Convoluted Convolved Fibonacci Numbers

Pieter Moree Max-Planck-Institut für Mathematik Vivatsgasse 7 D-53111 Bonn Germany

                moree@mpim-bonn.mpg.de

Abstract

The convolved Fibonacci numbers F_{j}^{(r)} are defined by (1−x−x^{2})^{−r} = ∑_{j≥0} F_{j+1}^{(r)}x^{j}. 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 \frac{1}{n} ∑_{d | n} μ(d)f(z^{d})^{n/d} (with f any formal series), which was introduced and studied in a companion paper by Moree.

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-known and easy to derive that for |z| < (\sqrt{5} − 1)/2, we have (1 − z − z^{2})^{−1} = ∑_{j=0}^{∞} F_{j+1}z^{j}. For any real number r the convolved Fibonacci numbers are defined by

\frac{1}{(1 − z − z^{2})^{r}} = ∑_{j=0}^{∞} F_{j+1}^{(r)}z^{j}.  (1)

The Taylor series in (1) converges for all z ∈ ℂ with |z| < (\sqrt{5} − 1)/2. In the remainder of this note it is assumed that r is a positive integer. Note that

F_{m+1}^{(r)} = ∑_{j_{1}+···+j_{r}=m} F_{j_{1}+1}F_{j_{2}+1} ⋯ F_{j_{r}+1},

where the sum is over all j_{1}, …, j_{r} with j_{t} ≥ 0 for 1 ≤ t ≤ r. We also have F_{m+1}^{(r)} = ∑_{j=0}^{m} F_{j+1}F_{m−j+1}^{(r−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 book by Riordan [9], who proposed as an exercise (at p. 89) to show that

F_{j+1}^{(r)} = ∑_{v=0}^{r} \binom{r + j − v − 1}{j − v} \binom{j − v}{v} .  (2)

However, the latter identity is false in general (it would imply F_{j+1}^{(1)} = F_{j+1} = j for j ≥ 2 for example). Using a result of Gould [1, p. 699] on Humbert polynomials (with n = j, 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 [j/2].

In Section 4 we give a formula expressing the convolved Fibonacci numbers in terms of Fibonacci- and Lucas numbers. Hoggatt and Bicknell-Johnson [2], using a different method, derived such a formula for F_{j+1}^{(2)}. However, in this note our main interest is in numbers G_{j+1}^{(r)} and H_{j+1}^{(r)} 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_{f}^{(r)}(z) = \frac{1}{r} ∑_{d | r} μ(d)f(z^{d})^{\frac{r}{d}} = ∑_{j=0}^{∞} m_{f}(j, r)z^{j},  (3)

where as usual μ denotes the Möbius function.

For every integer r ≥ 1 we put

G_{j+1}^{(r)} = m_{f}(j, r) with f = \frac{1}{1 − z − z^{2}} .

Similarly we put

H_{j+1}^{(r)} = (−1)^{r}m_{f}(j, r) with f = \frac{−1}{1 − z − z^{2}} .  (4)

On comparing (4) with (1) one sees that

G_{j+1}^{(r)} = \frac{1}{r} ∑_{d | gcd(r,j)} μ(d)F_{\frac{j}{d} +1}^{( \frac{r}{d})} and H_{j+1}^{(r)} = \frac{(−1)^{r}}{r} ∑_{d | gcd(r,j)} μ(d)(−1)^{\frac{r}{d}} F_{\frac{j}{d} +1}^{( \frac{r}{d})}.  (5)

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 k such that g^{k} ≡ 1 (mod p). Let d ≥ 1 be an integer. It can be shown that 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 + \frac{[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^{k}, L(s, χ^{k}), is defined, for Re(s) > 1, by ∑_{n=1}^{∞} χ^{k}(n)n^{−s}. It satisfies the Euler product

L(s, χ^{k}) = ∏_{p} \frac{1}{1 − χ^{k}(p)p^{−s}} .

We similarly define L(s, −χ^{k}) = ∏_{p}(1 + χ^{k}(p)p^{−s})^{−1}. The Artin constant, which appears in many problems involving the multiplicative order, is defined by

A = ∏_{p} (1 − \frac{1}{p(p − 1)}) = 0.3739558136 …

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

B_{χ} = A\frac{L(2, χ)}{L(3, −χ)} ∏_{r=1}^{∞} ∏_{j=3r+1}^{∞} L(j, (−χ)^{r})^{−e(j,r)},

where e(j, r) = G_{j−3r+1}^{(r)}.

