Transforming Recurrent Sequences by Using the Binomial and Invert Operators

Stefano Barbero, Umberto Cerruti, and Nadir Murru Department of Mathematics University of Turin via Carlo Alberto 8/10 Turin Italy

               stefano.barbero@unito.it
               umberto.cerruti@unito.it
                 nadir.murru@unito.it

Abstract

In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we determine how they change the characteristic polynomials. We show that these operators, with the aid of two other elementary operators (essentially the left and right shifts), can transform any impulse sequence (a linear recurrent sequence starting from (0, …, 0, 1)) into any other impulse sequence, by two processes that we call construction and deconstruction. Finally, we give some applications to polynomial sequences and pyramidal numbers. We also find a new identity on Fibonacci numbers, and we prove that r–bonacci numbers are a Bell polynomial transform of the (r − 1)– bonacci numbers.

1 Introduction

A rich research field arises from investigation on the operators acting over sequences. In particular, two of these operators, Binomial and Invert, have been deeply studied. So we want to point out some new aspects, giving an in–depth examination of their action over the set of linear recurrent sequences.

Definition 1. We define, over an integral domain R, the set S(R) of sequences a = (a_{n})_{n=0}^{+∞}, and the set W(R) of linear recurrent sequences.

Definition 2. We recall the definition of the well-known operators I and L, called Invert and Binomial respectively. They act on elements of S(R) as follows:

I(a) = b, ∑_{n=0}^{∞} b_{n}t^{n} = \frac{∑_{n=0}^{∞} a_{n}t^{n}}{1 − t ∑_{n=0}^{∞} a_{n}t^{n}}

L(a) = c, c_{n} = ∑_{i=0}^{n} \binom{n}{i}a_{i} .

The operators I and L can be iterated [9] and interpolated [1], becoming I^{(x)} and L^{(y)}, as we show in the next two definitions.

Definition 3. The Invert interpolated operator I^{(x)}, with parameter x ∈ R, transforms any sequence a ∈ S(R), having ordinary generating function A(t), into a sequence b = I^{(x)}(a) ∈ S(R) having ordinary generating function

B(t) = \frac{A(t)}{1 − xtA(t)} .  (1)

Definition 4. The Binomial interpolated operatorL^{(y)}, with parameter y ∈ R, transforms any sequence a ∈ S(R) into a sequence c = L^{(y)}(a) ∈ S(R), whose terms are

c_{n} = ∑_{i=0}^{n} \binom{n}{i}y^{n−i}a_{i} .

For our purposes we also introduce the following two operators

Definition 5. The right-shift operator σ and the left-shift operator ρ change any sequence a ∈ S(R) as follows:

σ(a) = (a_{1}, a_{2}, a_{3}, …) ρ(a) = (0, a_{0}, a_{1}, a_{2}, …) .

Remark 6. The sequence b = I^{(x)}(a) is characterized by the following recurrence

\{ \begin{matrix}b_{n} = a_{n} + x ∑_{j=0}^{n−1} a_{n−1−j}b_{j}; \\ b_{0} = a_{0}. \end{matrix}  (2)

This relation is a straightforward consequence of Definition 3. In fact, if we consider the sequence u = (1, 0, 0, …) and the convolutional product ∗ between sequences, (1) becomes

b = a ∗ (u − xρ(a))^{−1},

where (u − xρ(a))^{−1} is the convolutional inverse of u − xρ(a). So

b = a + xρ(a) ∗ b,

from which (2) follows easily.

In a previous paper [2], Barbero and Cerruti proved that the operators L^{(y)}, I^{(x)} generate a group which is commutative. We want to generalize these results, analyzing the action of these operators on the set W(R) of all linear recurrent sequences over R. We will prove that the group generated by L^{(y)} and I^{(x)} acts on W(R) preserving it, as we will show in the next section.

2 A commutative group action on the set of linear recurrent sequences

First of all, we focus our attention on the operator L^{(y)}, with the aim to know how it acts on the characteristic polynomial of a given sequence a ∈ W(R). In order to do so, we prove the

Lemma 7. Let us consider a field 𝔽 containing R in which we choose an element α and a

nonzero element y. For any polynomial P (t) = ∑_{j=0}^{r} x_{j}t^{j} over 𝔽 we have

∑_{i=0}^{m} \binom{m}{i}y^{i}α^{m−i}P (i) = (α + y)^{m}Q(m),

where Q(m) = ∑_{j=0}^{r} y_{j}m^{j} and y_{j} = y_{j}(x_{0}, ⋯, x_{r}, y, α) ∈ 𝔽.

Proof. We use the classical Newton formula

∑_{i=0}^{m} \binom{m}{i}y^{i}α^{m−i} = (α + y)^{m}.

Differentiating with respect to y, one gets

\frac{1}{y} ∑_{i=0}^{m} \binom{m}{i}y^{i}α^{m−i}i = m(α + y)^{m−1},

so

∑_{i=0}^{m} \binom{m}{i}y^{i}α^{m−i}i = ym(α + y)^{m−1} = \frac{y}{α + y}m(α + y)^{m} .

