On the Product Representation of Number Sequences, with Applications to the Family of

Generalized Fibonacci Numbers

Michelle Rudolph-Lilith

Unité de Neurosciences, Information et Complexité (UNIC) CNRS, 1 Ave de la Terrasse 91198 Gif-sur-Yvette France

               rudolph@unic.cnrs-gif.fr

Abstract

We investigate general properties of number sequences which allow explicit representation in terms of products. We find that such sequences form whole families of number sequences sharing similar recursive identities. Applying the proposed identities to power sequences and the sequence of Pochhammer numbers, we recover and generalize known recursive relations. Restricting to the cosine of fractional angles, we then study the special case of the family of k-generalized Fibonacci numbers, and present general recursions and identities which link these sequences.

1 Introduction

It has long been known that Fibonacci (\underline{A000045}) and Pell (\underline{A000129}) numbers, defined by

F_{0} = 0, F_{1} = 1, F_{n} = F_{n−1} + F_{n−2}  (1)

and

P_{0} = 0, P_{1} = 1, P_{n} = 2P_{n−1} + P_{n−2}  (2)

with n ≥ 2, respectively, can be represented in product form (see, e.g., [1, 2, 3, 4, 5]), specifically

F_{n} = ∏_{l=1}^{⌊ \frac{n−1}{2} ⌋} (3 + 2 cos (\frac{2lπ}{n} )) = ∏_{l=1}^{n−1} (1 − 2i cos (\frac{lπ}{n} ))  (3)

and

P_{n} = 2^{⌊ \frac{n}{2} ⌋} ∏_{l=1}^{⌊ \frac{n−1}{2} ⌋} (3 + cos (\frac{2lπ}{n} )) = ∏_{l=1}^{n−1} (2 − 2i cos [\frac{lπ}{n} ]) ,  (4)

where n ∈ ℕ, n ≥ 2. The above sequences F_{n} and P_{n} are specific examples of general Lucas sequences, the latter being defined by the recursive relation

L_{0}^{(m,p)} = 0, L_{1}^{(m,p)} = 1, L_{n}^{(m,p)} = mL_{n−1}^{(m,p)} − pL_{n−2}^{(m,p)}  (5)

for n ≥ 2 [4]. Already Zeitlin [3] showed that

L_{n}^{(m,p)} = p^{\frac{n−1}{2}} ∏_{l=1}^{n−1} (\frac{m}{\sqrt{p}} − 2 cos (\frac{lπ}{n} )) ,  (6)

m, p ∈ ℝ, provides a valid product representation of all members in general Lucas sequences. Later, expressions of the form (6) were used to obtain other explicit representations of the corresponding number sequences in terms of finite power series in the sequence parameters (see, e.g., [6, 7, 8, 9]), thus highlighting the importance and usefulness of such product representations for the investigation of number sequences.

In this contribution, we will show that product representations of number sequences can also be utilized to establish direct links between different sequences. To that end, we formulate

Definition 1. (family of number sequences) Let {x_{n,l}} with x_{n,l} ∈ ℂ and n, l ∈ ℕ be an arbitrary two-parameter set of numbers. The corresponding family of number sequences {X_{n,m}} is defined by the set of all X_{n,m} with

X_{n,m} = ∏_{l=1}^{n}(m + x_{n,l}),  (7)

where m ∈ ℂ labels the individual number sequences within the family, and n the members of each sequence.

In what follows, we will restrict to the subset of families which are constructed by m ∈ ℤ. A concrete example of such a family is given if we set x_{n,l} = −2i cos (\frac{lπ}{n+1}). In this case, using (6) with p = −1 and m ∈ ℤ, we have

X_{n,m} = ∏_{l=1}^{n} (m − 2i cos (\frac{lπ}{n+1})) = L_{n+1}^{(m,−1)},  (8)

thus X_{n,m} defines the family of generalized Fibonacci sequences F_{n}^{(m)} := L_{n}^{(m,−1)}, n ≥ 2 obeying the recursive relation

F_{0}^{(m)} = 0, F_{1}^{(m)} = 1, F_{n}^{(m)} = mF_{n−1}^{(m)} + F_{n−2}^{(m)}.  (9)

We will call this family the Fibonacci family, and explore some of its properties in Section 4.

The paper is organized as follows. In Section 2, we will prove various general properties of the number sequences X_{n,m} within a given family, with focus on linear recursive relations between the individual sequences. Two simple examples will be investigated in Section 3, and Section 4 will focus on the Fibonacci family defined in (8). Some generalizations will be discussed at the end.

2 Product representation of certain number sequences

For any given set of numbers {x_{n,l}}, we define

𝒳_{n} := ∑_{l=1}^{n} x_{n,l}.  (10)

We will first express 𝒳_{n} in terms of the associated number sequences X_{n,m} defined in (7).

Lemma 2. For any given set of numbers {x_{n,l}} with x_{n,l} ∈ ℂ, n, l ∈ ℕ, the sum over x_{n,l} is given by

𝒳_{n} = \frac{(−1)^{n}}{n!} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} l X_{n,l} − \frac{1}{2}n(n + 1),  (11)

where X_{n,m}, m ∈ ℤ denotes the number sequences associated with {x_{n,l}}.

