The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

Transcription

1 The Möbius inversion forula for Fourier series applied to Bernoulli and Euler polynoials Luis M Navas a, Francisco J Ruiz b,, Juan L Varona c,, a Departaento de Mateáticas, Universidad de Salaanca, Plaza de la Merced -4, Salaanca, Spain b Departaento de Mateáticas, Universidad de Zaragoza, Capus de la Plaza de San Francisco, Zaragoza, Spain c Departaento de Mateáticas y Coputación, Universidad de La Rioja, Calle Luis de Ulloa s/n, Logroño, Spain Counicated by Andrei Martínez-Finkelshtein Dedicated to Guillero López Lagoasino on the occasion of his sixtieth birthday Abstract Hurwitz found the Fourier expansion of the Bernoulli polynoials over a century ago In general, Fourier analysis can be fruitfully eployed to obtain properties of the Bernoulli polynoials and related functions in a siple anner In addition, applying the technique of Möbius inversion fro analytic nuber theory to Fourier expansions, we derive identities involving Bernoulli polynoials, Bernoulli nubers, and the Möbius function; this includes forulas for the Bernoulli polynoials at rational arguents Finally, we show soe asyptotic properties concerning Bernoulli and Euler polynoials Key words: Bernoulli polynoials, Euler polynoials, Fourier series, Möbius transfor, inversion forula, rational arguents, asyptotic properties 2000 MSC: Priary B68, Secondary 42A0, A25 Introduction The Bernoulli polynoials, which play an iportant role in Analytic Nuber Theory, are usually defined by eans of the generating function te tx e t = k=0 B k x) tk k! Corresponding author Eail addresses: navas@usales Luis M Navas), fjruiz@unizares Francisco J Ruiz), jvarona@uniriojaes Juan L Varona) URL: Juan L Varona) Research supported by grant MTM C03-03 of the DGI Published in Journal of Approxiation Theory 63 20), doi:006/jjat

2 The polynoial B k x) is onic and has degree k For exaple, B 0 x) =, B x) = x 2 Although there are no closed forulas for the kth Bernoulli polynoial, the uniqueness theore for power series expansions allows one to easily prove various properties regarding the Aong these we recall the following: and In 890, Hurwitz found the Fourier expansions B 2k x) = 2 )k 2k)! 2π) 2k B 2k+ x) = 2 )k 2k + )! 2π) 2k+ B k+x) = k + )B k x) ) B k x) = ) k B k x) 2) cos2πnx) n 2k, x [0, ), n= k ; 3) sin2πnx) n 2k+, x [0, ), k 0 4) The Bernoulli nubers are given by B k = B k 0) For odd indexes, we have n= B 2k+ 0) = 0, k 5) An iediate consequence of the Fourier expansions for the Bernoulli polynoials, probably the ost well-known, is the connection between the Bernoulli nubers and the values of the Rieann zeta function at the positive even integers Indeed, one only needs to set x = 0 in the first forula to obtain B 2k = 2 )k 2k)! 2π) 2k ζ2k) The following forula, using Möbius inversion, is also well-known: n= n 2k = ζ2k) = 2π) 2k 2 ) k 2k)! n= n 2k B 2k Indeed, Möbius inversion shows = ζs) n= n for any coplex s with Res) > s However, leaving s = 2k fixed, we can instead generalize the forula to cos2πx) = 2π) 2k 2 ) k 2k)! n= n 2k B 2k{nx}), 6) where {x} denotes the fractional part of x This expression, which provides a nice approxiation to the cosine by eans of polynoials, is a consequence of a ore general type of Möbius inversion, discussed by two of the authors in [0] In this case, it involves the inversion of Fourier series The analogous result for odd Bernoulli polynoials also holds In addition, we find a siilar forula involving Euler polynoials All of this is discussed in Section 2 2

