Convoluted Convolved Fibonacci Numbers

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1 Journal of Integer Sequences, Vol. 7 (2004, Article Convoluted Convolved Fibonacci Numbers Pieter Moree Max-Planc-Institut für Mathemati Vivatsgasse 7 D Bonn Germany moree@mpim-bonn.mpg.de Abstract The convolved Fibonacci numbers F (r are defined by (1 x x 2 r = 0 F (r +1 x. In this note we consider some related numbers that can be expressed in terms of convolved Fibonacci numbers. These numbers appear in the numerical evaluation of a constant arising in the study of the average density of elements in a finite field having order congruent to a (mod d. We derive a formula expressing these numbers in terms of ordinary Fibonacci and Lucas numbers. The non-negativity of these numbers can be inferred from Witt s dimension formula for free Lie algebras. This note is a case study of the transform 1 n series, which was introduced and studied in a companion paper by Moree. d n µ(df(zd n/d (with f any formal 1 Introduction Let {F n } n=0 = {0, 1, 1, 2, 3, 5,... } be the sequence of Fibonacci numbers and {L n } n=0 = {2, 1, 3, 4, 7, 11,... } the sequence of Lucas numbers. It is well-nown and easy to derive that for z < ( 5 1/2, we have (1 z z 2 1 = =0 F +1z. For any real number r the convolved Fibonacci numbers are defined by 1 (1 z z 2 = F (r r +1 z. (1 The Taylor series in (1 converges for all z C with z < ( 5 1/2. In the remainder of this note it is assumed that r is a positive integer. Note that F (r m+1 = F 1 +1F 2 +1 F r+1, r=m 1 =0

2 where the sum is over all 1,..., r with t 0 for 1 t r. We also have F (r m+1 = m =0 F +1F (r 1 m +1. Convolved Fibonacci numbers have been studied in several papers, for some references see, e.g., Sloane [10]. The earliest reference to convolved Fibonacci numbers the author is aware of is in a boo by Riordan [9], who proposed as an exercise (at p. 89 to show that F (r +1 = r v=0 ( ( r + v 1 v v v. (2 However, the latter identity is false in general (it would imply F (1 +1 = F +1 = for 2 for example. Using a result of Gould [1, p. 699] on Humbert polynomials (with n =, m = 2, x = 1/2, y = 1, p = r and C = 1 it is easily inferred that (2 holds true with upper index of summation r replaced by [/2]. In Section 4 we give a formula expressing the convolved Fibonacci numbers in terms of Fibonacci- and Lucas numbers. Hoggatt and Bicnell-Johnson [2], using a different method, derived such a formula for F (2 +1 and H (r +1. However, in this note our main interest is in numbers G(r +1 analogous to the convolved Fibonacci numbers, which we name convoluted convolved Fibonacci numbers, respectively sign-twisted convoluted convolved Fibonacci numbers. Given a formal series f(z C[[z]], we define its Witt transform as W (r f (z = 1 µ(df(z d r d = m f (, rz, (3 r d r where as usual µ denotes the Möbius function. For every integer r 1 we put Similarly we put G (r +1 = m f(, r with f = =0 H (r +1 = ( 1r m f (, r with f = On comparing (4 with (1 one sees that G (r +1 = 1 r d gcd(r, µ(df ( r d and H(r +1 d +1 = ( 1r r 1 1 z z z z 2. (4 d gcd(r, µ(d( 1 r d F ( r d +1. (5 d In Tables 1, 2 and 3 below some values of convolved, convoluted convolved, respectively signtwisted convoluted convolved Fibonacci numbers are provided. The purpose of this paper is to investigate the properties of these numbers. The next section gives a motivation for studying these numbers. 2 Evaluation of a constant Let g be an integer and p a prime not dividing g. Then by ord p (g we denote the smallest positive integer such that g 1 (mod p. Let d 1 be an integer. It can be shown that 2

