; so we have + t! + 2 t n t n + + t + t2 3! + + tn n+)! + n=0 = B 0 ) 0! + B )! t + + B n ) t n + In order to write the Taylor series epansion of the function e t e t with respect to t at the origin, we can compare the left member of previous equation with the right member, and multiplying the right member by the denominator of the fraction on the left member we obtain + t! + 2 t n t n + ) B 0 ) = + B ) t + + B n ) t n + 0!! + t + + t n ) n + )! + By multiplying the series on the right hand side of previous equation according to the Cauchy-product rules, this equation leads to the following system of infinite equations in the unnown c i ) = Bi) i!,i=0,, c 0 ) = c 0 ) + c ) =! c 0 ) 3! + c ) + c 2 ) = 2 c 0 ) n+)! + c ) + + c n ) = n t

10 0 F COSTABILE F DELL ACCIO M I GUALTIERI [0] The special form of the previous system lower triangular) allows us to wor out the unnown c n ) operating with the first n + equations only, by applying the Cramer method: c n ) = ! 3! 0 2 n )! n 2)! n n )! n )! n = ! 0 0 n )! n 2)! 0 n+)! n+)! n )! = ! 3! 0 2 n )! n 2)! n n+)! n )! n )! n n =, 2, From the above steps it follows that 8) B n ) = ! 3! 0 2 n )! n 2)! n n+)! n )! n )! n n =, 2, Finally, the determinant 2) can be obtained from previous determinant by means of a transposition and elementary row and column operations In fact the trans-

11 [] A new approach to Bernoulli polynomials position of 8) is 3! 0 n )! 0 0 n 2)! B n ) = )!! n n n )! n+)! n )! n =, 2, and multiplying the i-th row i = 2,, n by i 2)! and the j-th column j = 2,, n +byj )! we obtain! n )! 3! n+)! n )! 0 0!! 0! 0!n )! 0! n )!! n 2)! 0 0 B n )=!!!n 2)!!n )!! n =, 2, n )! n 2)!! n 2)! 2 n )! )! n n n )! that is eactly 2) after the echange of the first row with the last one REFERENCES [] PE Appell: Sur une classe de polynomes, Annales d ecole normale superieur, s 2, 9 882) [2] J Bernoulli: Ars conjectandi, Basel, pag 97, 73), posthumously published [3] F Costabile: Epansions of real functions in Bernoulli polynomials and applications, Conf Sem MatUniv Bari, N 273, 999) [4] L Euler: Methodus generalis summandi progressiones, Comment acad sci Petrop, 6 738) [5] NH Higham: Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 996 [6] A Hürwitz: Personal communication via George Polya, that Hurwitz used the Fourier series approach to Bernoulli polynomials in his lectures [7] D H Lehmer: A New Approach to Bernoulli Polynomials, Amer, Math Monthly ), [8] E Lucas: Théorie des Nombres, Paris 89, Chapter 4 [9] AI Marushevich: Theory of functions of a comple variable, vol I, Prentice- Hall, Inc 965

12 2 F COSTABILE F DELL ACCIO M I GUALTIERI [2] [0] JL Raabe: Zurucfuhrung einiger Summen und bestimmten Integrale auf die Jacob Bernoullische Function, Journal für die reine and angew math, 42 85), [] G Walz: Asymptotics and Etrapolation, Aademie Verlag, Berlin, 996 Lavoro pervenuto alla redazione il 09 novembre2004 ed accettato per la pubblicazione il 05 maggio 2005 Bozze licenziate il 6 gennaio 2006 INDIRIZZO DEGLI AUTORI: F Costabile F Dell Accio MI Gualtieri Dipartimento di Matematica Università degli Studi della Calabria via P Bucci cubo 30 A Rende CS) Italy {costabil; fdellacc;miggualtieri}@unicalit