3 In Section 3, we consider the relations obtained by evaluating 3) and its Möbius inverse 6) at rational values One obtains, in an eleentary anner, expressions for cobinations of sus of the for n r ) n 2k, n r ) n 2k where n r ) is a shorthand for n r od )) in ters of values of the Bernoulli polynoials at these rational arguents In Section 4, we obtain the asyptotic behavior of the Bernoulli polynoials on [0, ] fro forulas 3) and 4) This result is not new, however, our proof is sipler, and in addition we also study the rate of convergence Throughout the paper, we will use x to denote the integer part also known as floor) of x R ie, x is the largest integer x); then, the fractional part of x will be {x} = x x 2 Möbius inversion forulas 2 Arithetical Möbius inversion In Nuber Theory, an arithetical function is typically siply a function α : N R or C) An arithetical function α is copletely ultiplicative if it satisfies αn) = αn)α) for all n, N and is not the zero function Given two arithetical functions α and β, their Dirichlet convolution also called Dirichlet product) α β is defined by α βn) = n ) αd)β = n ) α βd) = αa)βb), d d d n d n where d n represents the su over all divisors d of n Dirichlet convolution is a coutative and associative operation on arithetical functions, with identity the delta function at, ie δ) = and δn) = 0 if n An arithetical function α is invertible with respect to Dirichlet convolution if and only if α) 0 In this case the unique function β such that α β = β α = δ is referred to as the Dirichlet) inverse of α, and we use the standard notation α to denote it A fundaental role in the theory is played by the Möbius function µ, which is the Dirichlet inverse of the constant function It is given by µ) =, = 0 if n has a squared factor, µp p 2 p k ) = ) k when p, p 2,, p k are distinct pries For a general invertible arithetical function α, its Dirichlet inverse ay be coputed recursively, but it is often difficult to deduce a siple closed expression for α An easy exception is when α is copletely ultiplicative, since then α = µα the pointwise product) The Möbius Inversion Forula ost often refers to the equivalence β = α α = β µ, which is an iediate algebraic consequence of the facts stated above, and which 3 ab=n

4 ay be written explicitly as βn) = αd) αn) = µn/d)βd) d n d n 22 Möbius inversion of Fourier series For our purposes, we need a variation on the inversion thee of a ore analytic nature, belonging to a class of forulas also referred to as Möbius inversion We shall restrict ourselves to the case of Fourier series Suppose we have a real-variable function f expanded in a Fourier series, fx) = αn)e 2πinx n= Regard the Fourier coefficients αn) as an arithetical function arithetical function β, and for the generalized convolution Now consider any β f)x) = β)fx) Substituting the Fourier series for f into this expression, one finds forally that β f)x) = = = β) αn)e 2πinx = l= n=l n=,n= αn)β)e 2πinx ) = αn)β) e 2πilx = α β)l)e 2πilx Thus β f is the Fourier series with coefficients given by the Dirichlet convolution of α and β Now, if α is invertible and we take β = α, so that α β = δ, this reduces to l= e 2πix = α f)x), 2) so we have an expansion of the exponential in ters of the function f whose Fourier series we started out with Note also that the Fourier series itself is the generalized convolution of the Fourier coefficients with the function Ex) = e 2πix, naely f = α E In fact forally one has in general that α β g) = α β) g for any function g and arithetical functions α, β, and the inversion relation f = α g g = α f Reark Justifying the foral steps above is not hard in the case of a bounded function such as e 2πix It is enough for the double series to converge absolutely, which is iplied by,n= αn)β) < Note that this is equivalent to l= α β )l) < In particular, the inversion forula 2) holds if l= α α )l) < In the case of a copletely ultiplicative function α, since α = µα and µ, it is sufficient to check if l= α α )l) < By taking real and iaginary parts, we obtain analogous forulas involving the functions sin2πx) and cos2πx) In this guise, the idea of applying Möbius inversion to 4

5 Fourier series goes at least as far back as Chebyshev [8] and appears recently in [9] in a study of a lattice proble in Physics Let us ention that the above results are special cases of a general theory which extends far beyond the case of Fourier series and which sees to originate with a littleknown idea of Cesàro [7], rediscovered on occasion, for exaple in [6] The interested reader ay consult [4] for an abstract forulation, [5] for a series of concrete exaples, [0] for an inversion forula involving Chebyshev polynoials, and [2] as a general reference for analytic nuber theory If we want to obtain concrete approxiation results fro forulas such as 2), we need to have an expression for the Dirichlet inverse of the Fourier coefficients that we can work with, and the best case of this occurs when they are copletely ultiplicative Now, this certainly does not happen in general For this reason, it is an interesting question to deterine which functions do indeed give copletely ultiplicative Fourier coefficients, at least odulo constant factors This happens, for instance, in the case of the square and triangular waves, which were the exaples studied by Chebyshev Perhaps surprisingly, this also happens with a well-known faily of functions: the Bernoulli polynoials 23 Möbius inversion of the Fourier series of the Bernoulli polynoials The Bernoulli polynoials B k x) play an iportant role in various expansions and approxiation forulas which are useful both in analytic nuber theory and in classical and nuerical analysis These polynoials can be defined by various ethods depending on the applications see [2] and the references therein) The Fourier expansions 3) and 4) are actually valid for 0 x, and the convergence is absolute and unifor on [0, ], except for B x) that requires 0 < x < For the tie being, we disregard B x) and also B 0 x) Fro 5) and 2), it is clear that B k 0) = B k ), k 2, so we can construct the periodic extension of B k x) on [0, ] to R by taking fractional parts {x} = x x and using B k {x}) instead of B k x); these [0, ]-periodic extensions are continuous Then, 3) and 4) for k becoe B 2k {x}) = 2 )k 2k)! 2π) 2k B 2k+ {x}) = 2 )k 2k + )! 2π) 2k+ cos2πnx) n 2k, n= x R, 22) sin2πnx) n 2k+, x R 23) n= Now, applying the real versions of 2), we obtain the following Theore 2 For every k, the functions cosine and sine expand in ters of the Bernoulli polynoials B 2k and B 2k+, respectively, as cos2πx) = )k 2π) 2k 22k)! sin2πx) = )k 2π) 2k+ 22k + )! n= B 2k {nx}) n 2k, x R, 24) n= 5 B 2k+ {nx}) n 2k+, x R 25)