Proof. We first construct a system of equations by explicitly factorizing (7) for successive m ∈ [1, n]. To that end, we define for p ∈ [1, n]

X_{n}^{(p)} := ∑_{l_{1}=1}^{n−p+1} ∑_{l_{2}=1}^{n−p+2} ⋯ ∑_{l_{p}=1}^{n} x_{n,l_{1}}x_{n,l_{2}} ⋯ x_{n,l_{p}},

where the summation is subject to the constraints l_{i+1} > l_{i} for all l_{i}, i ∈ [1, p]. Specifically, for p = 1, we have

X_{n}^{(1)} = ∑_{l_{1}=1}^{n} x_{n,l_{1}} = x_{n,1} + x_{n,2} + ⋯ + x_{n,n},

for p = 2

X_{n}^{(2)} = ∑_{l_{1}=1}^{n−1} ∑_{\begin{matrix}l_{2}=1 \\ l_{2}>l_{1} \end{matrix}}^{n} x_{n,l_{1}}x_{n,l_{2}}
= x_{n,1}x_{n,2} + x_{n,1}x_{n,3} + ⋯ + x_{n,n−1}x_{n,n},

and for p = 3

X_{n}^{(3)} = ∑_{l_{1}=1}^{n−2} ∑_{\begin{matrix}l_{2}=1 \\ l_{2}>l_{1} \end{matrix}}^{n−1} ∑_{\begin{matrix}l_{3}=1 \\ l_{3}>l_{2} \end{matrix}}^{n} x_{n,l_{1}}x_{n,l_{2}}x_{n,l_{3}}
= x_{n,1}x_{n,2}x_{n,3} + x_{n,1}x_{n,2}x_{n,4} + ⋯ + x_{n,n−2}x_{n,n−1}x_{n,n}.

With this, the product in (7) yields

X_{n,m} = (m + x_{n,1})(m + x_{n,2})(m + x_{n,3}) ⋯ (m + x_{n,n})
= m(m + x_{n,2})(m + x_{n,3}) ⋯ (m + x_{n,n}) + x_{n,1}(m + x_{n,2})(m + x_{n,3}) ⋯ (m + x_{n,n})
= m^{2}(m + x_{n,3}) ⋯ (m + x_{n,n}) + m(x_{n,1} + x_{n,2})(m + x_{n,3}) ⋯ (m + x_{n,n})
+ x_{n,1}x_{n,2}(m + x_{n,3}) ⋯ (m + x_{n,n})
= m^{n} + m^{n−1}(x_{n,1} + x_{n,2} + x_{n,3} + ⋯ + x_{n,n}) + ⋯ + x_{n,1}x_{n,2}x_{n,3} ⋯ x_{n,n}
= m^{n} + m^{n−1}X_{n}^{(1)} + ⋯ + X_{n}^{(n)},

from which we obtain for m ∈ [1, n] the following system of linear equations in X_{n}^{(p)}, p ∈ [1, n]:

X_{n,1} = 1^{n} + 1^{(n−1)}X_{n}^{(1)} + 1^{(n−1)}X_{n}^{(2)} + ⋯ + X_{n}^{(n)}
X_{n,2} = 2^{n} + 2^{(n−1)}X_{n}^{(1)} + 2^{(n−2)}X_{n}^{(2)} + ⋯ + X_{n}^{(n)}
X_{n,3} = 3^{n} + 3^{(n−1)}X_{n}^{(1)} + 3^{(n−2)}X_{n}^{(2)} + ⋯ + X_{n}^{(n)}
⋮
X_{n,n} = n^{n} + n^{(n−1)}X_{n}^{(1)} + n^{(n−2)}X_{n}^{(2)} + ⋯ + X_{n}^{(n)}.

This system can be written in more compact form as

X_{n,i} − i^{n} = ∑_{j=1}^{n} a_{ij}X_{n}^{(j)},  (12)

where a_{ij} = i^{n−j}, i, j ∈ [1, n]. What remains is to solve (12) for X_{n}^{(1)} = 𝒳_{n}. To that end, we note that a_{in} = 1, ∀i ∈ [1, n], which allows us to construct a new system of n − 1 equations by subtracting successive equations in (12). We obtain

X_{n,i+1} − X_{n,i} − ((i + 1)^{n} − i^{n}) = ∑_{j=1}^{n−1} a_{ij}^{(1)}X_{n}^{(j)},  (13)

where

a_{ij}^{(1)} = a_{i+1,j} − a_{ij} = ((i + 1)^{n−j} − i^{n−j}) = ∑_{l=0}^{1}(−1)^{l+1}\binom{1}{l}(i + l)^{n−j}

for i, j ∈ [1, n − 1]. Again, a_{i,n−1}^{(1)} = 1, ∀i ∈ [1, n − 1], and we can further reduce the system (13) by subtracting successive equations. After m repetitions, we have

(−1)^{m} ∑_{l=0}^{m}(−1)^{l}\binom{m}{l}X_{n,i+l} − (−1)^{m} ∑_{l=0}^{m}(−1)^{l}\binom{m}{l}(i + l)^{n} = ∑_{j=1}^{n−m} a_{ij}^{(m)}X_{n}^{(j)}  (14)

for i, j ∈ [1, n − m], where

a_{ij}^{(m)} = (−1)^{m} ∑_{l=0}^{m}(−1)^{l}\binom{m}{l}(i + l)^{n−j}

with a_{i,n−m}^{(m)} = m! for all i ∈ [1, n − m]. For m = n − 1, we finally obtain

(−1)^{n−1} ∑_{l=0}^{n−1}(−1)^{l}\binom{n − 1}{l}X_{n,1+l} − (−1)^{n−1} ∑_{l=0}^{n−1}(−1)^{l}\binom{n − 1}{l}(1 + l)^{n} = (n − 1)! X_{n}^{(1)}.

Changing the summation variable l → l + 1 in both sums, and observing that \binom{n−1}{l−1} = \frac{l}{n} \binom{n}{l} and

\frac{(−1)^{n}}{n!} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l}l^{n+1} = \frac{1}{2}n(n + 1)

[10, Equation (1.14)], we finally arrive at (11).