We can prove by induction that

∑_{i=0}^{m} \binom{m}{i}y^{i}α^{m−i}i^{s} = c_{s}(m, α, y)(α + y)^{m} ,  (3)

for every integer s ≥ 1, where c_{s}(m, α, y) is a coefficient depending on m, α, y and it can be viewed as a polynomial of degree s in the variable m. The basis of the induction has already been proved. So let us suppose that the above formula is true for every integer between 1 and s, and we prove it for s + 1. Differentiating (3) with respect to y, one gets

\frac{1}{y} ∑_{i=0}^{m} \binom{m}{i}y^{i}α^{m−i}i^{s+1} = c′_{s}(m, α, y)(α + y)^{m} + c_{s}(m, α, y)m(α + y)^{m−1},

∑_{i=0}^{m} \binom{m}{i}y^{i}α^{m−i}i^{s+1} = (yc′_{s}(m, α, y) + \frac{yc_{s}(m, α, y)m}{α + y}) (α + y)^{m} = c_{s+1}(m, α, y)(α + y)^{m},

where c′_{s}(m, α, y) is the derivative of c_{s}(m, α, y) with respect to y and c_{s+1}(m, α, y) is a polynomial of degree s + 1 in m, by the induction hypothesis. Therefore we can complete the proof of the lemma

∑_{i=0}^{m} \binom{m}{i}y^{i}P (i)α^{m−i} = ∑_{j=0}^{r} x_{j} ∑_{i=0}^{m} \binom{m}{i}y^{i}α^{m−i}i^{j} = (α+y)^{m} ∑_{j=0}^{r} x_{j}c_{j}(m, α, y) = Q(m)(α+y)^{m}.

In the next proposition we show how to write c_{s}(m, α, y) of (3), for every natural number s and m, in order to obtain Q(m).

Proposition 8. The polynomial c_{s}(m, α, y) has the following expression

c_{s}(m, α, y) = ∑_{h=0}^{s} (∑_{k=h}^{s} \{ \begin{matrix}s \\ k \end{matrix} \} [ \begin{matrix}k \\ h \end{matrix} ] (−1)^{k−h} (\frac{y}{α + y} )^{k})m^{h} .

Here \{ \begin{matrix}s \\ k \end{matrix} \} and [ \begin{matrix}k \\ h \end{matrix} ] are the Stirling numbers of second and first kind respectively, as

defined in Graham, Knuth and Patashnik [6].

Proof. We use the finite differences of a function f, defined as follows:

Δ^{n}f(t) = ∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}f(t + k),

which enable us to express f(t) in this way [6]

f(t) = Δ^{d}f(0)\binom{t}{d} + Δ^{d−1}f(0)\binom{t}{d − 1} + ⋯ + Δf(0)\binom{t}{1} + f(0)\binom{t}{0} .

In the special case f(i) = i^{s}, we obtain

i^{s} = ∑_{k=0}^{s} Δ^{k}f(0)\binom{i}{k} = ∑_{k=0}^{s} (∑_{j=0}^{k} \binom{k}{j}(−1)^{k−j}j^{s}) \binom{i}{k} = ∑_{k=0}^{s} k! \{ \begin{matrix}s \\ k \end{matrix} \} \binom{i}{k},  (4)

where we use the relation [6]

∑_{j=0}^{k} ( \begin{matrix}k \\ j \end{matrix} )(−1)^{k−j}j^{s} = k! \{ \begin{matrix}s \\ k \end{matrix} \} .

Now it is clear that

∑_{i=0}^{m} \binom{m}{i}α^{m−i}y^{i}i^{s} = ∑_{k=0}^{s} Δ^{k}f(0) ∑_{i=0}^{m} \binom{m}{i}\binom{i}{k}α^{m−i}y^{i}.  (5)

Considering the second sum at the right side, we have

∑_{i=0}^{m} \binom{m}{i}\binom{i}{k}α^{m−i}y^{i} = \binom{m}{k} ∑_{i=k}^{m} \binom{m − k}{i − k}α^{m−i}y^{i} = \binom{m}{k}y^{k}(α + y)^{m−k}.

If we put this relation into (5), using (4), we obtain

∑_{i=0}^{m} \binom{m}{i}α^{m−i}y^{i}i^{s} = ∑_{k=0}^{s} k! \{ \begin{matrix}s \\ k \end{matrix} \} \binom{m}{k} (\frac{y}{α + y} )^{k} (α+y)^{m} = ∑_{k=0}^{s} \{ \begin{matrix}s \\ k \end{matrix} \} (\frac{y}{α + y} )^{k} m\underline{k}(α+y)^{m},

where m\underline{k} denotes the falling factorial kth-power of m [6]. We can write

m\underline{k} = ∑_{h=0}^{k} [ \begin{matrix}k \\ h \end{matrix} ](−1)^{k−h}m^{h}.  (6)

Therefore the proof is complete, since we can use (6) in the previous equality.

Corollary 9. With the hypotheses of Lemma 7 and Proposition 8

Q(m) = ∑_{i=0}^{r} ∑_{h=0}^{i} ∑_{k=h}^{i} x_{i} \{ \begin{matrix}i \\ k \end{matrix} \} [ \begin{matrix}k \\ h \end{matrix} ] (−1)^{k−h} (\frac{y}{α + y} )^{k}m^{h}.