6 Proof This is a special case of 2) Up to constants, we are dealing with the arithetical functions α s n) = n s, which are copletely ultiplicative for any s C here s = 2k or 2k + ) Hence αs = µα s The constants siply affect inversion by cα s ) = c µα s and of course do not affect the convergence that justifies the inversion As for the convergence itself, by Reark, it is enough to show that n= α s α s )n) < This is true whenever σ = Res) >, since α s n) = α σ n) and α σ α σ )n) = k n k σ n/k) σ = k n n σ = dn)n σ, where dn) is the nuber of divisors of n A standard result fro Analytic Nuber Theory states that dn) = on ɛ ) for any ɛ > 0 Hence α σ α σ )n) = On σ+ɛ ) and so n= α σ α σ )n) converges by coparison with the zeta series 24 Special values In Analytic Nuber Theory one often obtains interesting arithetical results by evaluating relations involving transcendental functions at rational arguents Since e 2πix is a root of unity, hence an algebraic nuber, when x is rational, its iaginary part sin2πx) is also algebraic A nice expression for this algebraic nuber exists, for exaple, when it is constructible, in the sense of Galois Theory, and the ost faous case of this is x = /7, corresponding to the construction of the regular 7-gon found by Gauss Algebraically this eans that cos2π/7) has an expression in nested square roots Applying the first forula of Theore 2 to x = /7 and, for siplicity, to the lowest valid value of k, that is k =, yields, after evaluating B 2 r/7), where 289, if n 0 od 7), 93, if n ± od 7), 09, if n ±2 od 7), 37, if n ±4 od 7), βn) = 23, if n ±4 od 7), 7, if n ±5 od 7), 07, if n ±6 od 7), 3, if n ±7 od 7), 43, if n ±8 od 7) = 23 π n= βn) n 2 This is the kind of explicit forula one can obtain with these ethods Note the aount and variety of atheatics that goes into this result: the ideas of Bernoulli, Cesàro, Chebyshev, Dirichlet, Euler, Fourier, Galois, Gauss, Hurwitz and Möbius are all involved 6

7 25 The case of Euler polynoials In a way siilar to their cousins the Bernoulli polynoials, the Euler polynoials are defined by eans of the generating function 2e 2tx e t + = E k x) tk k!, k=0 which is convergent for t < π For 0 x 0 < x < in the case of E 0 x)), the Euler polynoials have Fourier expansions also siilar to those of the Bernoulli polynoials: E 2k x) = 4 )k 2k )! π 2k E 2k x) = 4 )k 2k)! π 2k+ n=0 n=0 Let us find the Möbius inverse of these series By denoting { 0 if j is even, α k j) = if is odd, j = 2n +, 2n+) k cos2n + )πx) 2n + ) 2k, k, 26) sin2n + )πx) 2n + ) 2k+, k 0 27) we can write E 2k x) in 26) as a constant ties j= α 2kj) cosjπx), and siilarly for 27) Moreover, the function α k is copletely ultiplicative, so its Dirichlet inverse is α k = µα k Extending E k x) fro [0, ] to R is only a bit ore coplicated than for B k x) The extension that is copatible with 26) and 27) is ) x E k {x}) Then, in a anner entirely siilar to Theore 2, we deduce the following Theore 22 For every k, the functions cosine and sine expand in ters of the Euler polynoials E 2k and E 2k, respectively, as cosπx) = )k π 2k 42k )! sinπx) = )k π 2k+ 42k)! n=0 n=0 µ2n + ) ) 2n+)x E 2k {2n + )x}) 2n + ) 2k, x R, µ2n + ) ) 2n+)x E 2k {2n + )x}) 2n + ) 2k+, x R 3 Sus of restricted zeta series and their Möbius inverses The evaluation of 22), 23) and their inverses 24), 25) at rational arguents x = r/ introduces a periodicity odulo into the sus which on rearrangeent by residue classes causes the following sus to appear: M k, r) = n r ) n k, Z k, r) = 7 n r ), r = 0,,, 3) nk

