Bernoulli, Ramanujan, Toeplitz and the triangular matrices

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1 Bernoulli, Ramanujan, Toeplitz and the triangular matrices Carmine Di Fiore, Francesco Tudisco, Paolo Zellini Dipartimento di Matematica, Università di Roma Tor Vergata April 4, 3 Abstract By using one of the definitions of the Bernoulli numbers, we prove that they solve particular odd and even lower triangular Toeplitz (ltt) systems of equations In a paper Ramanujan writes down a sparse lower triangular system solved by Bernoulli numbers; we observe that such system is equivalent to a sparse ltt system The attempt to obtain the sparse ltt Ramanujan system from the ltt odd and even systems, has led us to study efficient methods for solving generic ltt systems Such methods are here eplained in detail in case n, the number of equations, is a power of b, b, 3 and b generic Keywords: Bernoulli numbers; triangular Toeplitz matrices MSC: B68, Y55, 5A6, 5A9, 5A4, 5B5, 5B99, 65F5 Corresponding author ( address): difiore@matuniromait Introduction The j-th Bernoulli number, B j (), is a rational number defined for any j N, positive if j is odd and negative if j is even, whose denominator is known, in the sense that it is the product of all prime numbers p such that p divides j [4], and, instead, only partial information are known about the numerator [8], [39], [4] Shortly, B j (), j, could be defined by the well known Euler formula B j () ( ) j+ (j)! + (π) j k k, j involving the Zeta-Riemann function [3], [] May be the latter formula alone is sufficient to justify the past and present interest in investigating Bernoulli numbers (Bn) Note that an immediate consequence of the Euler formula is the fact that the B j () go to infinite as j diverges In literature one finds several identities involving Bn, and also several eplicit formulas for them, which may appear more eplicit than Euler formula since involve finite (instead of infinite) sums [3], [33], [4], [36], [7], [37], [6], [9] Some of such identities/formulas have been used to define algorithms for the computation of the numerators of the Bn It is however interesting to note that there are efficient algorithms for such computations which eploit directly the epression of the Bn in terms of the Zeta-Riemann function [], [38], [4], [43] See also [3], [4], [5] As it is noted in [33], the Bn appear in several fields of mathematics; in particular, the numerators of the Bn and their factors play an important role in advanced number theory (see [34], [35], [8], [8], [39]) So, wider and wider lists of the first Bn have been and are compiled, and also lists of the known factors of their numerators The updating of these lists requires the implementation of efficient primality-test/integer-factorization algorithms on powerful parallel computers For instance, by this way the numerator of B () first has been proved not prime, and then has been factorized as the product of five prime integers Two of such factors, respectively of 9 and 5 digits, have been found only very recently [9], [] A lower triangular Toeplitz (ltt) matri A is a matri such that a ij if i < j, and a i,j a i+,j+, for all i, j The product of two ltt matrices whatever order is used generates the same matri, and such matri is ltt Non singular ltt matrices have an inverse which is ltt, and thus is uniquely defined by

2 its first column Such remarks simply follow from the fact that the set of all ltt matrices is nothing else than the set {p(z)} of all polynomials in the lower-shift matri Z (δ i,j+ ), and the fact that {p(x)} is, for any choice of X, a commutative matri algebra closed under inversion Note that, given a n n ltt matri A, multiplying A by a vector (M), or solving a system whose coefficient matri is A (S), are both operations that can be performed in at most O(n log n) arithmetic operations, thus in an amount of operations significantly smaller than, for eample, the n(n + )/ multiplications required by the standard algorithms for lower triangular (non Toeplitz) matrices Such performances are possible by introducing alternative algorithms which eploit, first, the strict relationship between the Toeplitz structure and the discrete Fourier transform [3], and second, the fast implementation, known as FFT, of the latter However, for (M) and (S) it is not so clear what is the best possible alternative algorithm In particular, the algorithms performing the multiplication ltt matri vector hold unchanged if the ltt is replaced by a generic (full) Toeplitz matri; so one guesses that better algorithms may be introduced, ad hoc for the ltt case Analogously, a widely known eact algorithm able to solve ltt systems (or, more precisely, to compute the first column of the inverse of a ltt matri) in at most O(n log n) ao, has essentially a recursive character, which is not so convenient from the point of view of the space compleity [4] In order to avoid such drawback, however, one could use approimation inverse algorithm [], [] See also [5], [6], [7], [8], [9], [], and the references in [] In this paper we emphasize the connection (may be also noted elsewhere, see fi [4]) between Bernoulli numbers and lower triangular Toeplitz matrices This connection will finally result into new possible algorithms for computing simultaneously the first n Bernoulli numbers More precisely, we prove that the vector z ( B j () j /(j)! ) + j, R (B () ), solves three type I ltt semi-infinite linear systems A f, named even, odd and Ramanujan, respectively To such systems correspond other three systems, of type II, solved by the vector Z T z ( B j () j /(j)! ) + j Type I ad II ltt systems have been obtained as follows: - Introducing/considering three particular lower triangular systems solved by Bernoulli numbers The first two, which we may call almost-even and almost-odd, are introduced by eploiting a well known power series epansion involving Bernoulli polynomials It is interesting to note that the coefficient matrices of such systems are particular submatrices of the lt Tartaglia matri The third one, the almost-ramanujan system, is simply deduced from the equations, solved by the absolute values of the first Bn, listed by Ramanujan in the paper [3] - Noting that the almost-even, almost-odd, and almost-ramanujan systems are structured in such a way that their coefficient matrices can be forced to be Toeplitz This result follows, for the first two systems, from the matri series representation of the Tartaglia matri in terms of powers of a kind of regularly weighted lower shift matri, and, for the third one, by a remarkable remark proved in the case, and conjectured in the general case - Proving that each of the three ltt systems so obtained (even, odd and Ramanujan), which is solved by z (or Z T z), can be manipulated so to define a correspondent ltt system whose solution is Z T z (or z) The Ramanujan lt system in [3] has the remarkable peculiarity to have two null diagonals alternating the nonnull ones The same peculiarity is inherited by its Toeplitz version, obtained in this paper (see (), ()) For some time we have tried to obtain by linear algebra arguments the system in [3] as a consequence of our odd and even systems, also with the aim to learn a technique for introducing a system possibly more sparse than and as simple as the Ramanujan one and, above all, its Toeplitz version In order to do that, first of all it was necessary to nullify the second, the third, the fifth, the sith, the eigth, the nineth, and so on, diagonals of our odd and even systems At that time we conceived the idea of a fast direct (not recursive) solver of ltt systems In fact, the process of making null two diagonals every one, could be repeated, so to finally transform the initial ltt into the identity matri Moreover, each step of such sort of Gaussian elimination procedure could be realized by a left multiplication by a suitable ltt matri These remarks led us to conceive a O(n log 3 n) solver of ltt systems A f where A is n n with n 3 s, and then to etend the result, obtaining analogous low compleity algorithms, ad hoc for the cases n b s, b and b