2) We have

B_{χ} = A\frac{L(2, χ)L(3, χ)}{L(6, χ^{2})} ∏_{r=1}^{∞} ∏_{j=3r+1}^{∞} L(j, χ^{r})^{f(j,r)},

where f(j, r) = (−1)^{r−1}H_{j−3r+1}^{(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(j, r) appearing in the former double product are actually positive integers and that the f(j, r) are non-zero integers that satisfy sgn(f(j, r)) = (−1)^{r−1}. The proof makes use of properties of the Witt transform that was introduced by Moree [7].

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) = ∏_{j=0}^{∞} ∏_{r=1}^{∞}(1 − z^{j}y^{r})^{m_{f} (j,r)}.

Moreover, the numbers m_{f}(j, r) are integers. If

1 − yf(z) = ∏_{j=0}^{∞} ∏_{r=1}^{∞}(1 − z^{j}y^{r})^{n(j,r)},

for some numbers n(j, r), then n(j, r) = m_{f}(j, 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) = ∑_{j} a_{j}z^{j}.

1) We have

(−1)^{r}W_{−f}^{(r)}(z) = \begin{cases}W_{f}^{(r)}(z) + W_{f}^{(r/2)}(z^{2}), & if r ≡ 2 (mod 4); \\ W_{f}^{(r)}(z), & otherwise. \end{cases}

2) If f(z) ∈ ℤ[[z]], then so is W_{f}^{(r)}(z).

3) If f(z) ∈ ℤ_{≥0}[[z]], then so are W_{f}^{(r)}(z) and (−1)^{r}W_{−f}^{(r)}(z).

Suppose that {a_{j}}_{j=0}^{∞} is a non-decreasing sequence with a_{1} ≥ 1.

4) Then m_{f}(j, r) ≥ 1 and (−1)^{r}m_{−f}(j, r) ≥ 1 for j ≥ 1.

5) The sequences {m_{f}(j, r)}_{j=0}^{∞} and {(−1)^{r}m_{−f}(j, r)}_{j=0}^{∞} are both non-decreasing.

In Moree [7] several further properties regarding monotonicity in both the j and r direction are established that apply to both G_{j}^{(r)} and H_{j}^{(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_{j}^{(r)} and H_{j}^{(r)}

Since clearly F_{j+1}^{(r)} ∈ ℤ we infer from (5) that rG_{j+1}^{(r)}, rH_{j+1}^{(r)} ∈ ℤ. More is true, however:

Proposition 1 Let j, r ≥ 1 be integers. Then

1) G_{j}^{(r)} and H_{j}^{(r)} are non-negative integers.

2) When j ≥ 2, then G_{j}^{(r)} ≥ 1 and H_{j}^{(r)} ≥ 1.

3) We have

H_{j}^{(r)} = \begin{cases}G_{j}^{(r)} + G(r/2)_{\frac{j+1}{2}}, & if r ≡ 2 (mod 4)and j is odd; \\ G_{j}^{(r)}, & otherwise. \end{cases}

4) The sequences {G_{j}^{(r)}}_{j=1}^{∞} and {H_{j}^{(r)}}_{j=1}^{∞} are non-decreasing.

The proof easily follows from Theorem 3.

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 j ≥ 0 and r ≥ 1. We have

F_{j+1}^{(r)} = ∑_{{k=0 \atop r+k≡0 (mod 2)}}^{r−1} \binom{r + k − 1}{k} \binom{r − k + j − 1}{j} \frac{L_{r−k+j}}{5^{(k+r)/2}} +

∑_{{k=0 \atop r+k≡1 (mod 2)}}^{r−1} \binom{r + k − 1}{k} \binom{r − k + j − 1}{j} \frac{F_{r−k+j}}{5^{(k+r−1)/2}} .