Now we are ready to point out the effects of L^{(y)} on characteristic polynomials, as we claim in the following

Theorem 10. Let a ∈ W(R) be a linear recurrent sequence of degree r, with characteristic polynomial over R given by f(t) = t^{r} − ∑_{i=1}^{r} h_{i}t^{r−i} and zeros α_{1}, …, α_{r}, over a field 𝔽 containing R. Then c = L^{(y)}(a) is a linear recurrent sequence of degree r, with characteristic polynomial L^{(y)}(f(t)) = f(t − y) and zeros y + α_{1}, …, y + α_{r}, over the same field 𝔽.

Proof. If f(t) has distinct zeros α_{1}, …, α_{r}, by Binet formula

a_{n} = ∑_{i=0}^{r} A_{i}α_{i}^{n},

for some A_{i} ∈ 𝔽 derived from initial conditions. For all terms of c, we have

c_{n} = ∑_{i=0}^{n} \binom{n}{i}y^{n−i}a_{i} = ∑_{i=0}^{n} \binom{n}{i}y^{n−i} ∑_{j=0}^{r} A_{j}α_{j}^{i} = ∑_{j=0}^{r} A_{j}(α_{j} + y)^{n} .

If f(t) has a multiple zero α_{r} of multiplicity s, the Binet formula becomes

a_{n} = ∑_{i=0}^{r−s} A_{i}α_{i}^{n} + A_{r}(n)α_{r}^{n},

where A_{i} ∈ 𝔽, for i = 1, …, r − s, and A_{r}(n) is a polynomial of degree s − 1 over 𝔽. In this case

c_{n} = ∑_{j=1}^{r−s} A_{j}(α_{j} + y)^{n} + ∑_{i=0}^{n} \binom{n}{i}y^{i}A_{r}(n − i)α_{r}^{n−i},

and by Lemma 7

c_{n} = ∑_{j=1}^{r−s} A_{j}(α_{j} + y)^{n} + Q(n)(α_{r} + y)^{n},

where Q(n) is a polynomial of degree s − 1 in n, as shown in Proposition 8 and Corollary 9. In order to complete the proof, we have to observe that this method can be applied, even though there would be more than one multiple zero.

In the next corollary we write explicitly the characteristic polynomial L^{(y)}(f(t)) .

Corollary 11. Let f(t) = ∑_{k=0}^{r} h_{k}t^{r−k} be a polynomial over R, with h_{0} = 1. Then

L^{(y)}(f(t)) = ∑_{k=0}^{r} p_{k}(y, h_{1}, …, h_{k})t^{r−k},

where

p_{k}(y, h_{1}, …, h_{k}) = ∑_{i=0}^{k} \binom{r − i}{k − i}h_{i}(−y)^{k−i}.

Proof. Using the results of Theorem 10, we have

L^{(y)}(f(t)) = f(t − y) = ∑_{i=0}^{r} h_{i}(t − y)^{r−i}.

Developing (t − y)^{r−i} and rearranging sums, we obtain

L^{(y)}(f(t)) = ∑_{i=0}^{r} h_{i} ∑_{k=i}^{r} \binom{r − i}{k − i}(−y)^{k−i}t^{r−k} = ∑_{k=0}^{r} (∑_{i=0}^{k} \binom{r − i}{k − i}h_{i}(−y)^{k−i}) t^{r−k} .

Remark 12. The polynomials p_{k}(y, h_{1}, …, h_{k}) have coefficients related to triangular numbers, tetrahedral numbers and their generalizations. In order to make clear this relation, we introduce the sequences (T_{h}^{(k)})_{h=0}^{+∞} of figurate numbers. Starting from (T_{h}^{(1)})_{h=0}^{+∞} = {0, 1, 1, 1, 1, 1, …}, each term of (T_{h}^{(k)})_{h=0}^{+∞} corresponds to a partial sum of (T_{h}^{(k−1)})_{h=0}^{+∞}, for k = 1, 2, 3, ... .

For example, (T_{h}^{(2)})_{h=0}^{+∞} = {0, 1, 2, 3, 4, 5, …}, (T_{h}^{(3)})_{h=0}^{+∞} = {0, 1, 3, 6, 10, 15, …} (triangular numbers \underline{A000217}). The hth term of (T_{h}^{(k)}) is [5]

T_{h}^{(k)} = \binom{h + k − 2}{k − 1} .

In particular,

T_{k−i+1}^{(r−k+1)} = \binom{r − k + 1 + k − i + 1 − 2}{k − i} = \binom{r − i}{k − i},

therefore the polynomials p_{k}(y, h_{1}, …, h_{k}) are connected to figurate numbers

p_{k}(y, h_{1}, …, h_{k}) = ∑_{i=0}^{k} T_{k−i+1}^{(r−k+1)}h_{i}(−y)^{k−i} .

Now we study the action of I^{(x)} on W(R). We recall that any linear recurrent sequence a ∈ W(R), of degree r, with characteristic polynomial f(t) = t^{r} − ∑_{i=1}^{r} h_{i}t^{r−i}, has a rational generating function [6]

A(t) = \frac{u(t)}{f^{R}(t)} .

Here

f^{R}(t) = 1 − ∑_{i=1}^{r} h_{i}t^{i}

denotes the reflected polynomial of f(t). Moreover u(t) = ∑_{i=0}^{r−1} u_{i}t^{i} is a polynomial over R, wwhose coefficients u_{i} can be derived from initial conditions

(a_{0}, a_{1}, …, a_{r−1}) = (s_{0}, s_{1}, …, s_{r−1}).