3 generic Such eact algorithms are described in the present paper in detail, since we believe that, for their not recursive character and for their clearness, they could be competitive with any known O(n log n) ltt systems solver [], [3], [], [6], [5], [4], [7], [5] In particular, as a first test, the 3 s 3 s -algorithm could be applied to the Toeplitz versions (9), (), () of the Ramanujan system (6) [3], in order to compute the vector {z} n that contains the first n 3 s Bernoulli numbers in at most O(n log 3 n) ao (assuming already computed the entries of A and f) Note that the first step of the algorithm can be in this case skipped, as it has been already performed eplicitly by Ramanujan Lower triangular Toeplitz matrices (ltt) Let Z be the following n n matri Z Z is usually called lower-shift due to the effect that its multiplication by a vector v [v v v n ] T C n produces: Zv [ v v v n ] T Let L be the subspace of C n n of those matrices which commute with Z It is simple to observe that L is a matri algebra closed under inversion, that is if A, B L then AB L and if A L is nonsingular then A L Let us investigate the structure of the matrices in L Let A C n n Then AZ a a n a n a nn, ZA a a n a n a n n Forcing the equality between AZ and ZA we obtain the conditions a a 3 a n a n a n,n and a i,j+ a i,j, i,, n, j,, n, from which one deduces the structure of A L: A must be a lower triangular Toeplitz (ltt) matri, ie of the type A a a a a 3 a a a n a a () It follows that dim L n and that, by a well known general result [4], L can be represented as the set of all polynomials in Z, ie L {p(z) : p polynomials} Actually, by investigating the powers of Z one realizes that the matri A in () is eactly the polynomial n k a kz k Note also that, as a consequence of the above arguments, the inverse of a ltt matri is still ltt, thus it is completely determined as soon as its first column is known In the net section we will illustrate an efficient algorithm for the solution of a lower triangular Toeplitz linear system A f, A L, where n s We will show that such operation can be realized trough O(log n) matri-vector products, where the matrices involved are ltt and their dimension is j j, with j,, s Since such products require no more than cj j arithmetic operations (see Appendices A, B) the overall compleity of the proposed algorithm is O(n log n) 3

4 An algorithm for the solution of a lower triangular Toeplitz linear system of n equations, where n is a power of In this section we present an algorithm of compleity O(n log n) for the computation of such that A f, where A is a n n lower triangular Toeplitz matri, with n power of and [A] Preliminary Lemmas Given a vector v [v v v ] T, v i C (briefly v C N ), let L(v) be the semi-infinite lower triangular Toeplitz matri whose first column is v, ie L(v) v k Z k, Z k Lemma Let a, b, c be vectors in C N Then L(a)L(b) L(c) if and only if L(a)b c Proof If L(a)L(b) L(c), then the first column of L(a)L(b) must be equal to the first column of L(c), and these are the vectors L(a)b and c, respectively Conversely, assume that L(a)b c and consider the matri L(a)L(b) It is lower triangular Toeplitz being a product of lower triangular Toeplitz matrices, and, by hypothesis, its first column L(a)b coincides with the vector c, which in turn is the first column of the lower triangular Toeplitz matri L(c) The thesis follows from the fact that ltt matrices are uniquely defined by their first columns Given a vector v [v v v ] T C N, let E be the semi-infinite matri with entries or, which maps v into the vector Ev [v v v ] T : E In other words, the application of E to v has the effect of inserting a zero between two consecutive components of v It is easy to observe that E, E s, s, that is the application of E s to v has the effect of inserting s zeros between two consecutive components of v 4

5 Lemma Let u and v be vectors in C N with u v Then L(Eu)Ev EL(u)v, and, more in general, for each s N, L(E s u)e s v E s L(u)v Proof By inspecting the vectors L(Eu)Ev and EL(u)v one observes that they are equal By multiplying E on the left of the identity L(Eu)Ev EL(u)v and using the same identity also for the vectors Eu and Ev, in place of u and v respectively, one observes that it also holds L(E u)e v E L(u)v And so on The algorithm Let A be a n n ltt matri, with n power of and [A] Assume we want to solve the system A f The algorithm presented below eploits the fact that A is still a n n ltt matri Compute the first column of the ltt matri A by solving the particular linear system A e via the algorithm () of compleity O(n log n) shown in the net section, based upon Lemmas, and their repeated application Compute the ltt matri-vector product A f with no more than O(n log n) arithmetic operations (see Appendices A and B) 3 The computation of the first column of the inverse of a n n ltt matri, where n is a power of For the sake of readability here we present the algorithm for the computation of such that A e in the particular case n 8 When suitable we briefly discuss the general case n s, s N; nevertheless such case can be easily deduced from the considered one, and is reported in detail in Appendi C The algorithm consist of two parts In the first one particular ltt matrices are introduced and computed, with the property that their successive left multiplication by the matri A transforms A into the the identity matri In the second part such matrices are successively left multiplied by the vector e As it will be clear throughout what follows, the method is nothing more than a kind of Gaussian elimination, where diagonals are nullified instead of columns The overall cost of O(n log n) comes from the fact that at each step of the first part a half of the remaining non null diagonals are nullified, and from the fact that in the second part the computations can be simplified by eploiting the structure of e, which has only one nonzero component First of all observe that the 8 8 matri A can be thought as the upper-left submatri of a semi-infinite ltt matri L(a), whose first column is [ a a a 7 a 8 ] T Step Look for â such that L(a)â a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a â â â 3 â 4 â 5 â 6 â 7 a () a () a () 3 Ea () for some a () i C, and compute such a () i The computation of a () i requires, once â is known, one ltt 8 8 ( s s ) matri-vector product or, more precisely, two ltt 4 4 ( s s ) matri-vector products We will see that â is actually available with no computations 5