Lemma 2 provides, for any given n ≥ 1, an explicit representation of the sum over x_{n,l}, Equation (11), in terms of a finite linear combination of the number sequences X_{n,m}. With this, we can immediately formulate

Lemma 3. For any given set of numbers {x_{n,l}} with x_{n,l} ∈ ℂ, l ∈ ℕ and associated family of number sequences {X_{n,m}}, m ∈ ℤ, the sum over x_{n,l} obeys for any given n ∈ ℕ the identities

𝒳_{n} = \frac{(−1)^{n}}{n!} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} l X_{n,l+m} − \frac{1}{2}n(n + 1) − nm  (15)

and, for m ≠ 0,

𝒳_{n} = \frac{(−1)^{n}}{n! m^{n−1}} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} l X_{n,lm} − \frac{1}{2}n(n + 1)m.  (16)

Proof. We first prove (15). Let us define X_{n,m} and 𝒳_{n} for the set of numbers (m′ + x_{n,l}) as follows:

X_{n,m}^{(m′)} := ∏_{l=1}^{n} (m + (m′ + x_{n,l}))

𝒳_{n}^{(m′)} := ∑_{l=1}^{n}(m′ + x_{n,l})

for arbitrary m′ ∈ ℤ. From the first equation and definition (7), it follows immediately that X_{n,m}^{(m′)} = X_{n,m+m′} and 𝒳_{n}^{(m′)} = nm′ + 𝒳_{n}, which together with (11) yield (15).

The second relation (16) can be shown in a similar fashion. We define X_{n,m} and 𝒳_{n} for the set of numbers x_{n,l}/m′, m′ ∈ ℤ, m ≠ 0 as follows:

X̃_{n,m}^{(m′)} := ∏_{l=1}^{n} (m + \frac{x_{n,l}}{m′} )

𝒳̃_{n}^{(m′)} := ∑_{l=1}^{n} \frac{x_{n,l}}{m′} ,

from which follows that

X̃_{n,m}^{(m′)} = \frac{1}{m^{′n}} ∏_{l=1}^{n}(mm′ + x_{n,l}) = \frac{1}{m^{′n}} X_{n,mm′}  (17)

and 𝒳̃_{n}^{(m′)} = 𝒳_{n}/m′. Using again (11), we obtain (16).

Lemma 3 is interesting in various respects. It not just generalizes (11), but also shows that, for an given n ∈ ℕ, various combinations of X_{n,m} within a given family of number sequences must yield the same result 𝒳_{n}. This, in turn, allows us to construct general relations between X_{n,m}, which will hold for all families of number sequences {X_{n,m}} representable in product form (7), and constitutes the main result of this contribution. We can formulate

Proposition 4. The members X_{n,m} of a given family of number sequences {X_{n,m}}, m ∈ ℤ and n ∈ ℕ, obey the general recursive relation

X_{n,m+1} = (−1)^{n} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l − 1}X_{n,l+m−n} + n!  (18)

and are subject to the identity

\frac{1}{m^{n−1}} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} l X_{n,lm} = ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} l X_{n,l} + \frac{(−1)^{n−1}}{2} (1 − m)n(n + 1)!  (19)

for m ≠ 0.

Proof. The proof of (18) utilizes (15) for m → m − n + 1 and m → m − n, yielding

𝒳_{n} = \frac{(−1)^{n}}{n!} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} l X_{n,l+m−n+1} − \frac{1}{2}n(n + 1) − n(m − n + 1)

= \frac{(−1)^{n}}{n!} ∑_{l=1}^{n−1}(−1)^{l}\binom{n}{l} l X_{n,l+m−n+1} + \frac{1}{(n − 1)!}X_{n,m+1} − \frac{1}{2}n(n + 1) − n(m − n + 1)

and

𝒳_{n} = \frac{(−1)^{n}}{n!} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} l X_{n,l+m−n} − \frac{1}{2}n(n + 1) − n(m − n)

= \frac{(−1)^{n}}{n!} ∑_{l=1}^{n−1}(−1)^{l+1}\binom{n}{l + 1} (l + 1) X_{n,l+m−n+1} + \frac{(−1)^{n+1}}{(n − 1)!} X_{n,m−n+1}

−\frac{1}{2}n(n + 1) − n(m − n),

respectively, where in the last step l → l − 1 was used. Subtracting both identities and observing that (l + 1)\binom{n}{l+1} + l\binom{n}{l} = n\binom{n}{l}, we obtain

\frac{(−1)^{n+1}}{(n − 1)!} ∑_{l=1}^{n−1}(−1)^{l}\binom{n}{l}X_{n,l+m−n+1} + \frac{(−1)^{n+1}}{(n − 1)!} X_{n,m−n+1} − \frac{1}{(n − 1)!}X_{n,m+1} + n = 0,

from which

\frac{(−1)^{n+1}}{(n − 1)!} ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l}X_{n,l+m−n+1} − \frac{1}{(n − 1)!}X_{n,m+1} + n = 0

follows. After a change of the summation variable l → l + 1, the last relation yields (18). In a similar fashion, (16) yields together with (11)

\frac{(−1)^{n}}{n!} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} l X_{n,l} − \frac{1}{2}n(n + 1) = \frac{(−1)^{n}}{n! m^{n−1}} ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} l X_{n,lm} − \frac{1}{2}n(n + 1)m,

from which (19) follows.

Equation (18) in Proposition 4 provides general linear recursions in m ∈ ℤ for X_{n,m}. The form of these recursions depends on n, and contains an increasing number of terms for increasing n. Specifically, for any given n, (18) expresses X_{n,m} in terms of X_{n,m′} with m′ ∈ [m − n, m − 1]. Based on these recursions, using the generating function approach, we can deduce explicit identities which express X_{n,m} and X_{n,−m} for m ≥ n, in terms of X_{n,m′} with m′ ∈ [0, n − 1] and m′ ∈ [−n + 1, 0], respectively.