Theorem 13. Let a ∈ W(R) be a linear recurrent sequence of degree r, with characteristic

polynomial over R given by f(t) = t^{r} −∑_{i=1}^{r} h_{i}t^{r−i}, and generating function A(t) = \frac{u(t)}{f^{R}(t)} .

Then b = I^{(x)}(a) is a linear recurrent sequence, with characteristic polynomial

(f^{R}(t) − xtu(t))^{R},

whose coefficients are

h_{1} + xs_{0}, h_{i+1} + xs_{i} − x ∑_{j=1}^{i} h_{j}s_{i−j} for i = 1, …, r − 1 .  (7)

Proof. By Definition 3, b = I^{(x)}(a) has the rational generating function

B(t) = \frac{A(t)}{1 − xtA(t)} .

Substituting the rational generating function of a we obtain

B(t) = \frac{A(t)}{1 − xtA(t)} = \frac{\frac{u(t)}{f^{R}(t)}}{1 − xt\frac{u(t)}{f^{R}(t)}} = \frac{u(t)}{f^{R}(t) − xtu(t)} = \frac{u(t)}{[(f^{R}(t) − xtu(t))^{R}]^{R}} .

This relation immediately proves that I^{(x)}(a) is a linear recurrent sequence having characteristic polynomial (f^{R}(t) − xtu(t))^{R}. If we explicitly write this polynomial, we find

(f^{R}(t) − xtu(t))^{R} = (−(h_{r} + xu_{r−1})t^{r} − (h_{r−1} + xu_{r−2})t^{r−1} − ⋯ − (h_{1} + xu_{0})t + 1)^{R}, (8)

and clearly

(f^{R}(t) − xtu(t))^{R} = t^{r} − (h_{1} + xu_{0})t^{r−1} − ⋯ − (h_{r−1} + xu_{r−2})t − (h_{r} + xu_{r−1}). (9)

Now, to complete the proof, we only need to observe that

u_{0} = s_{0}, u_{i} = s_{i} − ∑_{j=1}^{i} h_{j}s_{i−j} i = 1, …, r − 1.

In fact the terms u_{i} can be easily found, by means of a simple comparison between coefficients of t on the left and right side of the equality

A(t)f^{R}(t) = u(t),

and using initial conditions (a_{0}, a_{1}, …, a_{r−1}) = (s_{0}, s_{1}, …, s_{r−1}).

Corollary 14. Any two polynomials f(t) = t^{r} − ∑_{i=1}^{r} h_{i}t^{r−i} and g(t) = t^{r} − ∑_{i=1}^{r} H_{i}t^{r−i} over R can be viewed as characteristic polynomials of two uniquely determined sequences a ∈ W(R) and b = I^{(x)}(a) ∈ W(R), respectively, where x is an invertible element of R.

Proof. As a consequence of (7) we can easily solve the system

h_{1} + xs_{0} = H_{1}, h_{i+1} + xs_{i} − x ∑_{j=1}^{i} h_{j}s_{i−j} = H_{i+1} i = 1, …, r − 1 ,

in order to determine the unique values of initial conditions on a such that b = I^{(x)}(a).

Remark 15. The previous theorem shows how the action of I^{(x)} on W(R) is related to the initial conditions on a. In particular, for any a ∈ W(R), the characteristic polynomial of b = I^{(x)}(a) can have degree less than the one of a. In other words, under a suitable choice of x, we can reduce the degree of polynomial (8) changing his explicit form (9). In fact, when u_{r−1} is an invertible element of R, we can choose x = −h_{r}(u_{r−1})^{−1} in order to annihilate the coefficient of t^{r} in (8).

Remark 16. The Invert operator also has a combinatorial interpretation. If b = I(a), then b_{n} is the number of ordered arrangements of postage stamps, of total value n + 1, that can be formed if we have a_{i} types of stamps of value i + 1, when i ≥ 0 [3]. We want to point out a connection between Invert and complete ordinary Bell polynomials, as they are presented in Port [7], arising from their combinatorial interpretations. So we recall that complete ordinary Bell polynomials are defined by (Port [7])

B_{n}(t) = ∑_{k=1}^{n} B_{n,k}(t),

where t = (t_{1}, t_{2}, …) and the partial ordinary Bell polynomials B_{n,k}(t) satisfy

(∑_{n≥1} t_{n}z^{n})^{k} = ∑_{n≥k} B_{n,k}(t)z^{n},

and

B_{n,k} (t) = ∑_{\begin{matrix}i_{1}+2i_{2}+···+ni_{n}=n \\ i_{1}+i_{2}+···+i_{n}=k \end{matrix}} \frac{k!}{i_{1}!i_{2}! ⋯ i_{n}!}t_{1}^{i_{1}}t_{2}^{i_{2}} ⋯ t_{n}^{i_{n}} .