6 Note that, due to Lemma, we have L(â)L(a) L(Ea () ), that is the ltt matri L(a) is transformed into a ltt matri which alternates to each nonnull diagonal a null one Step Look for â () such that a () a () L(Ea () )Eâ () a () a () a () a () a () 3 a () a () a () 3 a () a () â () â () â () 3 a () E a () for some a () i C, and compute such a () i The computation of a () i requires, once â () is known, one ltt 4 4 ( s s ) matri-vector product or, more precisely, two ltt ( s s ) matri-vector products Note that, due to Lemma, we have L(Eâ () )L(Ea () ) L(E a () ), that is the ltt matri L(a) is transformed into a ltt matri which alternates to each nonnull diagonal three null ones Also note that, due to Lemma, if L(a () )â () Ea () then L(Ea () )Eâ () E a () We will see that â () such that L(a () )â () Ea () is actually available with no computations Step 3 Look for â () such that L(E a () )E â () a () a () a () a () â () E 3 a (3) for some a (3) i C, and compute such a (3) i The computation of a (3) i requires, once â () is known, one ltt ( s s ) matri-vector product or, more precisely, two ltt ( s 3 s 3 ) matri-vector products That is, no operation in our case n 8, where no entry a (3) i, i, is needed Note that, due to Lemma, we have L(E â () )L(E a () ) L(E 3 a (3) ), that is the ltt matri L(a) is transformed into a ltt matri which alternates to each nonnull diagonal seven null ones Also note that, due to Lemma, if L(a () )â () Ea (3) then L(E a () )E â () E 3 a (3) We will see that â () such that L(a () )â () Ea (3) is actually available with no computations Proceed this way, if n s > 8 Otherwise stop, the first part of the algorithm is complete Summarizing, we have proved that L(E â () )L(Eâ () )L(â)L(a) L(E 3 a (3) ) () where the upper left 8 8 submatrices of L(a) and of L(E 3 a (3) ) are the initial lower triangular Toeplitz 6

7 matri A and the identity matri, respectively: a L(a) a 7 a, L(E 3 a (3) ) a 8 a 7 a a (3) The operations we did so far are: 8 8 ltt vector ltt vector (if A were n n with n s the operations required would have been: s s ltt vector ltt vector) Now let us move to our main purpose, compute the first column of A, and thus let us show the second part of the algorithm Consider the following semi-infinite linear system: L(a)z E v (3) where v is a generic semi-infinite vector in C N (if A is n n with n s, then the matri E in (3) must be raised to the power s rather than ) Such system can be rewritten as follows v [ ] A O {z} 8 z 8 v v that is, pointing out the upper part of the system, consisting of only 8 equations Before proceeding further, let us note that {z} 8 is such that A{z} 8 [v v ] T, v, v C Therefore the choices v and v, would make {z} 8 equal to the vector we are looking for, A e By using the identity () one immediately observes that the system L(a)z E v is equivalent to the following one [ ] [ ] I8 O {z}8 L(E 3 a (3) )z L(â)L(Eâ () )L(E â () )E v Due to Lemma we can rewrite the right hand side in a more convenient way: L(â)L(Eâ () )L(E â () )E v L(â)L(Eâ () )E L(â () )v L(â)EL(â () )EL(â () )v Therefore, the following identity holds: [ ] [ I8 O {z}8 ] L(â)EL(â () )EL(â () )v All the matrices involved on the right hand side are lower triangular Moreover, the upper left square 7

8 submatrices of E of dimensions 8 8, 4 4 have half of its columns null, for eample {E} 4, {E} 8 These two observations let us obtain an effective representation of {z} 8 : {z} 8 {L(â)} 8 {E} 8 {L(â () )} 8 {E} 8 {L(â () )} 8 {v} 8 {L(â)} 8 {E} 8,4 {L(â () )} 4 {E} 4, {L(â () )} {v} By using such formula, when v, v, the vector {z} 8 can be computed by performing the operations 4 4 ltt vector ltt vector (if A is n n with n s the operations required would have been 4 4 ltt vector + + s s ltt vector), that is, as many operations as the Gaussian elimination, the first part of the algorithm In conclusion, if cj j is an upper bound for the cost of the product j j ltt vector, then the overall cost of the shown algorithm is c s j jj O(s s ) O(n log n) for an n n matri A with n s We still have to prove that the vector â such that L(a)â Ea () is indeed available with no computations To this aim it is sufficient to observe that a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a L(a) ( e + ( ) i ) a ie i+ e + i This can be verified by a direct calculation 4 Observations on the algorithm s core i a a a 3 a 4 a 5 a 6 a 7 a 8 a 9 a a a 4 a a 3 + a a 6 a a 5 + a a 4 a 3 a 8 a a 7 + a a 6 a 3a 5 + a 4 i δ i mod ( ai + ( ) j ) a ja i j ei+ j, (4) Given the vector a the problem of the computation of â such that L(a)â Ea (), for some a () is indeed a polynomial arithmetic problem In fact, due to Lemma, the identity L(a)â Ea () is equivalent to the equality L(a)L(â) L(Ea () ), ie ( k a k Z k )( k â k Z k ) k a () k Zk Therefore the polynomial arithmetic problem can be stated as follows: 8

9 given a(z) + k a kz k, find a polynomial â(z) + k âkz k such that for some coefficients a () i â(z)a(z) a () + a () z + a () z4 + : a () (z) Such problem is a particular case of the more general problem: transform a full polynomial a(z) into a sparse polynomial a () (z) + k a() k zrk, for a fied r N It is possible to describe eplicitly a polynomial â(z) that realizes such transformation, in fact the following theorem holds Theorem 3 [] Given a(z) + k a kz k, set â(z) a(zt)a(zt ) a(zt r ) where t is a r-th principal root of the unity (t C, t r, t i for < i < r) Then â(z)a(z) a () + a () zr + a () zr + : a () (z) for some a () i Moreover, if the coefficients of a are real, then the coefficients of â are real Let us consider two Corollaries of such Theorem For r we have â(z) a( z), that is we regain the result (4) It is clear that a( z)a(z) a () + a () z + a () z4 + (compare with [] and the references therein) In this case the coefficients of â are available with no computations, we only need to compute the new coefficients a () i For r 3 we have â(z) a(zt)a(zt ), t e i π 3 By Theorem 3 the following equalities a(z)a(zt)a(zt ) a () + a () z3 + a () z6 + and L(â)L(a) L(Ea () ), E hold, and the coefficients of â(z) a(zt)a(zt ) are real, provided that the ones of a are This time, the coefficients of â are not easily readable from the coefficients of a In order to compute them we observe that the polynomial equality â(z) a(zt)a(zt ) is equivalent to the matri identity L(â) L(c)L(d), c k a k t k, d k a k t k, and therefore we get the following formula â L(c)d R[L(c)]R[d] I[L(c)]I[d] (5) where the last equality holds only if the coefficients of a are real Later on we will describe an algorithm for the solution of systems A f where A is 3 s 3 s ltt analogous to the one presented before, but using vectors â such that the components in positions, 3, 5, 6, 8, 9, of L(a)â are null Thanks to Theorem 3, we have an eplicit formula (5) for such vectors, in terms of the product of a triangular Toeplitz matri by a vector 3 Bernoulli numbers and triangular matrices, 3 Bernoulli numbers and polynomials The conditions B( + ) B() n n, B() d, B() polinomio 9