Corollary 5. For any given family of number sequences {X_{n,m}}, the following identities hold

X_{n,m} = ∑_{l=0}^{n−1}(−1)^{n+l} \frac{n − l}{l − m}\binom{m}{n}\binom{n}{l}X_{n,l} + \frac{m!}{(m − n)!}  (20)

X_{n,−m} = ∑_{l=0}^{n−1}(−1)^{n+l} \frac{n − l}{l − m}\binom{m}{n}\binom{n}{l}X_{n,−l} + (−1)^{n} \frac{m!}{(m − n)!}  (21)

for all n ∈ ℕ and m ∈ ℤ with m ≥ n.

Proof. We start with (18) for m + 1 → m, i.e.,

X_{n,m} = ∑_{l=1}^{n}(−1)^{n+l}\binom{n}{l − 1}X_{n,l+m−n−1} + n!,

and define the general generating function

A(z) := ∑_{m≥0} X_{n,m}z^{m}  (22)

for arbitrary z ∈ ℝ, |z| < 1, z ≠ 0. Multiplication of X_{n,m} with z^{m} and summation over m ≥ n yields

∑_{m≥n} X_{n,m}z^{m} = ∑_{m≥n} ∑_{l=1}^{n}(−1)^{n+l}\binom{n}{l − 1}X_{n,l+m−n−1}z^{m} + n! ∑_{m≥n} z^{m}.  (23)

For any given n ∈ ℕ, the term on the left-hand side equals

∑_{m≥n} X_{n,m}z^{m} = X_{n,n}z^{n} + X_{n,n+1}z^{n+1} + ⋯ = A(z) − ∑_{l=0}^{n−1} X_{n,l}z^{l},

and the second term on the right-hand side simplifies to

n! ∑_{m≥n} z^{m} = n! (z^{n} + z^{n+1} + ⋯) = n! \frac{z^{n}}{1 − z} .

To treat the first term on the right-hand side, we first exchange the order of both sums, and observe that

∑_{m≥n} X_{n,l+m−n−1}z^{m} = X_{n,l−1}z^{n} + X_{n,l}z^{n+1} + ⋯ = (A(z) − ∑_{m=0}^{l−2} X_{n,m}z^{m}) z^{n−l+1},

which can easily be shown by generalizing the result for consecutive l ≥ 2. For instance, for l = 1, we have

∑_{m≥n} X_{n,m−n}z^{m} = X_{n,0}z^{n} + X_{n,1}z^{n+1} + ⋯ = A(z)z^{n},

and for l = 2

∑_{m≥n} X_{n,m−n+1}z^{m} = X_{n,1}z^{n} + X_{n,2}z^{n+1} + ⋯ = (A(z) − X_{n,0}) z^{n−1}.

With this, (23) takes the form

A(z) − ∑_{l=0}^{n−1} X_{n,l}z^{l}

= (−1)^{n}A(z)z^{n} + ∑_{l=2}^{n}(−1)^{n+l}\binom{n}{l − 1} (A(z) − ∑_{m=0}^{l−2} X_{n,m}z^{m}) z^{n−l+1} + n! \frac{z^{n}}{1 − z} ,

from which we obtain

A(z) (1 − (−1)^{n}z^{n} − ∑_{l=2}^{n}(−1)^{n+l}\binom{n}{l − 1}z^{n−l+1})

= ∑_{l=0}^{n−1} X_{n,l}z^{l} − ∑_{l=2}^{n} ∑_{m=0}^{l−2}(−1)^{n+l}\binom{n}{l − 1}X_{n,m}z^{m+n−l+1} + n! \frac{z^{n}}{1 − z} .  (24)

The term on the left-hand side in the last equation can be further simplified by observing that

1 − (−1)^{n}z^{n} − ∑_{l=2}^{n}(−1)^{n+l}\binom{n}{l − 1}z^{n−l+1}

= 1 − ∑_{l=1}^{n}(−1)^{n+l}\binom{n}{l − 1}z^{n−l+1}

= 1 − ((−1)^{n+1}\binom{n}{0}z^{n} + (−1)^{n+2}\binom{n}{1}z^{n−1} + ⋯ + (−1)^{2n}\binom{n}{n − 1}z)
= 1 − (−1)^{2n} (\binom{n}{n − 1}z + (−1)^{−1}\binom{n}{n − 2}z^{2} + ⋯ + (−1)^{−(n−1)}\binom{n}{0}z^{n})

= 1 − ∑_{l=0}^{n−1}(−1)^{−l}\binom{n}{n − 1 − l}z^{l+1}

= ∑_{l=0}^{n}(−1)^{l}\binom{n}{l}z^{l}
= (1 − z)^{n},

where in the third step we reversed the order of the terms in the sum. Inserting the last relation back into (24) yields

A(z) = ∑_{l=0}^{n−1} X_{n,l} \frac{z^{l}}{(1 − z)^{n}} − ∑_{l=2}^{n} ∑_{m=0}^{l−2}(−1)^{n+l}\binom{n}{l − 1}X_{n,m} \frac{z^{m+n−l+1}}{(1 − z)^{n}} + n! \frac{z^{n}}{(1 − z)^{n+1}} . (25)

We now develop the z-terms in a power series around z = 0. Specifically, for the first term in (25), we have

\frac{z^{l}}{(1 − z)^{n}} = \frac{1}{(n − 1)!} z^{l} \frac{d^{n−1}}{dz^{n−1}} \frac{1}{1 − z}
= \frac{1}{(n − 1)!} z^{l} \frac{d^{n−1}}{dz^{n−1}} ∑_{m≥0} z^{m}

= ∑_{m≥n−1} \frac{m!}{(m − n + 1)!} \frac{1}{(n − 1)!}z^{m−n+l+1}

= ∑_{q≥l} \binom{q − l + n − 1}{n − 1}z^{q}.