In particular, 5F_{j+1}^{(2)} = (j + 1)L_{j+2} + 2F_{j+1} and

50F_{j+1}^{(3)} = 5(j + 1)(j + 2)F_{j+3} + 6(j + 1)L_{j+2} + 12F_{j+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:

(1 − αz)^{−r}(1 − βz)^{−r} =

∑_{k=0}^{r−1} \binom{−r}{k} \frac{α^{r}β^{k}}{(α − β)^{r+k}} (1 − αz)^{k−r} + ∑_{k=0}^{r−1} \binom{−r}{k} \frac{β^{r}α^{k}}{(β − α)^{r+k}} (1 − βz)^{k−r},

where ({t \atop k}) = 1 if k = 0 and ({t \atop k}) = t(t−1) ⋯ (t−k+1)/k! otherwise (with t any real number). Using the Taylor expansion (with t a real number)

(1 − z)^{t} = ∑_{j=0}^{∞}(−1)^{j} \binom{t}{j} z^{j},

we infer that (1 − αz)^{−r}(1 − βz)^{−r} = ∑_{j=0}^{∞} γ(j)z^{j}, where

γ(j) = ∑_{k=0}^{r−1} \binom{−r}{k} \frac{α^{r}β^{k}}{(α − β)^{r+k}} (−1)^{j} \binom{k − r}{j} α^{j}+

∑_{k=0}^{r−1} \binom{−r}{k} \frac{β^{r}α^{k}}{(β − α)^{r+k}} (−1)^{j} \binom{k − r}{j} β^{j}.

Note that 1 − z − z^{2} = (1 − αz)(1 − βz) with α = (1 + \sqrt{5})/2 and β = (1 − \sqrt{5})/2. On substituting these values of α and β and using that α − β = \sqrt{5}, αβ = −1, L_{n} = α^{n} + β^{n} and F_{n} = (α^{n} − β^{n})/\sqrt{5}, we find that

F_{j+1}^{(r)} = ∑_{{k=0 \atop r+k≡0 (mod 2)}}^{r−1}(−1)^{k} \binom{−r}{k} (−1)^{j} \binom{k − r}{j} \frac{L_{r−k+j}}{5^{(k+r)/2}} +

∑_{{k=0 \atop r+k≡1 (mod 2)}}^{r−1}(−1)^{k} \binom{−r}{k} (−1)^{j} \binom{k − r}{j} \frac{F_{r−k+j}}{5^{(k+r−1)/2}} .

On noting that (−1)^{k}({−r \atop k}) = ({r+k−1 \atop k}) and (−1)^{j}({k−r \atop j}) = ({r−k+j−1 \atop j}), the proof is completed. □

Let r ≥ 1 be fixed. From the latter theorem one easily deduces the asymptotic behaviour of F_{j+1}^{(r)} considered as a function of j.

Proposition 2 Let r ≥ 2 be fixed. Let [x] denote the integer part of x. Let α = (1 + \sqrt{5})/2. We have F_{j+1}^{(r)} = g(r)j^{r−1}α^{j} + O_{r}(j^{r−2}α^{j}), as j 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_{j+1}^{(r)} for fixed r

In this and the next section we consider the numbers H_{j+1}^{(r)} for fixed r, respectively for fixed j. Very similar results can of course be obtained for the convoluted convolved Fibonacci numbers G_{j+1}^{(r)}.

For small fixed r we can use Theorem 4 in combination with (5) to explicitly express H_{j+1}^{(r)} in terms of Fibonacci- and Lucas numbers. In doing so it is convenient to work 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_{j+1}^{(2)} = F_{j+1}^{(2)} + F_{\frac{j}{2} +1}χ(\frac{j}{2}) and 3H_{j+1}^{(3)} = F_{j+1}^{(3)} − F_{\frac{j}{3} +1}χ(\frac{j}{3}). By Theorem 4 it then follows for example that

150H_{j+1}^{(3)} = 5(j + 1)(j + 2)F_{j+3} + 6(j + 1)L_{j+2} + 12F_{j+1} − 50F_{\frac{j}{3} +1}χ(\frac{j}{3}).

The asymptotic behaviour, for r fixed and j 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_{j+1}^{(r)} = g(r)j^{r−1}α^{j}/r + O_{r}(j^{r−2}α^{j}).

6 The numbers H_{j+1}^{(r)} for fixed j

In this section we investigate the numbers H_{j+1}^{(r)} for fixed j. We first investigate this question for the convolved Fibonacci numbers.