8 where k, N, k, 2 and the sus are always over positive integers thus the su defining M k, 0) begins at n = ) These sus are related to the Prie Nuber Theore for arithetic progressions Using techniques fro Analytic Nuber Theory, the sus Z and M can be expressed in ters of L-series for Dirichlet characters Here we show that, starting fro the Fourier expansions of the Bernoulli polynoials and their Möbius inverses, certain sus and differences of M and Z over a syetric pair ±r of residue classes odulo can be evaluated explicitly by eleentary and coputationally feasible eans, using only linear algebra, as a consequence of another auxiliary result involving atrices defined by the Bernoulli polynoials B k and the cosine function This approach is in the spirit of one of the probles dealt with in [], which studies the case k =, and, as is aply discussed there, is interesting in its own right We concentrate on the case of an even power 2k ostly, since we ake use of the evaluation of ζ2k) in several places, but in the last part of this section we also derive soe results for an odd power 2k + The ethods eployed in this section ay also be used to derive results for Euler polynoials and Euler nubers which are analogous to those we obtain for Bernoulli polynoials and Bernoulli nubers We have chosen to illustrate the ethod with the latter to allow an easier coparison with results in the literature, and for reasons of space the corresponding forulas for Euler polynoials are left to the reader 3 Linear relations aong values at rational arguents Trivially, we have r=0 M k, r) = ζk), r=0 Z k, r) = ζk), Z k, 0) = k ζk) Let us introduce notation for the constants which appear in the Fourier expansions of the Bernoulli polynoials, Fixing k and, let and define, for odd, Ck) = )k 2π) 2k 22k)!, Dk) = )k 2π) 2k+ 22k + )! ω = /2 x i = Ck)M 2k, i) + M 2k, i)), i =,, ω, x ω+ = Ck)M 2k, 0), y i = Ck) Z 2k, i) + Z 2k, i)), i =,, ω, y ω+ = Ck) Z 2k, 0), and for even, the sae expressions except that at i = ω = /2 we take x /2 = Ck)M 2k, /2), y /2 = Ck) Z 2k, /2) and not twice this expression as the previous forulas would indicate Note that in fact ζ2k) = Ck)B 2k, and hence y ω+ = B 2k / 2k 32) 8

9 Proposition 3 Let N, 2, and k N Then cos2πr/) = B 2k r/) = ω { }) jr B 2k x j + B 2k x ω+, r = 0,,, ω, 33) ω cos 2π jr ) y j + y ω+, r = 0,,, ω 34) j= j= Proof 33) follows iediately by evaluating 24) at the arguents x = r/, r = 0,,, grouping the series by residues odulo and taking into account the syetry B 2k x) = B 2k x) 34) is obtained in exactly the sae way via 22) and cos2π x)) = cos2πx) Reark 2 The equations 34) give forulas for the values of Bernoulli polynoials at rational arguents, but they are not very satisfactory as they involve the ters y i for which there are no siple expressions with the exception of y ω+ = B 2k / 2k ) In fact, 34) are practically the content of the paper [3], where the atter is not taken any further; a posterior paper that deals with this atter and related topics is [5] Here we are going to show soe of their applications, as well as those of their Möbius inverses 33) 32 Evaluation of M 2k, 0) by eleentary ethods The next step is to show that one can evaluate x ω+ = Ck)M 2k, 0) explicitly This can be done with Dirichlet series and characters, but in fact it is not too difficult to give a nice alternative eleentary proof, as we proceed to show Theore 32 Let = p p 2 p l where the p i are distinct pries, and let k N Then x ω+ = )k 2π) 2k 22k)! n) 2k = l B 2k p 2k 35) i If is not squarefree, x ω+ = 0 trivially since = 0 for all n N n= Proof The stateent is equivalent to M 2k, 0) = ζ2k) l prove a general forula n 0 ) n s = n= i= i= n) s = ζs) l i= ps i ) p2k) We shall for any coplex nuber s with Res) > Consider everything fixed except and denote this su by S) Note that, by definition of µ, { ) l if no p i divides n, = 0 if soe p i divides n For l =, the case where is equal to a prie p, the forula siply states Sp) = ζs) p s ) and we have µpn) = or 0 according to n 0 p) or n 0 p) 9 i