In a similar fashion, one obtains

\frac{z^{m+n−l+1}}{(1 − z)^{n}} = ∑_{q≥m+n−l+1} \binom{q − m + l − 2}{n − 1}z^{q}

and

\frac{z^{n}}{(1 − z)^{n+1}} = ∑_{q≥n} \binom{q}{n}z^{q}.

With this, (25) takes the form

A(z) = ∑_{l=0}^{n−1} ∑_{q≥l} \binom{q − l + n − 1}{n − 1}X_{n,l} z^{q}  (26)

− ∑_{l=2}^{n} ∑_{m=0}^{l−2} ∑_{q≥m+n−l+1}(−1)^{n+l}\binom{n}{l − 1}\binom{q − m + l − 2}{n − 1}X_{n,m} z^{q} + n! ∑_{q≥n} \binom{q}{n}z^{q},

which, we recall, holds for any n ∈ ℕ, n > 0. What remains is to reorder the terms in the above sums, and collect all terms of equal power in z. The first term in (26) yields

∑_{l=0}^{n−1} ∑_{q≥l} \binom{q − l + n − 1}{n − 1}X_{n,l} z^{q}

= ∑_{q=0}^{n−1} (∑_{l=0}^{q} \binom{q − l + n − 1}{n − 1}X_{n,l}) z^{q} + ∑_{q≥n} (∑_{l=0}^{n−1} \binom{q − l + n − 1}{n − 1}X_{n,l}) z^{q}.

Similarly, observing that the second term in (26) vanishes for n = 1 and yields a power series with minimum degree of 1 for n ≥ 2, we obtain

∑_{l=2}^{n} ∑_{m=0}^{l−2} ∑_{q≥m+n−l+1}(−1)^{n+l}\binom{n}{l − 1}\binom{q − m + l − 2}{n − 1}X_{n,m} z^{q}

= ∑_{q=1}^{n−1} ( ∑_{m=n−q+1}^{n} ∑_{l=0}^{q−n+m−1}(−1)^{n+m}\binom{n}{m − 1}\binom{q − l + m − 2}{n − 1}X_{n,l}) z^{q}

+ ∑_{q≥n} (∑_{m=2}^{n} ∑_{l=0}^{m−2}(−1)^{n+m}\binom{n}{m − 1}\binom{q − l + m − 2}{n − 1}X_{n,l}) z^{q}

The first term in the last relation can further be simplified by changing the summation variable m → m − n + q − 1 and collecting all terms X_{n,l} for a given l:

∑_{m=n−q+1}^{n} ∑_{l=0}^{q−n+m−1}(−1)^{n+m}\binom{n}{m − 1}\binom{q − l + m − 2}{n − 1}X_{n,l}

= − ∑_{m=0}^{q−1} ∑_{l=0}^{m}(−1)^{m−q}\binom{n}{m + n − q}\binom{m + n − l − 1}{n − 1}X_{n,l}

= − ∑_{l=0}^{q−1} X_{n,l} ∑_{m=l}^{q−1}(−1)^{m−q}\binom{n}{m + n − q}\binom{m + n − l − 1}{n − 1}

= ∑_{l=0}^{q−1} \binom{q − l + n − 1}{n − 1}X_{n,l}.

Here, we utilized in the last step the binomial identity

∑_{k=0}^{n}(−1)^{k}\binom{x}{n − k}\binom{k + x − 1}{x − 1} = 0,  (27)

which can easily be shown by induction in the upper summation limit and using Brill’s sum formula [10, Equation (3.181)].

Now, collecting all terms for q < n and q ≥ n, we can then rewrite (26) as

A(z)

= X_{n,0} + ∑_{q=1}^{n−1} (∑_{l=0}^{q} \binom{q − l + n − 1}{n − 1} − ∑_{l=0}^{q−1} \binom{q − l + n − 1}{n − 1}) X_{n,l} z^{q} + ∑_{q≥n} \binom{q}{n}n! z^{q}

+ ∑_{q≥n} (∑_{l=0}^{n−1} \binom{q − l + n − 1}{n − 1} − ∑_{m=2}^{n} ∑_{l=0}^{m−2}(−1)^{n+m}\binom{n}{m − 1}\binom{q − l + m − 2}{n − 1}) X_{n,l} z^{q}

= ∑_{q=0}^{n−1} X_{n,q} z^{q} + ∑_{q≥n} \binom{q}{n}n! z^{q}

+ ∑_{q≥n} (∑_{l=0}^{n−1} \binom{q − l + n − 1}{n − 1}X_{n,l} − ∑_{l=0}^{n−2} X_{n,l} ∑_{m=l}^{n−2}(−1)^{n+m}\binom{n}{m + 1}\binom{q + m − l}{n − 1}) z^{q}

= ∑_{q=0}^{n−1} X_{n,q} z^{q} + ∑_{q≥n} \binom{q}{n}n! z^{q}

− ∑_{q≥n} (\binom{q}{n − 1}X_{n,n−1} − ∑_{l=0}^{n−2}(−1)^{n+l} \frac{n − l}{l − q}\binom{q}{n}\binom{n}{l}X_{n,l}) z^{q},  (28)

where in the penultimate step we, again, collected terms X_{n,l} for any given l by reordering the sum, and in the last step we utilized the binomial relation

∑_{m=l}^{n−2}(−1)^{n+m}\binom{n}{m + 1}\binom{q + m − l}{n − 1} = \binom{q − l + n − 1}{n − 1} − (−1)^{n+l} \frac{n − l}{l − q}\binom{q}{n}\binom{n}{l}.