The coefficient F_{j+1}^{(r)} of z^{j} in (1 − z − z^{2})^{−r} is equal to the coefficient of z^{j} in (1 + F_{2}z + F_{3}z^{3} + ⋯ + F_{j+1}z^{j})^{r}. By the multinomial theorem we then find

F_{j+1}^{(r)} = ∑ ; ∑_{k=1}^{j} kn_{k}=j \binom{r}{n_{1}, ⋯, n_{j}} F_{2}^{n_{1}} ⋯ F_{j+1}^{n_{j}},  (6)

where the multinomial coefficient is defined by

\binom{r}{m_{1}, ⋯, m_{s}} = \frac{r!}{m_{1}!m_{2}! ⋯ m_{s}!(r − m_{1} − ⋯ − m_{s})!}

and m_{k} ≥ 0 for 1 ≤ k ≤ s.

Example. We have

F_{5}^{(r)} = \binom{r}{4} + 2 \binom{r}{2, 1} + 4 \binom{r}{2} + 3 \binom{r}{1, 1} + 5 \binom{r}{1}
= \frac{7}{4}r + \frac{59}{24}r^{2} + \frac{3}{4}r^{3} + \frac{r^{4}}{24}.

This gives an explicit description of the sequence {F_{5}^{(r)}}_{r=1}^{∞}, which is sequence A006504 of Sloane’s OEIS [10].

The sequence \{({r \atop m_{1},··· ,m_{k}})\}_{r=0}^{∞} is a polynomial sequence where the degree of the polynomial is m_{1} + ⋯ + m_{k}. It follows from this and (6) that {F_{j+1}^{(r)}}_{r=0}^{∞} is a polynomial sequence of degree max{n_{1} + ⋯ + n_{j}| ∑_{k=1}^{j} kn_{j} = j} = j. The leading term of this polynomial in r is due to the multinomial term having n_{1} = j and n_{t} = 0 for 2 ≤ t ≤ j. All other terms in (6) are of lower degree. We thus infer that F_{j+1}^{(r)} = r^{j}/j! + O_{j}(r^{j−1}), r → ∞. We leave it to the reader to make this more precise by showing that the coefficient of r^{j−1} is 3/(2(j − 2)!). If n_{1}, …, n_{j} satisfy max{n_{1} + ⋯ + n_{j}| ∑_{k=1}^{j} kn_{j} = j} = j, then j!/(n_{1}! … n_{j}!) is an integral multiple of a multinomial coefficient and hence an integer. We thus infer that j!F_{j+1}^{(r)} is a monic polynomial in ℤ[r] of degree j. Note that the constant term of this polynomial is zero. To sum up, we have obtained:

Theorem 5 Let j, r ≥ 1 be integers. There is a polynomial

A(j, r) = r^{j} + \frac{3}{2}j(j − 1)r^{j−1} + ⋯ ∈ ℤ[r]

with A(j, 0) = 0 such that F_{j+1}^{(r)} = A(j, r)/j!.

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_{1}^{(r)} = \begin{cases}1, & if r ≤ 2; \\ 0, & otherwise, \end{cases}

furthermore H_{2}^{(r)} = 1. We have

2H_{3}^{(r)} = 3 + r − (−1)^{r/2}χ(\frac{r}{2}) and 6H_{4}^{(r)} = 8 + 9r + r^{2} − 2χ(\frac{r}{3}).

Also we have

24H_{5}^{(r)} = 42 + 59r + 18r^{2} + r^{3} − (18 + 3r)(−1)^{\frac{r}{2}} χ(\frac{r}{2}) and

120H_{6}^{(r)} = 264 + 450r + 215r^{2} + 30r^{3} + r^{4} − 24χ(\frac{r}{5}).

In general we have

H_{j+1}^{(r)} = ∑_{d | j, 2∤d}μ(d)χ(\frac{r}{d})\frac{A(\frac{j}{d}, \frac{r}{d})}{r(j/d)!} + ∑_{d | j, 2 | d} μ(d)(−1)^{r/2}χ(\frac{r}{d})\frac{A(\frac{j}{d}, \frac{r}{d})}{r(j/d)!}.

Let j ≥ 3 be fixed. As r tends to infinity we have

H_{j+1}^{(r)} = \frac{r^{j−1}}{j!} + \frac{3r^{j−2}}{2(j − 2)!} + O_{j}(r^{j−2}).