3 the set of primes p for which ord p (g is divisible by d has a density and this density can be explicitly computed. It is easy to see that the primes p for which ord p (g is even are the primes that divide some term of the sequence {g r + 1} r=0. A related, but much less studied, question is whether given integers a and d the set of primes p for which ord p (g a (mod d has a density. Presently this more difficult problem can only be resolved under assumption of the Generalized Riemann Hypothesis, see, e.g., Moree [6]. In the explicit evaluation of this density and also that of its average value (where one averages over g the following constant appears: B χ = p ( 1 + [χ(p 1]p, [p 2 χ(p](p 1 where χ is a Dirichlet character and the product is over all primes p (see Moree [5]. Recall that the Dirichlet L-series for χ, L(s, χ, is defined, for Re(s > 1, by n=1 χ (nn s. It satisfies the Euler product L(s, χ = 1 1 χ (pp. s p We similarly define L(s, χ = p (1 + χ (pp s 1. The Artin constant, which appears in many problems involving the multiplicative order, is defined by A = ( 1 1 = p(p 1 p The following result, which follows from Theorem 2 (with f(z = z 3 /(1 z z 2 and some convergence arguments, expresses the constant B χ in terms of Dirichlet L-series. Since Dirichlet L-series in integer values are easily evaluated with very high decimal precision this result allows one to evaluate B χ with high decimal precision. Theorem 1 1 We have L(2, χ B χ = A L(3, χ r=1 =3r+1 L(, ( χ r e(,r, where e(, r = G (r 3r+1. 2 We have L(2, χl(3, χ B χ = A L(6, χ 2 where f(, r = ( 1 r 1 H (r 3r+1. r=1 =3r+1 L(, χ r f(,r, Proof. Moree [5] proved part 2, and a variation of his proof yields part 1. In the next section it is deduced (Proposition 1 that the numbers e(, r appearing in the former double product are actually positive integers and that the f(, r are non-zero integers that satisfy sgn(f(, r = ( 1 r 1. The proof maes use of properties of the Witt transform that was introduced by Moree [7]. 3

4 3 Some properties of the Witt transform We recall some of the properties of the Witt transform (as defined by (3 and deduce consequences for the (sign-twisted convoluted convolved Fibonacci numbers. Theorem 2 [5]. Suppose that f(z Z[[z]]. Then, as formal power series in y and z, we have 1 yf(z = (1 z y r m f (,r. =0 r=1 Moreover, the numbers m f (, r are integers. If 1 yf(z = (1 z y r n(,r, =0 r=1 for some numbers n(, r, then n(, r = m f (, r. For certain choices of f identities as above arise in the theory of Lie algebras, see, e.g., Kang and Kim [3]. In this theory they go by the name of denominator identities. Theorem 3 [7]. Let r Z 1 and f(z Z[[z]]. Write f(z = a z. 1 We have { ( 1 r W (r f (z = W (r (r/2 f (z + W f (z 2, if r 2 (mod 4; (z, otherwise. W (r f 2 If f(z Z[[z]], then so is W (r f (z. 3 If f(z Z 0 [[z]], then so are W (r f (z and ( 1r W (r f (z. Suppose that {a } =0 is a non-decreasing sequence with a Then m f (, r 1 and ( 1 r m f (, r 1 for 1. 5 The sequences {m f (, r} =0 and {( 1 r m f (, r} =0 are both non-decreasing. In Moree [7] several further properties regarding monotonicity in both the and r direction are established that apply to both G (r and H (r. It turns out that slightly stronger results in this direction for these sequences can be established on using Theorem 8 below. 3.1 Consequences for G (r and H (r Since clearly F (r +1 Z we infer from (5 that rg(r +1, rh(r +1 Z. More is true, however: Proposition 1 Let, r 1 be integers. Then 1 G (r and H (r are non-negative integers. 2 When 2, then G (r 3 We have H (r = 4 The sequences {G (r 1 and H (r 1. { G (r + G (r/2 +1, if r 2 (mod 4and is odd; 2, otherwise. G (r The proof easily follows from Theorem 3. } =1 and {H (r } =1 are non-decreasing. 4

5 4 Convolved Fibonacci numbers reconsidered We show that the convolved Fibonacci numbers can be expressed in terms of Fibonacci and Lucas numbers. Theorem 4 Let 0 and r 1. We have r 1 F (r +1 = r+ 0 (mod 2 r 1 r+ 1 (mod 2 ( ( r + 1 r + 1 ( ( r + 1 r + 1 In particular, 5F (2 +1 = ( + 1L F +1 and Lr + 5 (+r/2 + Fr + 5 (+r 1/2. 50F (3 +1 = 5( + 1( + 2F ( + 1L F +1. Proof. Suppose that α, β C with αβ 0 and α β. Then it is not difficult to show that we have the following partial fraction decomposition: r 1 ( r (1 αz r (1 βz r = α r β (α β (1 r 1 r+ αz r + ( r β r α (β α r+ (1 βz r, where ( t = 1 if = 0 and ( t = t(t 1 (t +1/! otherwise (with t any real number. Using the Taylor expansion (with t a real number ( t (1 z t = ( 1 z, we infer that (1 αz r (1 βz r = =0 γ(z, where r 1 γ( = r 1 ( r ( r =0 α r β ( 1 (α β r+ β r α ( 1 (β α r+ ( r ( r β. α + Note that 1 z z 2 = (1 αz(1 βz with α = (1 + 5/2 and β = (1 5/2. On substituting these values of α and β and using that α β = 5, αβ = 1, L n = α n + β n and F n = (α n β n / 5, we find that r 1 F (r +1 = r+ 0 (mod 2 ( ( r r ( 1 ( 1 Lr (+r/2 5