10 uniquely define the function B() It is a particular degree n monic polynomial called n-th Bernoulli polynomial and usually denoted by the symbol B n () It is simple to compute the first Bernoulli polynomials: B (), B () ( ) + 6, B 3() ( )( ), B () is assumed equal to It can be proved that Bernoulli polynomials define the coefficients of the power series representation of several functions, for instance to our aim it is useful to recall that the following power series epansion holds: te t + e t n B n () t n (6) n! Moreover, Bernoulli polynomials satisfy many identities Among all we recall the following ones, concerning the value of their derivatives and their property of symmetry with respect to the line : B n() nb n (), B n ( ) ( ) n B n () It is simple to observe as a consequence of their definition and of the last identity that all the Bernoulli polynomials with odd degree (ecept B ()) vanish for On the contrary, the value that an even degree Bernoulli polynomial attains in the origin is different from zero and especially important In particular, recall the following Euler formula ζ(j) B j() (π) j, ζ(s) (j)! k which shows the strict relation between the numbers B j () and the values that the Riemann Zeta function ζ(s) attains over all even positive integer numbers j [], [3] For instance, from such relation and from the fact that ζ(j) if j +, one deduces that B j () tends to + almost the same way as (j)!/(π) j does Another important formula involving the values B j () is the Euler-Maclaurin formula [], which is useful for the computation of sums: if f is a smooth enough function over [m, n], m, n Z, then n f(r) n [f(m) + f(n)] + f() d + rm m k j k s, B j () (j)! [f (j ) (n) f (j ) (m)] + u k+, (7) where u k+ (k + )! (k)! (k + )! n m n m n f (k+) ()B k+ () d f (k) ()B k () d m f (k+) ()[B k+ () B k+ ()] d and B n is B n [,) etended periodically over R Let us recall that the Eulero-Maclaurin formula also leads to an important representation of the error of the trapezoidal rule I h h[ g(a) + n r g(a + rh) + g(b)], h b a n, in the approimation of the definite integral I b g() d Such representation, holding for a functions g which are smooth enough in [a, b], is obtained by setting m and f(t) g(a + th) in (7): I h I + k j h j B j () (j)! [g (j ) (b) g (j ) (a)] + r k+, r k+ g(k+) (ξ)h k+ (b a)b k+ (), (8) (k + )!

11 ξ (a, b) Such representation of the error, in terms of even powers of h, shows the reason why the Romberg etrapolation method for estimating a definite integral is efficient, when combined with trapezoidal rule From (8) it is indeed clear that Ĩh/ : ( I h/ I h )/( ) approimates I with an error of order O(h 4 ), whereas the error made by I h and I h/ is of order O(h ) For these and many other reasons (see for instance [8], [34], [33], [], [3]), the values B j () have their own name: Bernoulli numbers 3 Bernoulli numbers solve triangular Toeplitz systems From (6) it follows that Bernoulli numbers satisfy the following identity t + e t t + k B k () (k)! tk Multiplying the latter by e t, epanding e t in terms of powers of t, and setting to zero the coefficients of t i of the right hand side, i, 3, 4,, yields the following equations: j ] [ j + k ( ) j B k k (), j, 3, 4, (9) Now, putting together equations (9) for j even and for j odd, we obtain two lower triangular linear systems that uniquely define Bernoulli numbers: ( ) ( ) ( ) 4 4 B () ( ) ( ) ( ) B () B 4 () 4 ( ) ( ) ( ) ( ) B 6 () 3 4, 4 6 ( ) ( ) ( ) 3 3 B () ( ) ( ) ( ) B () B 4 () 4 ( ) ( ) ( ) ( ) B 6 () 3/ 5/ 7/ 4 6 From such systems we can for instance easily compute the first Bernoulli numbers:, 6, 3, 4, 3, 5 66, 69 73, 7 6, () Now we want to obtain an analytic representation for the coefficients matrices W e and W o of such linear systems To this end it is enough to observe that W e and W o are suitable submatrices of the Tartaglia matri X, which can be represented as a power series More precisely, set Y 3, φ 3,, 3 4, 3 5 6,, 56

12 and note that from the equality ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) X : 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) k k! Y k, ( ) which holds because [X] ij (i j)! [Y i j ] ij (i j)! j (i )(i ) i, j i n, it follows j that W e Z T φ (k + )! φk, W o k k (k + )! φk We can therefore rewrite the two linear systems solved by Bernoulli numbers as follows: B () B () (k + )! φk b q e, b B 4 () k B 6 (), /3 qe /5 /7, () k (k + )! φk b q o, q o / / / () Now, let us show that systems () and () are equivalent to two lower triangular Toeplitz linear systems Our aim is to replace φ, a matri whose subdiagonal entries are all different, by a matri whose subdiagonal entries are all equal Set D diag (d, d, d 3, ), d i By investigating the nonzero entries of the matri DφD, it is easy to observe that it can be forced to be equal to a matri of the form Z; just choose d k k d /(k )!, k,, 3, So, if D! 4! n (n )!, (3)