Noting that q = n+a with a ∈ ℕ, a ≥ 0, this relation is a direct consequence of the binomial identity

∑_{k=0}^{n}(−1)^{k}\binom{x}{n − k}\binom{k + x − 1 + a}{x − 1} = −\frac{a(n − x)}{x(n + a)}\binom{x}{n}\binom{x − 1 + a}{x − 1},  (29)

which itself is a generalization of (27) and can be shown by induction in the upper summation limit using again Brill’s sum formula [10, Equation (3.181)] and the binomial identities

\binom{n}{k} = \frac{n + 1 − k}{k} \binom{n}{k − 1}
\binom{n}{k} = \frac{n + 1 − k}{n + 1} \binom{n + 1}{k}.

From (28), we finally obtain

A(z) = ∑_{q=0}^{n−1} X_{n,q} z^{q} + ∑_{q≥n} (∑_{l=0}^{n−1}(−1)^{n+l} \frac{n − l}{l − q}\binom{q}{n}\binom{n}{l}X_{n,l} + \binom{q}{n}n!) z^{q},

which, after comparison with (22), yields (20). Equation (21) can be shown in a similar fashion.

Equation (18) also allows to deduce a number of general identities the number sequences X_{n,m} of any given family must obey. Specifically, we have

Corollary 6. For n, p, q ∈ ℕ, n ≥ p + 1 with p ≥ 1, 0 ≤ q < p and m ∈ ℤ, the sequences X_{n,m} of any given family of number sequences {X_{n,m}} are subject to the following identities:

∑_{l=0}^{n}(−1)^{l}\binom{n}{l} l^{q} X_{n−p,m−n+l} = 0  (30)

∑_{l=0}^{n}(−1)^{l}\binom{n}{l} l^{p} X_{n−p,m−n+l} − (−1)^{n}n! = 0.  (31)

Proof. We first show that (30) is valid for the special case q = 0 by induction in p. From (18), after change of the summation variable l → l + 1, we have for n → n − 1 ≥ 1 and arbitrary m

(−1)^{n} ∑_{l=0}^{n−1}(−1)^{l}\binom{n − 1}{l}X_{n−1,m−n+l+1} + (n − 1)! = 0,

and for m → m − 1

(−1)^{n} ∑_{l=0}^{n−1}(−1)^{l}\binom{n − 1}{l}X_{n−1,m−n+l} + (n − 1)! = 0.

Subtracting the last two identities yields

(−1)^{n} ∑_{l=0}^{n−2}(−1)^{l}\binom{n−1}{l}X_{n−1,m−n+l+1} − X_{n−1,m}

−(−1)^{n}X_{n−1,m−n} − (−1)^{n} ∑_{l=1}^{n−1}(−1)^{l}\binom{n−1}{l}X_{n−1,m−n+l} = 0.

Using \binom{n−1}{l−1} + \binom{n−1}{l} = \binom{n}{l}, we obtain

(−1)^{n+1} ∑_{l=0}^{n}(−1)^{l}\binom{n}{l}X_{n−1,m−n+l} = 0,

which proves (30) for the special case q = 0 for p = 1. In a similar fashion, assuming (30) is true for q = 0 and a given p ≥ 1, we subtract the resulting relations for n → n − 1, m arbitrary, and m → m − 1, and obtain

−(−1)^{n}X_{n−(p+1),m−n} + (−1)^{n+1} ∑_{l=1}^{n−1}(−1)^{l}\binom{n}{l}X_{n−(p+1),m−n+l} − X_{n−(p+1),m} = 0,

which yields

(−1)^{n+1} ∑_{l=0}^{n}(−1)^{l}\binom{n}{l}X_{n−(p+1),m−n+l} = 0

and, thus, proves (30) for q = 0 and p + 1.

Identity (30) for 0 < q < p can be shown by induction in q. Assuming that (30) is true for a given q ≥ 0 and all p ≥ q + 1, we obtain for n → n − 1 and m → m − 1

∑_{l=0}^{n−1}(−1)^{l}\binom{n − 1}{l} l^{q} X_{n−p,m−n+l} = 0.

Using the binomial identity \binom{n−1}{l} = \frac{n−l}{n} \binom{n}{l}, the last relation can be rewritten as

∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l} l^{q} X_{n−p,m−n+l} − \frac{1}{n} ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l} l^{q+1} X_{n−p,m−n+l} = 0

and further simplified to

−(−1)^{n}n^{q}X_{n−p,m} − \frac{1}{n} ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l} l^{q+1} X_{n−p,m−n+l} = 0,

from which finally

∑_{l=0}^{n}(−1)^{l}\binom{n}{l} l^{q+1} X_{n−p,m−n+l} = 0

follows, thus proving (30) for q < p.

Identity (31) can be shown in an equivalent fashion through induction in p, utilizing (18) and (30).

We finally note that the recursive relation (18) and identities listed in Corollaries 5 and 6 are general and hold for each family of number sequences {X_{n,m}}, thus suggesting that all families constructed from number sequences of the form (11) are governed by identical relationships between their members. In the next section, we will elaborate on this property, and briefly consider two simple examples, before illustrating the application to Fibonacci numbers in Section 4.

Table 1: The first members of the family of power sequences X_{n,m} = m^{n}, n ∈ ℕ and m ∈ ℤ, Equation (32) with c = 0.

n m 0 1 2  3  4  5  6  7
1  0 1 2  3  4  5  6  7
2  0 1 4  9  16  25  36  49
3  0 1 8 27  64  125  216  343
4  0 1 16 81 256 625 1296 2401
5  0 1 32 243 1024 3125 7776 16807
6  0 1 64 729 4096 15625 46656 117649
7  0 1 128 2187 16384 78125 279936 823543

⋮

3 Two simple examples of families of number sequences

3.1 The family of power sequences

Let x_{n,l} = c =∈ ℂ. In this case, we have

X_{n,m} = ∏_{l=1}^{n}(m + c) = (m + c)^{n}  (32)

𝒳_{n} = ∑_{l=1}^{n} c = nc.  (33)

The first few members of this family, for c = 0, are listed in Table 1. With the results presented in the previous section, we can immediately formulate

Corollary 7. The family of power sequences X_{n,m} = (m + c)^{n} obeys for all c ∈ ℂ

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}l(l + m + c)^{n} = \frac{1}{2}(−1)^{n}(2m + 2c + n + 1)nn!