6 r 1 r+ 1 (mod 2 On noting that ( 1 ( r = ( r+ 1 ( ( r r ( 1 ( 1 Fr + 5. (+r 1/2 and ( 1 ( r = ( r + 1, the proof is completed. Let r 1 be fixed. From the latter theorem one easily deduces the asymptotic behaviour of considered as a function of. F (r +1 Proposition 2 Let r 2 be fixed. Let [x] denote the integer part of x. Let α = (1 + 5/2. We have F (r +1 = g(rr 1 α + O r ( r 2 α, as tends to infinity, where the implicit error term depends at most on r and g(r = α r 5 [r/2] /(r 1!. 5 The numbers H (r +1 for fixed r In this and the next section we consider the numbers H (r +1 for fixed r, respectively for fixed. Very similar results can of course be obtained for the convoluted convolved Fibonacci numbers G (r +1. For small fixed r we can use Theorem 4 in combination with (5 to explicitly express H (r +1 in terms of Fibonacci- and Lucas numbers. In doing so it is convenient to wor with the characteristic function χ of the integers, which is defined by χ(r = 1 if r is an integer and χ(r = 0 otherwise. We demonstrate the procedure for r = 2 and r = 3. By (5 we find 2H (2 +1 = F ( F +1χ( and 3H( = F (3 +1 F +1χ(. By Theorem 4 it then follows for 3 3 example that 150H (3 +1 = 5( + 1( + 2F ( + 1L F +1 50F 3 +1χ( 3. The asymptotic behaviour, for r fixed and tending to infinity can be directly inferred from (5 and Proposition 2. Proposition 3 With the same notation and assumptions as in Proposition 2 we have H (r +1 = g(r r 1 α /r + O r ( r 2 α. 6 The numbers H (r +1 for fixed In this section we investigate the numbers H (r +1 for fixed. We first investigate this question for the convolved Fibonacci numbers. The coefficient F (r +1 of z in (1 z z 2 r is equal to the coefficient of z in (1 + F 2 z + F 3 z F +1 z r. By the multinomial theorem we then find F (r +1 = =1 n = ( r n 1,, n F n 1 2 F n +1, (6 6

7 where the multinomial coefficient is defined by ( r r! = m 1,, m s m 1!m 2! m s!(r m 1 m s! and m 0 for 1 s. Example. We have F (r 5 = ( r 4 ( r ( ( r r ( r , 1 2 1, 1 1 = 7 4 r r r3 + r4 24. This gives an explicit description of the sequence {F (r 5 } r=1, which is sequence A of Sloane s OEIS [10]. r The sequence {( m 1,,m } r=0 is a polynomial sequence where the degree of the polynomial is m m. It follows from this and (6 that {F (r +1 } r=0 is a polynomial sequence of degree max{n n =1 n = } =. The leading term of this polynomial in r is due to the multinomial term having n 1 = and n t = 0 for 2 t. All other terms in (6 are of lower degree. We thus infer that F (r +1 = r /! + O (r 1, r. We leave it to the reader to mae this more precise by showing that the coefficient of r 1 is 3/(2( 2!. If n 1,..., n satisfy max{n n =1 n = } =, then!/(n 1!... n! is an integral multiple of a multinomial coefficient and hence an integer. We thus infer that!f (r +1 is a monic polynomial in Z[r] of degree. Note that the constant term of this polynomial is zero. To sum up, we have obtained: Theorem 5 Let, r 1 be integers. There is a polynomial with A(, 0 = 0 such that F (r +1 = A(, r/!. A(, r = r ( 1r 1 + Z[r] Using this result, the following regarding the sign-twisted convoluted convolved Fibonacci numbers can be established. Theorem 6 Let χ(r = 1 if r is an integer and χ(r = 0 otherwise. We have { H (r 1, if r 2; 1 = 0, otherwise, furthermore H (r 2 = 1. We have 2H (r 3 = 3 + r ( 1 r/2 χ( r and 6H(r 4 = 8 + 9r + r 2 2χ( r