13 we have the equality DφD Z Then from () it follows Dq e k (k + )! Dφk D Db (k + )! (DφD ) k Db, k that is and analogously from () it follows k k (k + )! Zk Db Dq e, (4) k k (k + )! Zk Db Dq o (5) Summarizing, let z be the vector Db Then the vector {b} n whose entries are the first n Bernoulli numbers can be obtained by a two-phase procedure: compute the first n components of the solution of the lower triangular Toeplitz system (4) (or (5)), ie {z} n such that { + k k (k+)! Zk } n {z} n {Dq e } n (or { + k k (k+)! Zk } n {z} n {Dq o } n ) solve the linear system {D} n {b} n {z} n over the rational field Observe that the computation in phase can be done by means of the algorithm described alongside the previous section at a computational cost of O(n log n), and that such algorithm can be made numerically stable by a suitable choice of the parameter For instance, the choice (π) would ensure the sequence z n n (n )! B n (), n N, to be bounded; indeed in this case z n if n +, due to Euler formula So, in phase, one obtains n machine numbers which are good approimations over R of the quantities s B s ()/(s)!, s,,, n Then, in phase, one should find a way to deduce, from the machine numbers obtained, the rational Bernoulli numbers B s (), s,,, n 33 The Ramanujan lower triangular Toeplitz linear system solved by Bernoulli numbers In [3] Ramanujan writes eplicitly sparse equations solved by the Bernoulli numbers B (), B 4 (),, B () They are the first of an infinite set of sparse equations solved by all the Bernoulli numbers The Ramanujan equations all together form a lower triangular system which, according to our notations and definitions, can be rewritten as follows B () B 4 () B 6 () B 8 () B () B () B 4 () B 6 () B 8 () B () B () (6) 3

14 For eample, by using the last but three equations of such system, from the Bernoulli numbers B (),, B 6 () listed in (), the following further Bernoulli numbers can be easily obtained: B 8 () , B () , B () Let R be the semi-infinite coefficient matri of the above Ramanujan system By recalling the definition of the semi-infinite lower shift matri Z and of the semi-infinite vector b [B () B () B 4 () ] T, the Ramanujan system can be shortly indicated as R(Z T b) f, where f [f f f 3 ] T obviously denotes the right hand side vector in (6) Apparently the non-zero entries of R are not related with each other, and it seems so also for the entries of f That is, it seems to be not possible to guess, just by looking at the above equations, the twelfth equation of the Ramanujan system We can only guess that the non-zero entries of R are in the same positions as the non-zero entries of a lower triangular Toeplitz matri R of the form + k w kz 3k, and, may be, it is possible to guess the sign of the entries of f Actually it is not difficult to note that the following identity must hold RΛ Λ R, Λ Z T DZ! 4! 3 6! (7) where D is defined in (3) and R is the following lower triangular Toeplitz matri: 3 8!3 3 8!3 R 3k 3 (6k + )!(k + ) Z3k 8!3 6 k 4!5 3 8!3 6 4!5 3 8!3 6 4!5 3 8!3 9!7 6 4!5 3 8!3 9!7 6 4!5 3 8!3 In fact it is easy to check that the upper left submatri of RΛ coincides with the upper left submatri of Λ R Assuming that the conjecture (7) is true, we have that R(Z T b) f iff RΛ (ΛZ T b) f iff Λ R(Z T Db) f iff R(Z T Db) Z T DZf (8) Thus, the vector Z T Db solves a lower triangular Toeplitz system which is more sparse than the ltt systems (4), (5), since in its coefficient matri two null diagonals alternate the nonnull ones Such Ramanujan ltt system will be defined more precisely in the following (see (), ()) 34 A unifying theorem with 6 ltt linear systems solved by Bernoulli numbers In this section we collect in a Theorem three ltt linear systems solved by the vector D b, say of type I, and the corresponding ltt linear systems solved by the vector Z T D b, say of type II (D is the matri D 4

15 in (3)) In fact, till now, we have only found two systems of type I, the even and odd systems (4) and (5), and, partially, one system of type II, the Ramanujan ltt system (8) (note that for the latter system only the coefficient matri has been written eplicitly) In the following, first we state a Proposition which allows one to state a system of type II from a system of type I, and viceversa Then we state the Theorem, with the si ltt linear systems solved by Bernoulli numbers, and we prove it by applying the Proposition to the even, odd, and Ramanujan ltt systems found till now, and, in the same time, by completing the definition of the Ramanujan ltt system Proposition 3 Let Z n and Z n be the upper-left (n ) (n ) and n n submatrices of the semiinfinite lower-shift matri Z, respectively Assume that, for some α j and f j (or w j ), the following equality holds: B () η n! α jzn j B() µ w! 4! B4() f α (! 4! j f + B () α w ) 4! w, n B ((n ))! (n )() n ((n ))! fn α n n ((n ))! wn where η and µ are arbitrary parameters Then we have n B()! α jz j! 4! n B4() f 4! f j n B ((n ))! (n )() n ((n ))! fn (! w 4! w n ((n ))! wn Also the converse is true provided that α B () η + B ()µ (or α B () w ) [ ] T Proof Eploit the equality Z n The details are left to the reader e Z n Theorem 3 Z Set, a a a a, + L(a) a i Z i Let d(z) be the diagonal matri with z i as diagonal entries Set b [B () B () B 4 () ] T, i B() a a a a a a a 3 a a a D diag ( i, i,,, ), R, (i)! where B i (), i,,,, are the Bernoulli numbers Then the vectors D b and Z T D b solve the following ltt linear systems α α α n L(a) (D b) D q, (9) L(a) (Z T D b) d(z)z T D q, () where the vectors a (a i ) + i, q (q i) + i, and z (z i) + i, can assume respectively the values: a R i i δ i mod 3 (i + )!( 3 i + ), qr i (i + )(i + ) ( δ 3 i mod ), i,,, 3, 3 zi R δ i mod 3 i,, 3,, 3i +, 5 ) ()

16 a o i a e i i (i + )!, qe i, i,,, 3, i + zi e i () i +, i,, 3,, i (i + )!, i,,, 3,, qo, qo i, i,, 3, zi o i (3) i +, i,, 3, Proof From the Ramanujan semi-infinite ltt linear system (8), we obtain the following finite linear system! B ()! n j α jz j 4! n B f 4() 4! f, α j δ j j mod 3, (j+)!( 3 j+) n ((n ))! B (n )() n ((n ))! f n (4) f 6, f 3, f 3 4, f 4 45, f 5 3, f , f 7, f 8, 36 f , f 3, f 55, Then, by Proposition 3, if η + B ()µ α B (), we have that B () η n α jzn j B()!! 4! B4() f 4! j f n B ((n ))! (n )() n ((n ))! fn or, more precisely, that (I + 8!3 3 Z 3 + 4!5 6 Z 6 +!7 9 Z 9 + ) B () B()! B4() 4! 3 B6() 6! 4 B8() 8! 5 B()! 6 B()! + B ()! 6 ( ) 4! ! 4 8!3 4 8! 45 5 ( )! !5 7 4! 8 ( ) 6! !7! 3 ( )! 55 6! 9 8! µ α α α n, ( )! ( )! 3 ( ) 4! ( ) 6! 74 5! ( 4 8! ( 95 ) ) ( )! ! ( 58 ) 8 6! ( ) ( ) 8! 9! ( )! ( ) 3 38 The latter equality is a clever remark that allows us to prove that D b must solve the following Ramanujan ltt system of type I: α j Z j D b D q R, (5) j j α j δ j mod 3 (j + )!( 3 j + ), qr j (j + )(j + ) ( δ 3 j mod 3 ), j,,, 3, 6