Proof. Using that ∑_{d | n} μ(d) = 0 if n > 1, it is easy to check that

H_{1}^{(r)} = \frac{(−1)^{r}}{r} ∑_{d | r} μ(r)(−1)^{r/d−1} = \begin{cases}1, & if r ≤ 2; \\ 0, & if r > 2. \end{cases}

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_{j}^{(r)}, G_{j}^{(r)} and H_{j}^{(r)} to hold true.

Proposition 4

1) Let j ≥ 2. Then {F_{j}^{(r)}}_{r=1}^{∞} is a strictly increasing sequence.

2) Let r ≥ 2. Then {F_{j}^{(r)}}_{j=1}^{∞} is a strictly increasing sequence.

The proof of this is easy. For the proof of part 2 one can make use of the following simple observation.

Lemma 1 Let f(z) = ∑_{j} a(j)z^{j} ∈ ℝ[[z]] be a formal series. Then f(z) is said to have k-nondecreasing coefficients if a(k) > 0 and a(k) ≤ a(k + 1) ≤ a(k + 2) < . . .. If a(k) > 0 and a(k) < a(k + 1) < a(k + 2) < …, then f is said to have k-increasing coefficients. If f, g are k-increasing, respectively l-nondecreasing, then fg is (k + l)-increasing. If f is k-increasing and g is l-nondecreasing, then f + g is max(k, l)-increasing. If f is k-increasing, then ∑_{j≥1} b(j)f^{j} with b(j) ≥ 0 and b(1) > 0 is k-increasing.

We conclude this paper by establishing the following result:

Theorem 7

1) Let j ≥ 4. Then {G_{j}^{(r)}}_{r=1}^{∞} is a strictly increasing sequence.

2) Let r ≥ 1. Then {G_{j}^{(r)}}_{j=2}^{∞} is a strictly increasing sequence.

3) Let j ≥ 4. Then {H_{j}^{(r)}}_{r=1}^{∞} is a strictly increasing sequence.

4) Let r ≥ 1. Then {H_{j}^{(r)}}_{j=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 invoke results on the monotonicity of these quantities to establish the result.

7.1 Circular words and Witt’s dimension formula

We will make 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

\frac{n!}{n_{1}! ⋯ n_{r}!},

where n_{1} + ⋯ + n_{r} = n. Let M(n_{1}, …, n_{r}) denote the number of circular words of length n_{1} + ⋯ + n_{r} = n and minimal period n such that the letter x_{i} appears exactly n_{i} times. This leads to the formula

\frac{n!}{n_{1}! ⋯ n_{r}!} = ∑_{d | gcd(n_{1},...,n_{r})} \frac{n}{d}M(\frac{n_{1}}{d} , \frac{n_{2}}{d} , …, \frac{n_{r}}{d}).  (7)

whence it follows by Möbius inversion that

M(n_{1}, …, n_{r}) = \frac{1}{n} ∑_{d | gcd(n_{1},...,n_{r})} μ(d) \frac{\frac{n}{d}!}{\frac{n_{1}}{d}! ⋯ \frac{n_{r}}{d}!}.  (8)

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_{1} + ⋯ + 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_{1} + ⋯ + n_{r}. Let

V_{1}(n_{1}, …, n_{r}) = \frac{(−1)^{n_{1}}}{n} ∑_{d | gcd(n_{1},...,n_{r})} μ(d)(−1)^{\frac{n_{1}}{d}} \frac{\frac{n}{d}!}{\frac{n_{1}}{d}! ⋯ \frac{n_{r}}{d}!}.

Then

V_{1}(n_{1}, …, n_{r}) = \begin{cases}M(n_{1}, …, n_{r}) + M(\frac{n_{1}}{2} , …, \frac{n_{r}}{2} ), & if n_{1} ≡ 2 (mod 4) and 2 | gcd(n_{1}, …, n_{r}); \\ M(n_{1}, …, n_{r}) & otherwise. \end{cases}

The numbers V_{1}(n_{1}, …, n_{r}) can also be interpreted as dimensions (in the context of free Lie superalgebras), see, e.g., Petrogradsky [8].

The numbers M and V_{1} enjoy 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_{1} + ⋯ + n_{r} ≥ 1 and strictly increasing if n_{1} + ⋯ + n_{r} ≥ 3.