8 Also we have 24H (r 5 = r + 18r 2 + r 3 (18 + 3r( 1 r 2 χ( r 2 and 120H (r 6 = r + 215r r 3 + r 4 24χ( r 5. In general we have H (r +1 = d, 2 d µ(dχ( r d A( d, r d r(/d! + Let 3 be fixed. As r tends to infinity we have d, 2 d H (r +1 = r 1 + 3r 2! 2( 2! + O (r 2. Proof. Using that d n µ(d = 0 if n > 1, it is easy to chec that H (r 1 = ( 1r r µ(r( 1 r/d 1 = d r µ(d( 1 r/2 χ( r d A( d, r d r(/d!. { 1, if r 2; 0, if r > 2. The remaining assertions can be all derived from (5, (6 and Theorem 5. 7 Monotonicity Inspection of the tables below suggests monotonicity properties of F (r, G (r true. Proposition 4 1 Let 2. Then {F (r } r=1 is a strictly increasing sequence. 2 Let r 2. Then {F (r } =1 is a strictly increasing sequence. and H (r to hold The proof of this is easy. For the proof of part 2 one can mae use of the following simple observation. Lemma 1 Let f(z = a(z R[[z]] be a formal series. Then f(z is said to have -nondecreasing coefficients if a( > 0 and a( a( + 1 a( + 2 <.... If a( > 0 and a( < a( + 1 < a( + 2 <..., then f is said to have -increasing coefficients. If f, g are -increasing, respectively l-nondecreasing, then f g is ( + l-increasing. If f is -increasing and g is l-nondecreasing, then f + g is max(, l-increasing. If f is -increasing, then 1 b(f with b( 0 and b(1 > 0 is -increasing. We conclude this paper by establishing the following result: 8

9 Theorem 7 1 Let 4. Then {G (r 2 Let r 1. Then {G (r 3 Let 4. Then {H (r 4 Let r 1. Then {H (r } r=1 is a strictly increasing sequence. } =2 is a strictly increasing sequence. } r=1 is a strictly increasing sequence. } =2 is a strictly increasing sequence. The proof rests on expressing the entries of the above sequences in terms of certain quantities occurring in the theory of free Lie algebras and circular words (Theorem 8 and then invoe results on the monotonicity of these quantities to establish the result. 7.1 Circular words and Witt s dimension formula We will mae use of an easy result on cyclic words. A word a 1 a n is called circular or cyclic if a 1 is regarded as following a n, where a 1 a 2 a n, a 2 a n a 1 and all other cyclic shifts (rotations of a 1 a 2 a n are regarded as the same word. A circular word of length n may conceivably be given by repeating a segment of d letters n/d times, with d a divisor of n. Then one says the word is of period d. Each word belongs to an unique smallest period: the minimal period. Consider circular words of length n on an alphabet x 1,..., x r consisting of r letters. The total number of ordinary words such that x i occurs n i times equals n! n 1! n r!, where n n r = n. Let M(n 1,..., n r denote the number of circular words of length n n r = n and minimal period n such that the letter x i appears exactly n i times. This leads to the formula n! n 1! n r! = d gcd(n 1,...,n r whence it follows by Möbius inversion that M(n 1,..., n r = 1 n n d M(n 1 d, n 2 d,..., n r. (7 d d gcd(n 1,...,n r µ(d n 1 n! d (8 d! nr!. d Note that M(n 1,..., n r is totally symmetric in the variables n 1,..., n r. The numbers M(n 1,..., n r also occur in a classical result in Lie theory, namely Witt s formula for the homogeneous subspaces of a finitely generated free Lie algebra L: if H is the subspace of L generated by all homogeneous elements of multidegree (n 1,..., n r, then dim(h = M(n 1,..., n r, where n = n n r. In Theorem 8 a variation of M(n 1,..., n r appears. Lemma 2 [7]. Let r be a positive integer and let n 1,..., n r be non-negative integers and put n = n n r. Let V 1 (n 1,..., n r = ( 1n 1 n µ(d( 1 n! 1 d d n n 1 d! nr!. d d gcd(n 1,...,n r 9