The last relation provides a straightforward combinatorial interpretation for complete ordinary Bell polynomials. In fact B_{n}(t) corresponds to the sum of monomials t_{i_{1}} ⋯ t_{i_{k}} (written with repetition of t_{i_{j}}), whose coefficients are the number of way to write the word t_{i_{1}} ⋯ t_{i_{k}}, when ∑_{j=1}^{k} i_{j} = n. For example,

B_{4}(t) = t_{4} + 2t_{1}t_{3} + t_{2}t_{2} + 3t_{1}t_{1}t_{2} + t_{1}t_{1}t_{1}t_{1} = t_{4} + 2t_{1}t_{3} + t_{2}^{2} + 3t_{1}^{2}t_{2} + t_{1}^{4}.

Now, if we consider the sequence b = I(a), a ∈ S(R), we can put together the combinatorial aspects of Invert and complete ordinary Bell polynomials

b_{n} = B_{n+1}(a).

3 Construction and Deconstruction of impulsequences

In this section we explain how previous results can create an interesting set of tools, which will enable us to play with sequences a ∈ W(R), of order r, having the r–impulse (0, 0, …, 0, 1) as initial conditions. Let us call such sequences impulsequences. Considering the impulsequence a, with characteristic polynomial f(t) of degree r, from Theorem 10 and Theorem 13, respectively, we have

L^{(z)}(f(t)) = f(t − z),  (10)

and

I^{(z)}(f(t)) = f(t) − z.  (11)

Using these relations we can construct every impulsequence, from the startsequence

u = (1, 0, 0, 0, …).

Viceversa it is possible to deconstruct every impulse sequence into u.

The sense of these claims will soon be clear. We also need the two operators σ and ρ, introduced in Definition 5. They act on a linear recurrent sequence changing its characteristic polynomial, dividing or multiplying it by t, respectively, when possible. So let us start a L–construction from u. The startsequence u = r^{(0)} has characteristic polynomial f(t) = t, with zero {0}. The operator L^{(z_{1})}, applied to r^{(0)}, translates by z_{1} the zero of f(t). The characteristic polynomial becomes f(t − z_{1}) = t − z_{1} and so

v^{(1)} = L^{(z_{1})}(r^{(0)}) = (1, z_{1}, z_{1}^{2}, z_{1}^{3}, …).

Using the operator ρ, we obtain

ρ(v^{(1)}) = (0, 1, z_{1}, z_{1}^{2}, z_{1}^{3}, …) = r^{(1)}.

The sequence r^{(1)} has characteristic polynomial t(t−z_{1}) = t^{2} −z_{1}t with zeros {0, z_{1}}. Acting on r^{(1)} with another operator L^{(z_{2})}, we obtain the sequence v^{(2)} with characteristic polynomial

(t − z_{2})(t − (z_{2} + z_{1})) = t^{2} − (z_{1} + 2z_{2})t + z_{2}(z_{1} + z_{2}),

and zeros {z_{2}, z_{2} + z_{1}}, i.e., we obtain

v^{(2)} = L^{(z_{2})}(r^{(1)}) = (0, 1, z_{1} + 2z_{2}, …).

For every α, β in a field 𝔽 containing R, setting z_{2} = α, z_{1} + z_{2} = β, it is possible to L–construct any impulsequence of order 2, with the 2–impulse (0, 1) as initial conditions. Repeating this process we can L–construct any impulsequence of order r. The explained process is naturally invertible using the operator σ. So it is possible to L–deconstruct every impulsequence into the startsequence.

Example 17. The L–deconstruction of the Fibonacci sequence \underline{A000045}. The characteristic polynomial of F = (0, 1, 1, 2, 3, 5, 8, …) is t^{2} − t − 1, with zeros

\{\frac{1 + \sqrt{5}}{2}, \frac{1 − \sqrt{5}}{2}\} .

Appliying L^{(− \frac{1+\sqrt{5}}{2} )} we obtain

L^{(− \frac{1+\sqrt{5}}{2} )}(F) = (0, 1, −\sqrt{5}, 5, …) = r^{(1)},

with characteristic polynomial whose zeros are translated by −\frac{1+\sqrt{5}}{2}, i.e., \{0, −\sqrt{5}\}. Using the operator σ we eliminate 0 from the last sequence and from the set of zeros of its characteristic polynomial. The new sequence is

σ(r^{(1)}) = (1, −\sqrt{5}, 5, …)

and the characteristic polynomial has the only zero \{−\sqrt{5}\}. In order to eliminate \{−\sqrt{5}\} and to obtain the startsequence, we have to use the operator L^{(^{\sqrt{5}})}, as folows:

L^{(^{\sqrt{5}})}(σ(r^{(1)})) = u.

Inverting this process we can L–construct the Fibonacci sequence. Our purpose is to transform the characteristic polynomial t of u = r^{(0)} = (1, 0, 0, 0, …) into the polynomial t^{2}−t−1.

L^{(−\sqrt{5})}(r^{(0)}) = (1, −\sqrt{5}, 5, …)

has characteristic polynomial t + \sqrt{5} with zero \{−\sqrt{5}\} and

ρ (L^{(−\sqrt{5})}(r^{(0)})) = (0, 1, −\sqrt{5}, 5, …) = r^{(1)},

has characteristic polynomial t(t + \sqrt{5}) with zeros \{0, −\sqrt{5}\}. In order to obtain the poly-

nomial t^{2} − t − 1, with zeros \{\frac{1 + \sqrt{5}}{2}, \frac{1 − \sqrt{5}}{2}\}, finally we have to use L^{( \frac{1+\sqrt{5}}{2} )} and we

obtain

L^{( \frac{1+\sqrt{5}}{2} )}(r^{(1)}) = F.