17 Note that from the eplicit epression of q R just obtained, it follows an eplicit epression for the entries f i of the original Ramanujan system (6), ie f i (i + )(i + ) ( δ 3 i mod 3 δ i mod 3 i,, 3, 3i + ), Note also that (4) can be rewritten as n α j Z j n I n D b diag (z i, i,,, n )In D q R j for suitable z i (the meaning of In is clear from the contet) Such z i are easily obtained by imposing the equality 3 ( δ i mod 3 δ i mod 3 3 i + ) z 3 i( δ i mod 3 ), which leads to the formula: 3 i+ 3 δ i mod 3 z i δ i mod 3 δ i mod 3 3 i + So, the ltt type I and type II systems (9), () and () hold Now let us consider the finite versions of the even and odd systems (4) and (5), n j and apply to them Proposition 3: n j n j j (j + )! Zj n j (j + )! Zj ni nd b I nd q e, j (j + )! Zj n! B() 4! B4() n ((n ))! B (n )()! B() 4! B4() n ((n ))! B (n )() From the above identities it follows that n j n j j (j + )! Zj n j (j + )! Zj n! B() 4! B4() n ((n ))! B (n )()! B() 4! B4() n ((n ))! B (n )() n j! 3 4! 5 n ((n ))!! 4! n ((n ))! j (j + )! Zj ni nd b I nd q o, n! 43 4! ! 87 B() B() n n ((n ))! n(n )! 3 3 4! ! 7 n n 3 ((n ))! (n ) So, also even and odd type II linear systems (), () and (3) hold 4! 6! n (n)! 3! 5! n (n )! 4! 4 6! 3 6 8! n (n ) (n)! 3! 3 5! 3 5 7! n n 3 (n )!, i diag ( i +, i n )I nd q e, diag ( i i +, i n )I nd q o 7

18 35 On the need of a new algorithm for the solution of ltt linear systems Now it is clear that the first n Bernoulli numbers b i, unless the factors (D ) ii, solve lower triangular Toeplitz systems A f, where A is the n n upper left submatri of the semi-infinite matri L(a) in (9) (or ()) Of course one can compute the (D b) i via the algorithm described in Section, well defined for n s By representing the first column of the lower triangular Toeplitz matri A in a row, the first part of such algorithm, ie the part in which A is transformed into the identity matri, can be schematized through the following steps: (O(n log n)) (four steps if, for eample, n 6) It is clear that the algorithm works very well if applied to the even and odd type I and type II ltt systems, but it does not appear the best possible algorithm to solve the Ramanujan type I and type II ltt linear systems, for instance the system (5) A better algorithm would clearly be one whose first part could be schematically represented as follows: (O(n log 3 n)) (three steps if, for eample, n 7) In other words, each step would consist of nullifying /3 of the still remaining nonzero diagonals, instead of nullifying half of them Such algorithm, moreover, would require one step less when applied to the Ramanujan ltt linear systems (9), (), (), (5) Now, it is possible to introduce such algorithm, well defined for n 3 s ; it is presented in the following last section of this work Then, in Appendi C, a general ltt linear system solver is described, well defined when n b s, which includes the previously shown cases n s, n 3 s as particular cases 4 An algorithm for solving a system of n linear lower triangular Toeplitz equations, with n power of 3 In this section it is shown an algorithm which computes such that A f, being A a lower triangular n n Toeplitz matri with n power of 3 and [A] Its computational cost is O(n log 3 n) We need to rewrite Lemma in a version suitable for the case n power of 3 Given a vector v [v v v ] T C N, let E be the semi-infinite - matri which maps v into the vector Ev [v v v ] T : E 8

19 In other words, the action of E over v has the effect of introducing two zeros between two successive components of v Observe that E, E s, 3 s, that is, the action of E s over v has the effect of introducing 3 s zeros between two successive components of v Lemma 4 Let u, v be vectors of C N with u v Then L(Eu)Ev EL(u)v, and, more in general, for any s N we have L(E s u)e s v E s L(u)v Proof Proceed as in the case n power of 4 The algorithm Let A be a ltt n n matri with n power of 3 and [A] We want to solve the system A f by a procedure consisting of two parts: Compute the first column of the ltt n n matri A, ie solve the particular ltt linear system A e by using the algorithm of computational cost O(n log 3 n) shown in the following section, based upon the successive application of Lemmas and 4 Compute the ltt matri vector product A f performing no more than O(n log 3 n) arithmetic operations (see Appendices A and B) 4 The computation of the first column of the inverse of a ltt n n matri with n power of 3 For the sake of simplicity let us present the algorithm for the computation of such that A e when n 9, underlining, however, what are the significant changes in the general case n 3 s, s N See the Appendi C, if interested in the details of the general case The algorithm is similar to the one shown for n power of The overall cost O(n log 3 n) of the algorithm comes from the fact that, at each step of the first part, /3 of the nonzero diagonals are nullified, and from the fact that the second part can be simplified by noting that the vector e has only one nonzero component First of all observe that the 9 9 matri A can be seen as the upper left submatri of a semi-infinite lower triangular Toeplitz matri L(a) whose first column is [ a a a 7 a 8 a 9 ] T 9

20 Step Find â such that L(a)â a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a â â â 3 â 4 â 5 â 6 â 7 â 8 a () a () Ea () for some a () i C and compute such a () i The computation of a () i requires, once â is known, one 9 9 (3 s 3 s ) ltt matri vector product or, more precisely, three 3 3 (3 s 3 s ) ltt matri vector products; the computation of â requires one 9 9 (3 s 3 s ) ltt matri vector product (see (5)) Note that, due to Lemma we have then that L(â)L(a) L(Ea () ), that is the ltt matri L(a) is transformed into a ltt matri which alternates to each nonzero diagonal two null diagonals Step Find â () such that a () a () L(Ea () )Eâ () a () a () a () a () a () a () a () â () â () E a () for some a () i C and compute such a () i The computation of a () i requires, once â () is known, one 3 3 (3 s 3 s ) ltt matri vector product or, more precisely, three (3 s 3 s ) ltt matri vector products That is, no operation in our case n 9, where no entry a () i, i, is needed Note that due to Lemma, we have that L(Eâ () )L(Ea () ) L(E a () ), ie the ltt matri L(a) is transformed into a ltt matri which alternates to each nonzero diagonal eight null diagonals Also note that, due to Lemma 4, if L(a () )â () Ea () then L(Ea () )Eâ () E a () The computation of â () such that L(a () )â () Ea () requires one 3 3 (3 s 3 s ) ltt matri vector product (see (5)) Proceed this way, if n 3 s > 9 Otherwise the first part of the algorithm is complete Summarizing, we have shown that, L(Eâ () )L(â)L(a) L(E a () ) (6) where the upper left 9 9 submatrices of L(a) and of L(E a () ) are the lower triangular Toeplitz matri