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}l(lm + c)^{n} = \frac{1}{2}(−1)^{n}m^{n−1} (2c + m(n + 1)) nn!

∑_{l=0}^{n}(−1)^{l}\binom{n}{l}(m + c + 1 − l)^{n} = n!

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}l ((lm + c)^{n} − m^{n−1}(l + c)^{n}) = \frac{1}{2}(−1)^{n−1}m^{n−1}(1 − m)n(n + 1)!

for all n ∈ ℕ and m ∈ ℤ,

∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l}\frac{n − l}{l − m}(c + l)^{n} = (−1)^{n} (\frac{(m − n)!}{m!}(c + m)^{n} − 1) n!

∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l}\frac{n − l}{l − m}(c − l)^{n} = (−1)^{n} (\frac{(m − n)!}{m!}(c − m)^{n} − (−1)^{n}) n!

for all n, m ∈ ℕ with m ≥ n, and

∑_{l=0}^{n}(−1)^{l}\binom{n}{l}(n − l)^{q}(m + c − l)^{n−p} = 0  for 0 ≤ q < p

∑_{l=0}^{n}(−1)^{l}\binom{n}{l}(n − l)^{p}(m + c − l)^{n−p} = n!

for all n, p ∈ ℕ with n ≥ p + 1, p ≥ 1 and m ∈ ℤ.

Proof. All identities in Corollary 7 are a direct consequence of Lemmata 2 and 3, Proposition 4 and Corollaries 5 and 6, using (32) and (33).

We note that Corollary 7 yields a number of interesting combinatorial, in particular binomial, identities and their generalizations. Specifically, for c = 0, Gould (1.13), (1.14) and (1.47) are recovered [10]. Furthermore, for any fixed n, X_{n,m} yields the sequence of nth powers of subsequent integers m. The third relation in Corollary 7, for c = 0, provides then the general form of the nth-order linear homogeneous recursions in m with constant coefficients for such sequences:

(m + 1)^{n} = ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l + 1}(m − l)^{n} + n!  (34)

∀m ∈ ℤ. For example, restricting to m ≥ 0, we obtain for n = 2 the sequence of square numbers a_{m} = m^{2}, obeying the known linear recursion

a_{0} = 0, a_{1} = 1, a_{m+1} = 2a_{m} − a_{m−1} + 2, m ≥ 1

(\underline{A000290}, M. Kristof, 2005), for n = 3 the sequence of cubes a_{m} = m^{3}, obeying

a_{0} = 0, a_{1} = 1, a_{2} = 2^{3}, a_{m+1} = 3a_{m} − 3a_{m−1} + a_{m−2} + 6, m ≥ 2

(\underline{A000578}, A. King, 2013), and for n = 4 the sequence a_{m} = m^{4}, subject to the 4th-order linear recursion

a_{0} = 0, a_{1} = 1, a_{2} = 2^{4}, a_{3} = 3^{4}, a_{m+1} = 4a_{m} − 6a_{m−1} + 4a_{m−2} − a_{m−3} + 24, m ≥ 3

(\underline{A000583}, A. King, 2013).

3.2 The family of Pochhammer numbers

Let x_{n,l} = l ∈ ℕ. In this case, we have

X_{n,m} = ∏_{l=1}^{n}(m + l) = (m + 1)_{n}  (35)

𝒳_{n} = ∑_{l=1}^{n} l = \frac{1}{2}n(n + 1),  (36)

where (a)_{n} = Γ(a + n)/Γ(a) denotes the Pochhammer symbol. The first members of this family of sequences are visualized in Table 2. With the results presented in the last section, we can immediately formulate

Corollary 8. The family of Pochhammer sequences X_{n,m} = (m + 1)_{n} obeys

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}l(l + m)_{n} = (−1)^{n}(m + n)nn!

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}(lm)_{n+1} = \frac{1}{2}(−1)^{n}m^{n}(1 + m)n(n + 1)!

∑_{l=0}^{n}(−1)^{l}\binom{n}{l}(l + m − n + 1)_{n} = (−1)^{n}n!

∑_{l=1}^{n}(−1)^{l}\binom{n}{l} ((lm)_{n+1} − m^{n}(l)_{n+1}) = \frac{1}{2}(−1)^{n+1}m^{n}(1 − m)n(n + 1)!

for all n ∈ ℕ and m ∈ ℤ,

∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l}\frac{n − l}{l − m}(1 + l)_{n} = (−1)^{n} (\frac{(m − n)!}{m!}(1 + m)_{n} − 1) n!

∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l}\frac{n − l}{l − m}(1 − l)_{n} = (−1)^{n} (\frac{(m − n)!}{m!}(1 − m)_{n} − (−1)^{n}) n!

for all n, m ∈ ℕ with m ≥ n, and

∑_{l=0}^{n}(−1)^{l}\binom{n}{l}l^{q}(m − n + l + 1)_{n−p} = 0  for 0 ≤ q < p

∑_{l=0}^{n}(−1)^{l}\binom{n}{l}l^{p}(m − n + l + 1)_{n−p} = n!

for all n, p ∈ ℕ with n ≥ p + 1, p ≥ 1 and m ∈ ℤ.