10 Then V 1 (n 1,..., n r = { M(n 1,..., n r + M( n 1 2,..., nr 2 if n 1 2 (mod 4 and 2 gcd(n 1,..., n r ; M(n 1,..., n r otherwise. The numbers V 1 (n 1,..., n r can also be interpreted as dimensions (in the context of free Lie superalgebras, see, e.g., Petrogradsy [8]. The numbers M and V 1 enoy certain monotonicity properties. Lemma 3 [7]. Let r 1 and n 1,..., n r be non-negative numbers. 1 The sequence {M(m, n 1,..., n r } m=0 is non-decreasing if n n r 1 and strictly increasing if n n r 3. 2 The sequence {V 1 (m, n 1,..., n r } m=0 is non-decreasing if n n r 1 and strictly increasing if n n r 3. Using (7 one infers (on taing the logarithm of either side and expanding it as a formal series that 1 z 1 z r = (1 z n 1 1 zr nr M(n 1,,n r, (9 n 1,...,n r=0 where (n 1,..., n r = (0,..., 0 is excluded in the product. From the latter identity it follows that 1 + z 1 z 2 z r = n (1 z 1 1 zr nr M(n 1,...,n r 2 n 1 n 1,...,nr=0 whence, by Lemma 2, 1 + z 1 z 2 z r = n 1,...,n r=0 Theorem 8 Let r 1 and 0. We have [/2] G (r +1 = n 1,...,nr=0 2 n 1 M(r,, 2 and H (r ( 1 z 2n 1 1 zr 2nr 1 z n 1 1 zr nr M(n1,...,n r, (1 z n 1 1 zr nr ( 1n 1 V 1 (n 1,...,n r. (10 +1 = [/2] V 1 (r,, 2. Proof. By Theorem 2 and the definition of G (r we infer that 1 y 1 z z = 2 =0 r=1 (1 z y r G(r The left hand side of the latter equality equals (1 z z 2 y/(1 z z 2. On invoing (9 with z 1 = z, z 2 = z 2 and z 3 = y the claim regarding G (r follows from the uniqueness assertion in Theorem

11 The proof of the identity for H (r +1 is similar, but maes use of identity (10 instead of (9. Proof of Theorem 7. 1 For 5, + 2 [/2] 3 and hence each of the terms M(r,, 2 with 0 [/2] is strictly increasing in r by Lemma 3. For 3 4, by Lemma 3 again, all terms M(r,, 2 with 0 [/2] are non-decreasing in r and at least one of them is strictly increasing. The result now follow by Theorem 8. 2 In the proof of part 1 replace the letter M by V 1. 3 For r = 1 we have G (1 +! = F +1 and the result is obvious. For r = 2 each of the terms M(r,, 2 with 0 [/2] is non-decreasing in. For 2 one of these is strictly increasing. Since in addition G (2 2 < G (2 3 the result follows for r = 2. For r 3 each of the terms M(r,, 2 with 0 [/2] is strictly increasing in. The result now follows by Theorem 8. 4 In the proof of part 3 replace the letter G by H and M by V 1. Acnowledgement. The wor presented here was carried out at the Korteweg-de Vries Institute (Amsterdam in the NWO Pioneer-proect of Prof. E. Opdam. I am grateful to Prof. Opdam for the opportunity to wor in his group. I lie to than the referee for pointing out reference [1] and the falsity of (2. Wolfdieter Lang indly provided me with some additional references. References [1] H. W. Gould, Inverse series relations and other expansions involving Humbert polynomials, Due Math. J. 32 (1965, [2] V.E. Hoggatt Jr. and M. Bicnell-Johnson, Fibonacci convolution sequences, Fibonacci Quart. 15 (1977, [3] S.-J. Kang and M.-H. Kim, Free Lie algebras, generalized Witt formula, and the denominator identity, J. Algebra 183 (1996, [4] P. Moree, Approximation of singular series and automata, Manuscripta Math. 101 (2000, [5] P. Moree, On the average number of elements in a finite field with order or index in a prescribed residue class, preprint arxiv:math.nt/ , to appear in Finite Fields Appl.. [6] P. Moree, On the distribution of the order and index of g (mod p over residue classes, preprint arxiv:math.nt/ , submitted for publication. [7] P. Moree, The formal series Witt transform, preprint ArXiv:math.CO/ , submitted for publication. 11

12 [8] V.M. Petrogradsy, Witt s formula for restricted Lie algebras, Adv. Appl. Math. 30 (2003, [9] J. Riordan, Combinatorial Identities, Wiley, [10] N.J.A. Sloane, Sequences A00096, A001628, A001629, A001870, A006504, On-Line Encyclopedia of Integer Sequences, nas/sequences/. 12

13 8 Tables Table 1: Convolved Fibonacci numbers F (r r\ Table 2: Convoluted convolved Fibonacci numbers G (r r\ Table 3: Sign twisted convoluted convolved Fibonacci numbers H (r r\

14 2000 Mathematics Subect Classification: Primary 11B39 ; Secondary 11B83. Keywords: circular words, monotonicity, Witt s dimension formula, Dirichlet L-series. (Concerned with sequences A000096, A006504, A001628, A001870, A Received November ; revised version received April ; Published in Journal of Integer Sequences, April Return to Journal of Integer Sequences home page. 14