2) The sequence {V_{1}(m, n_{1}, …, n_{r}}_{m=0}^{∞} is non-decreasing if n_{1} + ⋯ + n_{r} ≥ 1 and strictly increasing if n_{1} + ⋯ + n_{r} ≥ 3.

Using (7) one infers (on taking the logarithm of either side and expanding it as a formal series) that

1 − z_{1} − ⋯ − z_{r} = ∏_{n_{1},...,n_{r}=0}^{∞}(1 − z_{1}^{n_{1}} ⋯ z_{r}^{n_{r}})^{M(n_{1},··· ,n_{r})},  (9)

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},...,nr=0 \atop 2 | n_{1}}}^{∞}(1 − z_{1}^{n_{1}} ⋯ z_{r}^{n_{r}})^{M(n_{1},...,n_{r})} ∏_{{n_{1},...,nr=0 \atop 2∤n_{1}}}^{∞} (\frac{1 − z_{1}^{2n_{1}} ⋯ z_{r}^{2n_{r}}}{1 − z_{1}^{n_{1}} ⋯ z_{r}^{n_{r}}} )^{M(n_{1},...,n_{r})} ,

whence, by Lemma 2,

1 + z_{1} − z_{2} − ⋯ − z_{r} = ∏_{n_{1},...,n_{r}=0}^{∞}(1 − z_{1}^{n_{1}} ⋯ z_{r}^{n_{r}})^{(−1)^{n_{1}}V_{1}(n_{1},...,n_{r})}.  (10)

Theorem 8 Let r ≥ 1 and j ≥ 0. We have

G_{j+1}^{(r)} = ∑_{k=0}^{[j/2]} M(r, k, j − 2k) and H_{j+1}^{(r)} = ∑_{k=0}^{[j/2]} V_{1}(r, k, j − 2k).

Proof. By Theorem 2 and the definition of G_{j}^{(r)} we infer that

1 − \frac{y}{1 − z − z^{2}} = ∏_{j=0}^{∞} ∏_{r=1}^{∞}(1 − z^{j}y^{r})^{G_{j+1}^{(r)}}.

The left hand side of the latter equality equals (1 − z − z^{2} − y)/(1 − z − z^{2}). On invoking (9) with z_{1} = z, z_{2} = z^{2} and z_{3} = y the claim regarding G_{j}^{(r)} follows from the uniqueness assertion in Theorem 2.

The proof of the identity for H_{j+1}^{(r)} is similar, but makes use of identity (10) instead of (9). □

Proof of Theorem 7. 1) For j ≥ 5, k + j − 2k ≥ j − [j/2] ≥ 3 and hence each of the terms M(r, k, j − 2k) with 0 ≤ k ≤ [j/2] is strictly increasing in r by Lemma 3. For 3 ≤ j ≤ 4, by Lemma 3 again, all terms M(r, k, j − 2k) with 0 ≤ k ≤ [j/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_{j+!}^{(1)} = F_{j+1} and the result is obvious. For r = 2 each of the terms M(r, k, j − 2k) with 0 ≤ k ≤ [j/2] is non-decreasing in j. For j ≥ 2 one of these is strictly increasing. Since in addition G_{2}^{(2)} < G_{3}^{(2)} the result follows for r = 2. For r ≥ 3 each of the terms M(r, k, j − 2k) with 0 ≤ k ≤ [j/2] is strictly increasing in j. 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}’. □

Acknowledgement. The work presented here was carried out at the Korteweg-de Vries Institute (Amsterdam) in the NWO Pioneer-project of Prof. E. Opdam. I am grateful to Prof. Opdam for the opportunity to work in his group. I like to thank the referee for pointing out reference [1] and the falsity of (2). Wolfdieter Lang kindly provided me with some additional references.

References

[1] H. W. Gould, Inverse series relations and other expansions involving Humbert polynomials, Duke Math. J. 32 (1965), 697–711.

[2] V.E. Hoggatt Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fibonacci Quart. 15 (1977), 117–122.

[3] S.-J. Kang and M.-H. Kim, Free Lie algebras, generalized Witt formula, and the denominator identity, J. Algebra 183 (1996), 560–594.

[4] P. Moree, Approximation of singular series and automata, Manuscripta Math. 101 (2000), 385–399.

[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/0212220, 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/0211259, submitted for publication.