It is possible to find an explicit expression of v^{(k)} = L^{(z_{k})}(r^{(k−1)}) as shown in the following

Proposition 18. After k steps of L-construction with parameters z_{1}, z_{2}, …, z_{k} ∈ 𝔽, 𝔽 a field, we obtain the recurrent sequence v^{(k)} = L^{(z_{k})}(r^{(k−1)}) with characteristic polynomial

(t − z_{k})(t − z_{k} − z_{k−1}) ⋯ (t − z_{k} − z_{k−1} − ⋯ − z_{1})  (12)

and we have

(v^{(k)})_{n} = ∑_{h_{k−1}=k−1}^{n} ∑_{h_{k−2}=k−2}^{h_{k−1}−1} ⋯ ∑_{h_{1}=1}^{h_{2}−1} \binom{n}{h_{k−1}}\binom{h_{k−1} − 1}{h_{k−2}} ⋯ \binom{h_{2} − 1}{h_{1}}z_{k}^{n−h_{k−1}}z_{k−1}^{h_{k−1}−h_{k−2}−1} ⋯ z_{1}^{h_{1}−1}

(13)

Proof. Using an inductive argument, as previously explained, we have v^{(1)} = L^{(z_{1})}(r^{(0)}), when k = 1, with characteristic polynomial f(t) = t − z_{1}. If we suppose v^{(k−1)} = L^{(z_{k−1})}(r^{(k−2)}) with characteristic polynomial

(t − z_{k−1})(t − z_{k−1} − z_{k−2}) ⋯ (t − z_{k−1} − z_{k−2} − ⋯ − z_{1}),

we have ρ(v^{(k−1)}) = r^{(k−1)} with characteristic polynomial

t(t − z_{k−1})(t − z_{k−1} − z_{k−2}) ⋯ (t − z_{k−1} − z_{k−2} − ⋯ − z_{1})

and finally v^{(k)} = L^{(z_{k})}(r^{(k−1)}) with characteristic polynomial given by (12). Now let us consider v^{(k)}. From Definition 4 and Definition 5, observing that the first k − 1 terms of v^{(k)} are zeros

(v^{(k)})_{n} = ∑_{h_{k−1}=k−1}^{n} \binom{n}{h_{k−1}}z_{k}^{n−h_{k−1}}(ρ(v^{(k−1)}))_{h_{k−1}} = ∑_{h_{k−1}=k−1}^{n} \binom{n}{h_{k−1}}z_{k}^{n−h_{k−1}}(v^{(k−1)})_{h_{k−1}−1}.

Iterating this process of substitution, for 1 ≤ j ≤ k − 1, we have

(v^{(k−j)})_{h_{k−j}} = ∑_{h_{k−j−1}=k−j−1}^{h_{k−j}−1} \binom{h_{k−j} − 1}{h_{k−j−1}}z_{k−j}^{h_{k−j}−h_{k−j−1}−1}(v^{(k−j−1)})_{h_{k−j}−1}

we find easily (13).

Remark 19. As an immediate consequence of the previous proposition we observe that for any a ∈ W(R) of order 2, with characteristic polynomial f(t), we have

a_{n} = ∑_{i=1}^{n} \binom{n}{i}α^{n−i}(β − α)^{i−1} = ∑_{i=1}^{n} \binom{n}{i}α^{n−i}(−\sqrt{Δ})^{i−1}, ∀n ≥ 1,

where α, β are the zeros of f(t) and Δ its discriminant. In fact, the sequence a corresponds to a particular sequence v^{(2)} constructed from u setting z_{2} = α, z_{1} + z_{2} = β. We also have incidentally retrieved the classical Binet formula

∑_{i=1}^{n} \binom{n}{i}α^{n−i}(β − α)^{i−1} = \frac{1}{β − α} ∑_{i=0}^{n} \binom{n}{i}α^{n−i}(β − α)^{i} − \frac{α^{n}}{β − α} = \frac{β^{n} − α^{n}}{β − α} .

The processes of I–construction and I–deconstruction are similar to L–construction and L–deconstruction. Starting from u = r^{(0)}, and using the operator I^{(z_{1})}, we obtain the sequence with characteristic polynomial t − z_{1}, which is the same result obtained using L^{(z_{1})}. Applying ρ we have the sequence r^{(1)}, which we can write explicitly as follows:

r^{(1)} = (0, 1, z_{1}, z_{1}^{2}, z_{1}^{3}, …),

and using the operator I^{(z_{2})}, the resulting sequence

I^{(z_{2})}(r^{(1)}) = (0, 1, z_{1}, z_{1}^{2} + z_{2}, …)

has characteristic polynomial t(t − z_{1}) − z_{2} = t^{2} − z_{1}t − z_{2}. Repeating or inverting this process we can I–construct or I–deconstruct any impulsequence of order r. So we have all the necessary tools to transform any impulsequence a into any other impulsequence by means of a convenient composition of operators σ, ρ, I^{(z)}, L^{(z)}!