21 initially given A and the identity matri I, respectively, a L(a) a 7 a, L(E a () ) a 8 a 7 a a 9 a 8 a 7 a a (), and the operations we did so far are: two products 9 9 ltt matri vector + one product 3 3 ltt matri vector (if A were n n with n 3 s the operations required would have been: two products 3 s 3 s ltt matri vector + + two products 9 9 ltt matri vector + one product 3 3 ltt matri vector) Now let us move to our purpose, compute the first column of A, and thus let us show the second part of the algorithm Consider the following semi-infinite linear system: L(a)z Ev (7) where v is a generic semi-infinite vector in C N (if A is n n with n 3 s, then the matri E in (7) must be raised to the power s ) Such system can be rewritten as follows v [ ] A O {z} v 9 z 9, v v 3 that is, pointing out the upper part of the system, consisting of only 9 equations Before proceeding further, let us note that {z} 9 is such that A{z} 9 [v v v ] T, v, v, v C Therefore the choices v and v v, would make {z} 9 equals to the vector we are looking for, A e By using the identity (6) one immediately observes that the system L(a)z Ev is equivalent to the following one: [ ] [ ] I9 O {z}9 L(E a () )z L(â)L(Eâ () )Ev Due to Lemma 4 we can rewrite the right hand side in a more convenient way: L(â)L(Eâ () )Ev L(â)EL(â () )v Therefore, the following identity holds: [ ] [ I9 O {z}9 ] L(â)EL(â () )v

22 The matrices involved on the right hand side are all lower triangular submatrices of E of dimensions 9 9, 3 3 have /3 of its columns null, {E} 3, {E} 9 These two observations let us obtain an effective representation of {z} 9 : Moreover the upper left square {z} 9 {L(â)} 9 {E} 9 {L(â () )} 9 {v} 9 {L(â)} 9 {E} 9,3 {L(â () )} 3 {v} 3 By using such formula, when v, v v, the vector {z} 9 can be computed by performing a 9 9 ltt matri vector product (if A is n n with n 3 s the operations required would have been one product 9 9 ltt matri vector + + one product 3 s 3 s ltt matri vector), that is, about the same amount of operations required by the Gaussian elimination implemented in the first part of the algorithm In conclusion, if cj3 j is an upper bound for the cost of the product 3 j 3 j ltt matri vector, then the overall cost of the shown algorithm is c s j j3j O(s3 s ) O(n log 3 n), in case the dimension of the ltt system is n 3 s Finally observe that a formula more eplicit than (5) can be given for the entries of a vector â such that L(a)â Ea () It is reported here below: L(a)â [ â i a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a () a () a () 3 i r a r a i r + δ i mod a i + 3 { s 3 i 6 s 6 i 6 ] T Ea (), a a + a a 3 a a a 4 a a 3 + a a 5 + a a 4 a a 3 a 6 a a 5 a a 4 + a 3 a 7 a a 6 + a a 5 a 3 a 4 a 8 + a a 7 a a 6 a 3 a 5 + a 4 a 9 a a 8 a a 7 + a 3 a 6 a 4 a 5 a a a 9 + a a 8 a 3 a 7 a 4 a 6 + a 5 a + a a a a 9 a 3 a 8 + a 4 a 7 a 5 a 6 a i 3 i+3 +3sa 3s i odd a i 6 i+6 +3sa 3s i even, i,,, 3, 4, 5, (8) Such formula for â i was found by us looking for the simplest vector â such that (L(a)â) i, i, 3, 5, 6, 8, 9,, and it was found before knowing the eistence of Theorem 3 and of its easy consequence (5) As anyone can realize, (8) is just the scalar version of formula (5) May be an eplicit formula could be given also for the entries of the corresponding vector a () For instance we have: a () 3a 3 3a a + a 3, a() 3a 6 3a a 5 3a a 4 + 3a 3 3a a a 3 + 3a a 4 + a 3, a () 3 3a 9 3a a 8 3a a 7 + 6a 3 a 6 3a a a 6 3a a 3 a 5 3a a 3 a 4 + 3a a 7 + 3a a 4 3a 4 a 5 + 3a 5 a + a 3 3

23 The reader could try to obtain the epression for the generic a () i in terms of the a j, ie the scalar version of the vector identity Ea () L(a)â L(a)L(c)d, c k a k t k, d k a k t k, t e iπ/3 A concluding remark We conclude with a remark on the history of the results enclosed in this work Once the ltt even and odd systems (4), (5) were obtained, we tried to eploit them to retrieve by linear algebra arguments the sparse system, solved by Bernoulli numbers, we observed in the paper [3] of Ramanujan (see (6)) In order to do that, first of all it was necessary to nullify the second, third, fifth, sith, eighth, ninth, and so on, diagonals of our even and odd systems So, we naturally conceived the ltt linear systems solvers here presented, and, in particular, the one nullifying at each step /3 of the remaining non null diagonals Note that our original aim, ie write an eplicit formula for the vectors w er, w or C N such that L(a e )w er L(a o )w or a R, with a R, a e, a o defined in (), (), (3), has not been reached in this work We leave to the reader the interesting eercise to find the vectors w er and w or Appendi A The ltt matri-vector product The product of a n n lower triangular Toeplitz matri times a vector can be computed with much less than the n(n + )/ multiplications and (n )n/ additions required by the obvious algorithm The two alternative algorithms here described use the strong relation eisting between Toeplitz matrices and the circulant and ( )-circulant [3] matri algebras in order to perform the operation ltt matri vector via a small number of discrete Fourier transforms, and thus in O(n log n) arithmetic operations Preliminaries Let Π ± be the n n matri Π ± Z T ± e n e T, where Z is the n n lower-shift matri Then Π F D ω n F, Π (D ρ n F )ρd ω n (D ρ n F ) (9) where F is the following (symmetric) unitary Fourier matri F n W, W (ω (i )(j ) ) n i,j, ω such that ωn, ω i, < i < n, D ω n diag (, ω,, ω n ), ρ is such that ρ n, ρ i, < i < n, and D ρ n diag (, ρ,, ρ n ) From (9) it follows that for the circulant and ( )-circulant matrices whose first row is a T [a a a n ], that is for the matrices C(a) : n k a kπ k and C (a) : n k a kπ k, the following representations hold C(a) F d(f T a)d(f T e ) F, C (a) F d(f T a)d(f T e ) F, F D ρ n F, where d(z) denotes the diagonal matri whose diagonal elements are the entries of the vector z Given z C n, the matri-vector product F z is called discrete Fourier transform (DFT) of z Note that the Fourier matri satisfies the equalities F JΠ and F JΠ F, where J is the counter-identity, ie the permutation matri obtained by reversing the columns of the identity matri So, the inverse discrete Fourier transform of a vector z, F z, is simply a permutation of the DFT of z The DFT of z can be performed through a method, known as FFT, whose computational cost is O(n log b n), when n is a power of a number b (see Appendi B) It follows that the same order of arithmetic operations is enough to compute the matri-vector products C(a)z e C (a)z, for any a C n We are now ready to illustrate two procedures for the computation of the product of a Toeplitz matri T (t i j ) n i,j times a vector Obviously such procedures can be applied to our case, where t k, k < We stress the fact that more efficient methods for the computation of ltt matri-vector products may eist and they would be welcome, being such products the basic operations required by the algorithms presented 3