[7] P. Moree, The formal series Witt transform, preprint ArXiv:math.CO/0311194, submitted for publication.

[8] V.M. Petrogradsky, Witt’s formula for restricted Lie algebras, Adv. Appl. Math. 30 (2003), 219–227.

[9] J. Riordan, Combinatorial Identities, Wiley, 1968.

[10] N.J.A. Sloane, Sequences A00096, A001628, A001629, A001870, A006504, On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/˜ njas/sequences/.

8 Tables

Table 1: Convolved Fibonacci numbers F_{j}^{(r)}

\begin{matrix}r\j & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 1 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 \\ 2 & 1 & 2 & 5 & 10 & 20 & 38 & 71 & 130 & 235 & 420 & 744 \\ 3 & 1 & 3 & 9 & 22 & 51 & 111 & 233 & 474 & 942 & 1836 & 3522 \\ 4 & 1 & 4 & 14 & 40 & 105 & 256 & 594 & 1324 & 2860 & 6020 & 12402 \\ 5 & 1 & 5 & 20 & 65 & 190 & 511 & 1295 & 3130 & 7285 & 16435 & 36122 \end{matrix}

Table 2: Convoluted convolved Fibonacci numbers G_{j}^{(r)}

\begin{matrix}r\j & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 1 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 \\ 2 & 0 & 1 & 2 & 5 & 9 & 19 & 34 & 65 & 115 & 210 & 368 \\ 3 & 0 & 1 & 3 & 7 & 17 & 37 & 77 & 158 & 314 & 611 & 1174 \\ 4 & 0 & 1 & 3 & 10 & 25 & 64 & 146 & 331 & 710 & 1505 & 3091 \\ 5 & 0 & 1 & 4 & 13 & 38 & 102 & 259 & 626 & 1457 & 3287 & 7224 \\ 6 & 0 & 1 & 4 & 16 & 51 & 154 & 418 & 1098 & 2726 & 6570 & 15308 \\ 7 & 0 & 1 & 5 & 20 & 70 & 222 & 654 & 1817 & 4815 & 12265 & 30217 \\ 8 & 0 & 1 & 5 & 24 & 89 & 309 & 967 & 2871 & 8043 & 21659 & 56123 \\ 9 & 0 & 1 & 6 & 28 & 115 & 418 & 1396 & 4367 & 12925 & 36542 & 99385 \\ 10 & 0 & 1 & 6 & 33 & 141 & 552 & 1946 & 6435 & 20001 & 59345 & 168760 \end{matrix}

Table 3: Sign twisted convoluted convolved Fibonacci numbers H_{j}^{(r)}

\begin{matrix}r\j & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 1 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 \\ 2 & 1 & 1 & 3 & 5 & 11 & 19 & 37 & 65 & 120 & 210 & 376 \\ 3 & 0 & 1 & 3 & 7 & 17 & 37 & 77 & 158 & 314 & 611 & 1174 \\ 4 & 0 & 1 & 3 & 10 & 25 & 64 & 146 & 331 & 710 & 1505 & 3091 \\ 5 & 0 & 1 & 4 & 13 & 38 & 102 & 259 & 626 & 1457 & 3287 & 7224 \\ 6 & 0 & 1 & 5 & 16 & 54 & 154 & 425 & 1098 & 2743 & 6570 & 15345 \\ 7 & 0 & 1 & 5 & 20 & 70 & 222 & 654 & 1817 & 4815 & 12265 & 30217 \\ 8 & 0 & 1 & 5 & 24 & 89 & 309 & 967 & 2871 & 8043 & 21659 & 56123 \\ 9 & 0 & 1 & 6 & 28 & 115 & 418 & 1396 & 4367 & 12925 & 36542 & 99385 \\ 10 & 0 & 1 & 7 & 33 & 145 & 552 & 1959 & 6435 & 20039 & 59345 & 168862 \end{matrix}

2000 Mathematics Subject Classification: Primary 11B39 ; Secondary 11B83. Keywords: circular words, monotonicity, Witt’s dimension formula, Dirichlet L-series.

(Concerned with sequences \underline{A000096}, \underline{A006504}, \underline{A001628}, \underline{A001870}, \underline{A001629}.)

Received November 12 2003; revised version received April 20 2004; Published in Journal of Integer Sequences, April 26 2004.

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