10 Thus Sp) = p s n 0 p) n s = p s ζs) Sp)), = p s n n s n 0 p) n s and hence Sp) = ζs) p s ) follows iediately The general case can be proved by induction on l, applying the inclusion-exclusion forula In general, Sp p 2 p l ) will appear as a cobination of itself and all the sus Sp i p i2 p it ) with distinct indices i j and 0 t < l For exaple, for l = 2, if = pq with p, q distinct pries, we have Spq) = p s q s n 0 p), n 0 q) n s = p s q s ζs) Sp) Sq) + Spq) ), hence, substituting the values for Sp) and Sq), we can solve for Spq), yielding the corresponding forula 33 Solution of the Bernoulli syste of linear equations Once we have eleentary forulas for the ters x ω+ Theore 32) and y ω+ equation 32)), the syste of equations 33) and 34) ay be considered as involving only the unknowns x j and y j for j =,, ω, ω { }) jr cos2πr/) B 2k x ω+ = B 2k x j, r =,, ω, 36) j= B 2k r/) B ω 2k 2k = cos 2π jr ) y j, r =,, ω 37) j= The associated atrices will be denoted by { })) ω ij B 2k, = B 2k, Cos = i,j= cos 2π ij )) ω 38) i,j= Let us show, with a sall caveat, that these atrices are regular Indeed we copute the deterinant of the cosine atrix explicitly in closed for Theore 33 Let N Then detcos 2+ ) = ) ) )/2 2, detcos 2 ) = ) + 2 ) )/2 2 In the odd case, if U is the square atrix of order whose entries are all, then Cos 2+ = U + Cos 2+) ; 0

11 and, in the even case, Cos 2 = a i,j), where 2 + cosπij/)), if i, j <, a i,j = + cosπij/)), if j < i = or i < j =, + cosπij/)), if i = j = 2 Proof We assue is odd; the even case is siilar Replace the last row with the su of every row and ove it up to the first row It is easy to show that ω cos 2π jr ) { /2, if is odd, = ) j )/2, if is even, r= hence we obtain cos 2π 4π 2π 2+ ) cos 2+ ) cos 2+ ) cos 2 )π 4 )π 2 )π 2+ ) cos 2+ ) cos 2+ ) cos 2π 4π 2+ ) cos 2+ ) cos 22 π 2+ ) = cos 2 ) 2π 4π 2π 2+ ) cos 2+ ) cos 2+ ) cos 2 )π 4 )π 2 )π 2+ ) cos 2+ ) cos 2+ ) Now, since coskx) = 2 k cos k x + i<k α k,i cosix), k 2, for appropriate coefficients α k,i, it follows iediately that the last deterinant is the sae as the following Vanderonde deterinant: ) cos 2π 4π 2π 2+ ) cos 2+ ) cos 2+ ), cos 2π 2+ ) cos 4π 2+ ) cos 2π 2+ ) and hence, we obtain the forula detcos 2+ ) = ) i<j ) )) 2πj 2πi cos cos Since cosine is decreasing on 0, π) it is easy to calculate the sign of this deterinant We copute its absolute value by squaring the atrix and taking square roots Indeed, it is straightforward to copute the atrix Cos 2 2+ by expressing the resulting sus of products of cosines as the real parts of geoetric series of coplex exponentials We oit the details The result is 2 4 /2 /2 /2 2 4 /2 /2 /2 2 4