4 Applications to integer sequences

4.1 Binomial anti–mean transform

In this section we study the action of the operator L^{(− \frac{h}{2} )}, which we call Binomial anti–mean transform in analogy to the definition of the Binomial mean transform [9].

Let W = W (s_{0}, s_{1}, h, k) be a linear recurrent sequence of degree 2 over R. Assume that its characteristic polynomial is t^{2} − ht + k = (t − α)(t − β) and the initial conditions are s_{0}, s_{1} ∈ R. We know that L^{(y)}(W) has characteristic polynomial

t^{2} − (h + 2y)t + y^{2} + hy + k = (t − (α + y))(t − (β + y)).

If we consider y = −\frac{h}{2} , the binomial anti–mean transform

L^{(− \frac{h}{2} )}(W) = C

recurs with polynomial t^{2} − \frac{Δ}{4} , where Δ = h^{2} − 4k. For the sequence C we have

\begin{cases}C_{0} = s_{0} \\ C_{1} = \frac{δ}{2} \\ C_{n} = \frac{Δ^{⌊ \frac{n}{2} ⌋}}{2^{n}}δ^{n mod 2}s_{0}^{1−n mod 2}, ∀n ≥ 2 \end{cases}

where δ = 2s_{1} − s_{0}h. Using Definition 4, we obtain

∑_{i=0}^{n} \binom{n}{i} (−\frac{h}{2})^{n−i} W_{i} = C_{n}.

If s_{0} = 0, then C_{2n} = 0 ∀n. This is the case of Fibonacci numbers F = W (0, 1, 1, −1)

∑_{i=0}^{2n} \binom{2n}{i} (−\frac{1}{2})^{2n−i} F_{i} = 0.

4.2 r–bonacci numbers

The operator Invert acts on the startsequence giving

I(r^{(0)}) = (1, 1, 1, 1, …),

with characteristic polynomial t − 1. Using the operator ρ we obtain

r^{(1)} = ρ(I(r^{(0)})) = (0, 1, 1, 1, …),

with characteristic polynomial t^{2} − t. Now, if we use Invert again, it is easy to observe that the result is F = F^{(2)}, the Fibonacci sequence \underline{A000045}

I(r^{(1)}) = F^{(2)}.

The Tribonacci sequence \underline{A000073}, F^{(3)} = (0, 0, 1, 1, 2, 4, 7, 13, …), has characteristic polynomial t^{3} − t^{2} − t − 1 and initial conditions (0, 0, 1). So applying the operators ρ and I to the Fibonacci sequence we obtain the Tribonacci sequence

I(ρ(F^{(2)})) = F^{(3)}.

From the Tribonacci sequence it is possible to obtain the Tetranacci sequence \underline{A000078} F^{(4)} = (0, 0, 0, 1, 1, 2, 4, 8, …), as follows:

I(ρ(F^{(3)})) = F^{(4)}.

In general, let us start from the r–bonacci sequence F^{(r)}, whose recurrence polynomial is t^{r} −t^{r−1} −t^{r−2} −⋯−1 and initial conditions are the r–impulse. If we compose the operators ρ and I, by Definition 5 and by (11), it is simple to observe that the new sequence has characteristic polynomial t^{r+1} − t^{r} − t^{r−1} − ⋯ − 1 and initial conditions the (r + 1)–impulse, i.e., we obtain the (r + 1)–bonacci sequence F^{(r+1)} = I(ρ(F^{(r)})). Moreover, by Remark 16, we can see that the (r + 1)–bonacci numbers are the complete ordinary Bell polynomials of the r–bonacci numbers

F_{n}^{(r+1)} = B_{n+1}(0, F_{0}^{(r)}, F_{1}^{(r)}, …, F_{n}^{(r)}) .

Finally, by Remark 6 we can use (2), with x = 1, and we find another relation between r–bonacci numbers

F_{n+1}^{(r)} = F_{n}^{(r−1)} + ∑_{i=1}^{n−1} F_{i−1}^{(r)}F_{n−i−1}^{(r−1)} .

4.3 Polynomial sequences

If we consider f : ℕ → ℂ and d ∈ ℕ we have the following equivalent conditions [10]

•

∑_{n≥0} f(n)t^{n} = \frac{P (t)}{(1 − t)^{d+1}} ,  (14)

where P (t) is a polynomial over ℂ with deg(P) ≤ d

• ∀m ≥ 0

Δ^{d+1}f(m) = ∑_{i=0}^{d+1} (−1)^{d+1−i} ( \begin{matrix}d + 1 \\ i \end{matrix} ) f(m + i) = 0,  (15)

• f(n) is a polynomial function of n, deg(f(n)) ≤ d and deg(f(n)) = d ⇔ P (1) ≠ 0. It is well known that it is possible to expand f(n) in terms of the basis n^{i}, 0 ≤ i ≤ d, but an alternative way is to use the basis \binom{n}{i}, 0 ≤ i ≤ d. In this case

f(n) = ∑_{i=0}^{d} (Δ^{i}f(0)) ( \begin{matrix}n \\ i \end{matrix} ) .  (16)

Now we see another point of view to obtain (16), based on the L^{(y)} operator. Let us consider a polynomial f(t) over ℂ such that deg(f) ≤ d. From (15), where we set m = 0, we conclude that the characteristic polynomials of

(f(n)) = (f(0), f(1), f(2), …)

and

(Δ^{n}f(0)) = (Δ^{0}f(0), Δ^{1}f(0)Δ^{2}f(0), …)

are, respectively, (t − 1)^{d+1} and t^{d+1}. Using (10) it is clear that

L^{(−1)}((t − 1)^{d+1}) = t^{d+1}.