24 throughout this work In fact, in the previous sections we have seen that the solution of a triangular Toeplitz linear system of n equations, with n power of (3), can be reduced to the computation of O(log n) (O(log 3 n)) matri-vector products, where the matri involved is Toeplitz triangular and its dimension varies, reducing by a factor / (/3) each time Thus it would be suitable to have a method which performs such products in the most efficient way Procedure I (T embedded into a circulant) Consider a generic Toeplitz 4 4 matri T and a 4 vector v Then T can be seen as the upper left submatri of a 8 8 circulant matri C, and the following representation holds for the vector T v: t t t t 3 t 3 t t v t t t t t 3 t 3 t v t t t t 3 v T v t t t t v { t t t t t t 3 t 3 v t t t t v t 3 t t t t t t 3 v 3 } t 3 t t t t t t 3 { [ ] v } C 4 4 t 3 t t t v 3 t 3 t 3 t t t t t t t 3 t 3 t t t t t t t 3 t 3 t t t where the symbol {z} 4 denotes the 4 vector whose entries are the first four components of z If T is n n and v is n, then the observation still holds, and can be generalized: t t T v { [ ] v } C n, C C(a) bnf bn d(f bn a)fbn, H a t n+ (b )n (b )n+ t n t If n is a power of b (b, 3, ), from such formula one immediately deduces a procedure of cost O(n log b n) for the computation of the product of a n n Toeplitz matri times a vector (see Appendi B) Procedure II (T written as the sum of a circulant and a (-)-circulant) Set a [a a n ] T and a [a a n] T where a i (t i+ + t n i+ ), a i (t i+ t n i+ ), i,, n (t n ) Then, the following representation holds for our Toeplitz matri T (t i j ) n i,j : T C(a) + C (a ) F d(f T a)d(f T e ) F + F d(f T a )d(f T e ) F Again, if n is a power of b (b, 3, ), from this formula one immediately deduces a procedure of cost O(n log b n) for the computation of the product of a n n Toeplitz matri times a vector (see Appendi B) Appendi B The FFT algorithm Proposition 4 ((FFT)) Let n be a power of b (b, 3, ) Given z C n, the DFT of z can be computed in at most O(n log b n) arithmetic operations 4

25 Proof Let n be such that b n Since ω (i )(k ) is the (i, k) element of W and z k is the k-th element of z C n, we have (W z) i n k ω(i )(k ) z k n/b j ω(i )(bj b) z bj b+ + n/b j ω(i )(bj b+) z bj b+ + + n/b j ω(i )(bj b+b ) z bj b+b n/b j (ωb ) (i )(j ) z bj b+ + n/b j ω(i )(b(j )+) z bj b+ + + n/b j ω(i )(b(j )+b ) z bj b+b n/b j (ωb ) (i )(j ) z bj b+ + ω i n/b j (ωb ) (i )(j ) z bj b+ + + ω (i )(b ) n/b j (ωb ) (i )(j ) z bj b+b Note that ω is actually a function of n, in fact ω is such that ω n and ω i, < i < n So, a better notation for ω is ω n Then ω b ωn b is such that (ωn) b n/b and (ωn) b i, < i < n/b; in other words ωn b ω n/b (namely ωn b is the n/b-th principal root of the unity) Thus we have the identity (W n z) i n/b j ω(i )(j ) n/b + + ω (i )(b ) n It follows that, for i,, n b, z (W n z) i (W n/b z b+ ) i + ωn i z n b+ n/b z bj b+ + ωn i j ω(i )(j ) n/b z bj b+ n/b j (ω n/b) (i )(j ) z bj b+b, i,,, n (W n/b z z b+ z n b+ Moreover, letting i n b + k, k,, n b, in (3), we obtain (W n z) n b +k n/b j ω n b (j ) n/b ω (k )(j ) n/b ω n (k )(b ) n/b + + ω n b (b ) n n/b j ω(k )(j ) n/b n ω n (k )(b ) n/b + + ω n b (b ) n (W n/b z z b+ z n b+ + + ω n b (b ) n z bj b+ + ω n b n ω k n j ω n b (j ) n/b z bj b+ + ω n b n ω k ) k + ω n b n ωn k ω n (k )(b ) (W n/b ) i + + ω n (i )(b ) n/b j ω n b (j ) n/b z bj ω (k )(j ) n/b n/b j ω(k )(j ) n/b z bj b+ z bj j ω(k )(j ) n/b (W n/b z b z b z n z z b+ z n b+ ) k ) k, k,, n b, (W n/b z b z b z n ) i ω (k )(j ) n/b z bj b+ where ω n b n ω b Proceeding in this way, one obtains formulas for (W n z) r n b +k, r,,, b, k,, n b Such scalar equalities can be written in a more compact form: I D D b W n/b W n z I ω b D (ω b D) b W n/b Q, (3) I ω b b D (ω b b D) b W n/b where D ω n ω n b n, (3) 5