12 and the deterinant of this atrix is easily found to be detcos 2 2+) = ) 4 Putting everything together one finally obtains the first forulas in the stateent of the theore Finally, it is easy to see that the inverse of the atrix Cos 2 2+ is Cos 2 2+ = If we ignore the constant and ultiply this last atrix by Cos 2+ then, taking into account that the colun sus in this atrix are all equal to 2, it is clear that the i, j)th entry in the product is 2 + 2πij 2 cos 2+ ) Hence, Cos 2+ = Cos 2 2+ Cos 8 2+ = U + ) 2 Cos 2+ The explicit forulas in Theore 33 allow us to solve the syste 37) for any We shall briefly sketch the result for odd The even case is siilar, but the expressions that appear are longer The ain point here, in any case, is not the resulting explicit forulas for syetric cobinations of Z series, which ay be obtained by several other ethods, but rather those for M series see Reark 3 below) So, when is odd, Theore 33 iplies that the solutions of the syste of linear equations 37) are y r = 4 ω i= )) ) 2πir i + cos B 2k B ) 2k 2k for each r =,, ω following result This expression ay be siplified considerably by using the Proposition 34 Let 2 and k N Then ω ) i B 2k = i= { B2k 2 B 2k 2 Proof This follows fro the ultiplication forula 2k ), if is odd, 2k + 2 2k 2 ), if is even B n x) = n j=0 B n x + j/) see, for instance, [, forula 230, p 804]), which is easily proven by using the generating function and the cyclotoic equation 2

13 We then obtain the following explicit expression: Proposition 35 Let be an odd integer equal to or greater than 3 and k N Then n 2k = )k 2π) 2k 2 2 2k)! B 2k + 4 )/2 ) ) 2πir i cos B 2k n ±r ) for each r =,, )/2 As explained above, a siilar forula would arise for even, but we oit it Reark 3 We have not found the forulas in Theore 33 in the literature On the other hand, Proposition 35 can be obtained directly fro 22) by a straightforward arguent consisting of inverting the discrete Fourier transfor of -periodic even sequences see [3]) The ethod described in [3] allows one to su periodic Dirichlet series in general and does not require separate arguents according to the parity of However, the approach in [3] does not reveal the regularity of 37) and the explicit forula for the inverse of the cosine atrix In addition, since is not periodic, the ethod does not provide eleentary expressions for the x r starting fro 24), ie for the series n ±r ) n 2k The approach we give here shows that these sus ay be obtained in a siilar anner, that is, by proving the regularity of 36), using a siilar arguent to that which we have given for n 2k, n ±r ) except one ust replace the atrix Cos with B 2k, To this end, we note that by the results to be proved independently in the next section regarding the asyptotic behavior of the Bernoulli polynoials, we have ) k 2π) 2k li k 22k)! i= ) ω detb 2k,) = detcos ), and hence the regularity of the atrix Cos iplies that of B 2k,, at least for k sufficiently large this is the sall caveat we entioned) We conjecture that this is true for all k N, although a direct approach along the lines of that used for the cosine atrices does not see straightforward For instance, in [], the rank of the atrix of fractional parts {ij/}) i,j= is found to depend on the nuber of divisors of Since B x) = x /2, this is related to the odd exponent case k = of our proble To su up, recalling what x j is, we ay state the following Proposition 36 Let, k N, 3, k 0 The value of n ±r ) n 2k, r =,, /2, is π 2k ties a rational linear cobination of the values of cos2πx) at the rational arguents x = j/, j = 0,,, /2 3

14 Thus, for exaple, we have n ± 9) n 2 = 9 π 2 34 Soe rearks on the odd power case: 2k cos π 9 2 cos 4π ) 9 Since B 2k+ x) = B 2k+ x), in the odd power case we need to consider differences instead of sus Let 3 and define y i = Dk) M 2k +, i) M 2k +, i)), i =,, ω, where ω = /2 if is odd just as for the even power case of 2k) and ω = /2 if is even one less equation than for 2k); in short ω = )/2 We have two systes of linear equations analogous to 37) and 36), with atrices { })) ω ij B 2k+ ) = B 2k+, Sin = sin 2π ij )) ω i,j= i,j= This case is sipler because the square of the sine atrix is easily seen to be diagonal Proposition 37 For 3, Sin 2 = 4 I ω, where I ω is the identity atrix or order ω Reasoning in the sae way as in the even power case, we arrive at the following Proposition 38 Let, k N, 3 Then n r ) n 2k+ for each r =,, ω n r ) n 2k+ = 4 ) k 2π) 2k+ 22k + )! Proposition 39 Let, k N, 3, k 0 The value of n r ) n 2k+ n r ) ω i= n 2k+ sin 2π ir ) ) i B 2k+, is π 2k+) ties a rational linear cobination of the values sin2πx) at the rational arguents x = j/, 0 j 4