Consequently, we have the following one-click deconstruction

L^{(−1)}((f(n))) = (Δ^{n}f(0)),

which implies

L^{(1)}((Δ^{n}f(0))) = L((Δ^{n}f(0))) = (f(n))

and directly gives (16).

4.4 Pyramidal numbers

Let us consider the sequences

(0, 1, q − 1, 2q − 3, 3q − 5, …, 1 + (q − 2)n, …) .

For all q ≥ 2, these sequences have characteristic polynomial (t − 1)^{2}. Their partial sums are the polygonal numbers. For example, when q = 3, we have the triangular numbers (0, 1, 3, 6, 10, 15, …) \underline{A000217}, when q = 4 the square numbers (0, 1, 4, 9, 16, …) \underline{A000290}, and so on. The polygonal numbers have characteristic polynomial (t − 1)^{3}. The nth term of such sequences is given by

P_{q}^{(2)}(n) = \frac{1}{2}(q − 2)n^{2} + \frac{1}{2}(4 − q)n.  (17)

If we calculate the partial sums of polygonal numbers, we obtain the pyramidal numbers. Their characteristic polynomial is (t − 1)^{4}, and the nth term of such sequences is given by P_{q}^{(3)}(n), a third degree polynomial. Proceeding with these partial sums, we can define pyramidal numbers for higher dimensions. The figurate numbers, introduced in Remark 12, are a particular case of these numbers. These sequences recur with a polynomial (t − 1)^{d+1} and the terms of these sequences can be evaluated by P_{q}^{(d)}(n), a polynomial of degree d. So we can apply the one–click deconstruction.

For example, let us consider polygonal numbers P_{q}^{(2)}(n). As we have observed, the polygonal numbers recur all with polynomial (t − 1)^{3}, and equality (17) holds. If we think of (17) as a function f(n) in n, then we can apply the one–click deconstruction to the polygonal numbers

L^{(−1)}((P_{q}^{(2)}(n))) = (0, 1, q − 2, 0, 0, 0, …) .

Indeed, in this case Δ^{0}f(0) = 0, Δf(0) = 1, Δ^{2}f(0) = q − 2 . Consequently

L^{(1)}((0, 1, q − 2, 0, 0, 0, …)) = (P_{q}^{(2)}(n)) .  (18)

In the same way, starting from 1, we obtain

L^{(1)}((1, q − 1, q − 2, 0, 0, 0, …)) = (P_{q}^{(2)}(n)), n ≥ 1 .  (19)

The OEIS [8] shows special cases of (18) and (19), without proof. See for q = 4 Sloane’s \underline{A000566} (Paul Barry, 2003), and for q = 4, 5, 6, 8, 9, 10 Sloane’s \underline{A000290}, \underline{A000326}, \underline{A000384}, \underline{A000567}, \underline{A001106}, \underline{A001107} (Gary W. Adamson, 2007-2008).

References

[1] R. Bacher, Sur l’inversion des séries, Prépubl. Inst. Fourier 589 (2003), http://www-fourier.ujf-grenoble.fr/PUBLIS/publications/REF 589.pdf.

[2] S. Barbero and U. Cerruti, Catalan moments on the occasion the Thirteenth International Conference on Fibonacci Numbers and Their Applications, Congr. Numer. 201 (2010), 187–209.

[3] M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Algebra Appl., 226–228 (1995), 57–72.

[4] J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, 1995.

[5] L. E. Dickson, History of the Theory of Numbers, Vol 2: Diophantine Analysis, Chelsea Publishing Company, 1971.

[6] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: a Foundation for Computer Science, Addison–Wesley Publishing Co., 1988.

[7] D. Port, Polynomial maps with applications to combinatorics and probability theory, http://dspace.mit.edu/handle/1721.1/28041.

[8] N. J. A. Sloane, The On–Line Encyclopedia of Integer Sequences. Published electronically at, http://www.research.att.com/ njas/sequences, 2010.

[9] M. Z. Spivey and L. L. Steil, The k–binomial transforms and the Hankel transform, J. Integer Seq. 9 (2006), Article 06.1.1.

[10] R. P. Stanley Enumerative Combinatorics Vol. I, Cambridge Stud. Adv. Math. 49, 1997, pp. 208–209.

2000 Mathematics Subject Classification: Primary 11B37; Secondary 11B39. Keywords: Bell polynomials, binomial operator, Fibonacci numbers, impulse sequences, invert operator, pyramidal numbers, recurrent sequences.

(Concerned with sequences \underline{A000045}, \underline{A000073}, \underline{A000078}, \underline{A000217}, \underline{A000290}, \underline{A000292}, \underline{A000326}, \underline{A000384}, \underline{A000566}, \underline{A000567}, \underline{A001106}, and \underline{A001107}.)

Received September 14 2009; revised versions received February 15 2010; March 9 2010; June 29 2010; July 10 2010. Published in Journal of Integer Sequences, July 16 2010.

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