15 4 Asyptotic forulas for the Bernoulli and Euler polynoials Let us recall that we require knowing the asyptotic behavior of the Bernoulli and Euler polynoials in order to coplete the results of the previous section, by showing that the atrices we defined there in ters of the values at rational nubers of a given Bernoulli polynoial B k, are invertible for k 0 This asyptotic behavior is wellknown see [4]): and ) k 2π) 2k li B 2k x) = cos2πx), x [0, ], 4) k 22k)! ) k 2π) 2k+ li B 2k+ x) = sin2πx), x [0, ], 42) k 22k + )! ) k π 2k li k 42k )! E 2k x) = cosπx), x [0, ], 43) ) k π 2k+ li E 2k x) = sinπx), x [0, ], 44) k 42k)! the convergence being unifor on [0, ] Indeed, the result generalizes to C, with unifor convergence on copact sets Restricting ourselves to [0, ], as we have done throughout the paper, we observe that the asyptotic behavior of these polynoial failies is an iediate consequence of their Fourier expansions Moreover, the Fourier series allow one to obtain uch ore inforation regarding the degree of approxiation Thus, in this section, we will use the Fourier expansions and eleentary estiates to obtain not only the asyptotic behavior, but also explicit bounds for the differences between the polynoials and their liits, as well as for the ratios of successive differences To siplify notation here and in the results that follow, we let and B 2k x) = )k 2π) 2k 22k)! B 2k x), B2k+ x) = )k 2π) 2k+ B 2k+ x), 22k + )! Ẽ 2k x) = )k π 2k 42k )! E 2k x), Ẽ 2k x) = )k π 2k+ E 2k x) 42k)! Proposition 4 The Bernoulli polynoials satisfy B 2k x) cos2πx) < 2k + 2k B 2k+ x) sin2πx) < k + k, 22k x [0, ], k, 45), 22k+ x [0, ], k 46) Proof For 2 and s R, s >, consider the tail of the zeta series, Z s) = n= n We have the eleentary estiate s Z s) = n= n s < dx x s = s ) ) s 5

16 This estiate can be iproved by feeding it back into itself: n s = s + n= n=+ n s < s + s ) s = s + s s 47) Consider now the even index case s = 2k Separating the first ter, cos2πx), in the Fourier series 3), the reaining ters are bounded in absolute value by the tail Z 2 2k), and thus 45) follows iediately fro 47) In the sae anner, 46) follows fro 4) Proposition 42 The Euler polynoials satisfy Ẽ2k x) cosπx) < 2k + 2k, x [0, ], k, 22k+ Ẽ2kx) sinπx) < k +, x [0, ], k k 22k+2 Proof The only difference between the proof of this proposition and that of Proposition 4 is that here we need to estiate the tails of the odd zeta series Zs) = for and s > A good bound is obtained by siply observing that n= 2n+) s 2Zs) = 2+) + s 2+) + < s 2) + s 2+) + = Z s 2 s), and hence, by 47), n= 2n + ) s < 2 s + 2 s,, s > 48) 2) s This is a slightly better bound than that obtained fro the estiate Z s) < 2) s ) s + = 2 s Z s)) The result now follows fro 48) in the sae anner as Proposition 4 follows fro 47) Obviously, the asyptotic forulas 4), 42), 43) and 44), as well as the unifor convergence, follow iediately fro the previous two propositions With the sae technique we can also deterine the asyptotic rates of decrease of the error at each successive step in the approxiation of both the sine and the cosine by eans of the Bernoulli and Euler polynoials Naely, we have li k li k B 2k+2 x) cos2πx) = B 2k x) cos2πx) B 2k+3 x) sin2πx) B 2k+ x) sin2πx) = for the Bernoulli polynoials, and Ẽ 2k+ x) cosπx) li k Ẽ 2k x) cosπx) = Ẽ 2k+2 x) sinπx) li = k Ẽ 2k x) sinπx) { /4 if x [0, ] \ { 8, 3 8, 5 8, 7 8 }, /9 { if x = 8, 3 8, 5 8 or 7 8, /4 if x 0, ) \ { 4, 2, 3 4 }, /9 if x = 4 or 3 4, { /9 if x [0, ] \ { 6, 2, 5 6 }, /25 { if x = 6 or 5 6, /9 if x 0, ) \ { 3, 2 3 }, /25 if x = 3 or 2 3, for the Euler polynoials As with the asyptotic behavior, these results are an iediate consequence of the sharper explicit estiates for these quotients given below 6

17 Theore 43 For the Bernoulli and Euler polynoials, one has the following estiates Let x [0, ] \ { 8, 3 8, 5 8, 7 2k+2 8 } For k 0, specifically, when 2k 2 2k 3) < cos4πx), the quotient B 2k+2 x) cos2πx) lies between the two bounds B 2k x) cos2πx) 4 2k+4 sec4πx) 2 2k+2 2k+ 3) 2 2k, 3) ± sec4πx) 2k+2 2k where the signs are to be taken respectively on top for the lower bound) and botto for the upper bound) If x = 8, 3 8, 5 8 or 7 8, then for k 2, the quotient lies between the two bounds 9 2 2k+5 3 2k+2 2k+ 4) ± 2 2k+3 3 2k 4) 2 Let x 0, ) \ { 4, 2, 3 2k+3 4 } For k 0, specifically, when 2k 2 2k+ 3) < sin4πx), the quotient B 2k+3 x) sin2πx) lies between the two bounds B 2k+ x) sin2πx) 4 2k 2k+5 csc4πx) 2 2k+3 2k+2 3) 2 2k+ 3) ± csc4πx) 2k+3 2k If x = 4 or 3 4, then for k 2, the quotient lies between the two bounds 9 k+3 3 2k+3 k+ 4) ± k+2 3 2k+ 4) k 3 Let x [0, ] \ { 6, 2, 5 6 } For k 0, specifically, when 2 2k+3 2k 3 2k 4) < cos3πx), the quotient Ẽ2k+x) cosπx) lies between the two bounds Ẽ 2k x) cosπx) 9 sec3πx) 2 2k+5 3 2k+2 2k+ 4) ± sec3πx) 3 2k 4) 2 2k+3 2k If x = 6 or 5 6, then for k 2, the quotient lies between the two bounds 25 2k k+2 2k+ 6) ± 3 2k+5 5 2k 6) 4 Let x 0, ) \ { 3, 2 3 } For k 0, specifically, when 2 k+2 k 3 2k+ 4) < sin3πx), the quotient Ẽ2k+2x) sinπx) lies between the two bounds Ẽ 2k x) sinπx) 2k 9 csc3πx) 2 k+3 3 2k+3 k+ 4) ± csc3πx) 2 k+2 3 2k+ k 4) 7

18 If x = 3 or 2 3, then for k 2, the quotient lies between the two bounds 25 k k+3 k+ 6) ± 3 k+2 5 2k+ 6) Note that the reaining excluded values of x correspond to the cases when the polynoial and all ters in its Fourier series are null) Proof Since the techniques are the sae in all cases, we will only outline the proof of the first stateent, for the even Bernoulli polynoials Let k x) = B 2k x) cos2πx) As with the first asyptotic results, the leading ter in the Fourier series doinates the reaining ones Thus we separate k x) = B 2k x) cos2πx) = cos4πx) 2 2k + k n=3 cos2πnx) n 2k, where the leading ter is l k x) = cos4πx) By 47), the tail is bounded uniforly in x 2 2k by ɛ k = 2k+2 2k 3 2k Thus we have the approxiation k x) l k x) < ɛ k The condition x [0, ] \ { 8, 3 8, 5 8, 7 8 } eans siply that cos4πx) 0, or equivalently, l kx) 0, and we then verify that, for such a fixed x, the error ter ɛ k is always eventually saller than l k x) in absolute value In this particular case, ɛ k < l k x) translates to 2k+2 2k 2 2k 3) < cos4πx), which clearly holds when k 0 This iplies that for k 0, k x) and l k x) have the sae sign, which is the sign of cos4πx) In particular, the succesive quotients k+x) k x) are positive Then, by the triangle inequality, 0 < l k x) ɛ k < k x) < l k x) + ɛ k and hence, for k 0, l k+ x) ɛ k+ < k+ x) l k x) + ɛ k k x) = k+x) < l k+x) + ɛ k+, k x) l k x) ɛ k which, after soe algebraic anipulation, yields the bounds given in the stateent of the results above In the exceptional cases, x = 8, 3 8, 5 8 or 7 8, since l kx) = 0, we take the next ter in the series as leading ter, naely l k x) = cos6πx) 3 2k This works because in fact cos6πx) = / 2 for all these x Changing ɛ k to the estiate 47) for the new tail and proceeding as before, gives the set of exceptional bounds also stated above The other cases are dealt with in the sae anner, identifying the corresponding k, l k, ɛ k, noting that for the Euler polynoials we use the bound 48) instead of 47) Acknowledgeent We wish to thank the referee who suggested the general ideas in our proofs of Proposition 4, Proposition 42 and Theore 43, that allow one to obtain explicit constants for bounding the rate of convergence 8

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