Table 2: The first members of the family of Pochhammer sequences X_{n,m} = (m + 1)_{n}, n ∈ ℕ, m ∈ ℤ, Equation (35).

n m 0  1  2  3  4  5  6  7
1  1  2  3  4  5  6  7  8
2  2  6  12  20  30  42  56  72
3  6  24  60  120  210  336  504  720
4  24 120  360  840  1680  3024  5040  7920
5  120 720 2520 6720 15120 30240 55440  95040
6  720 5040 20160 60480 151200 332640 665280 1235520
7  5040 40320 181440 604800 1663200 3991680 8648640 17297280
⋮

Proof. All identities in Corollary 8 are a direct consequence of Lemmata 2 and 3, Proposition 4 and Corollaries 5 and 6, using (32) and (33) and the Pochhammer identity l(lm + 1)_{n} = (lm)_{n+1}/m, valid ∀m ∈ ℤ.

As in the case of the family of power sequences, for any given n, the relations listed in Corollary 8 provide links between Pochhammer numbers (m)_{n} for different m. Specifically, the third identity yields the general linear recursive rule for sequences defined by (m + 1)_{n} for any fixed n, namely

(m + 1)_{n} = ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l + 1}(m − l)_{n} + n!,  (37)

which is valid ∀m ∈ ℤ. Restricting again to m ≥ 0, n = 2 yields the sequence of Oblong numbers a_{m} = m(m + 1) (\underline{A002378}), subject to the recursion

a_{0} = 0, a_{1} = 2, a_{m+1} = 2a_{m} − a_{m−1} + 2, m ≥ 1,

for n = 3 we obtain the sequence a_{m} = m(m + 1)(m + 2), obeying

a_{0} = 0, a_{1} = 3!, a_{2} = 4!, a_{m+1} = 3a_{m} − 3a_{m−1} + a_{m−2} + 6, m ≥ 2

(\underline{A007531}, Z. Seidov, 2006), and for n = 4 the sequence of products of four consecutive integers a_{m} = m(m + 1)(m + 2)(m + 3) (\underline{A052762}) with

a_{0} = 0, a_{1} = 4!, a_{2} = 5!, a_{3} = \frac{1}{2}6!, a_{m+1} = 4a_{m} − 6a_{m−1} + 4a_{m−2} − a_{m−3} + 24, m ≥ 3.

We note that already for n = 2 and n = 4, the recursions obtained here differ in form from those provided in OEIS [11] for the corresponding sequences.

4 The family of k-generalized Fibonacci numbers

In the remainder of this contribution, we will apply the general results presented in Section 2 to a less trivial case, namely the generalized Fibonacci sequences defined in (9). To that end, we set

x_{n,l} = −2i cos (\frac{lπ}{n + 1}) ,  (38)

from which, using (8), immediately follows that

X_{n,m} = ∏_{l=1}^{n} (m − 2i cos (\frac{lπ}{n + 1})) = F_{n+1}^{(m)}.  (39)

Furthermore, noting that cos (\frac{lπ}{n+1}) , l ∈ [0, n] are the zeros of Chebyshev polynomials of the second kind

U_{n}(x) = \frac{1}{2\sqrt{x^{2} − 1}} ((x + \sqrt{x^{2} − 1})^{n+1} − (x − \sqrt{x^{2} − 1})^{n+1})

(see, e.g., [13, Chapter 22], [14, §8.94]), we have

𝒳_{n} = −2i ∑_{l=1}^{n} cos (\frac{lπ}{n + 1}) = 0,  (40)

where the orthogonality relations for Chebyshev polynomials were used. The first members of the family of sequences formed by (39) are visualized in Table 3. Specifically, the second column m = 1 contains the original Fibonacci sequence F_{n} (\underline{A000045}), Equation (1), and the third column m = 2 the sequence of Pell numbers P_{n} (\underline{A000129}), Equation (2), for n ≥ 1.

While individual columns yield subsequent sequences of generalized Fibonacci numbers, each row, for fixed n, generates new integer sequences whose elements are all generalized Fibonacci numbers. Specifically, for n = 2, we obtain, for m ≥ 0, the sequence a_{m} = m^{2} + 1 (\underline{A002522}), for n = 3 the sequence a_{m} = m^{3} + 2m (\underline{A054602}) and for n = 4 the sequence a_{m} = m^{4}+3m^{2}+1 (\underline{A057721}). In general, for any given n ∈ ℕ, integer sequences are obtained whose explicit form is given in terms of Fibonacci polynomials (see, e.g., [6]), polynomials of nth order in the sequence index m of the form

a_{m} = ∑_{l=0}^{⌊ \frac{n}{2} ⌋} \binom{n − l}{l}m^{n−2l} = F_{n+1}^{(m)}.  (41)

From Lemmata 2 and 3, we can immediately formulate Table 3: The first members of the family of generalized Fibonacci numbers X_{n,m} = F_{n+1}^{(m)}, n ∈ ℕ, m ∈ ℤ, defined explicitly in (39) and subject to the recursive relation (9).

n m 0 1 2  3  4  5  6  7
1  0 1 2  3  4  5  6  7
2  1 2 5 10  17  26  37  50
3  0 3 12 33  72  135  228  357
4  1 5 29 109 305 701 1405 2549
5  0 8 70 360 1292 3640 8658 18200
6  1 13 169 1189 5473 18901 53353 129949
7  0 21 408 3927 23184 98145 328776 927843
⋮

Proposition 9. The family of generalized Fibonacci sequences {F_{n}^{(m)}}, defined in (39), obeys the following identities:

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}lF_{n+1}^{(l+m)} = \frac{1}{2}(−1)^{n}(2m + n + 1)nn!  (42)

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}lF_{n+1}^{(lm)} = \frac{1}{2}(−1)^{n}m^{n}n(n + 1)!  (43)

for all n ∈ ℕ and m ∈ ℤ.

Proof. Both identities are a direct consequence of (15) and (16), using (39) and (40).

Equation (43) can be used to link generalized Fibonacci numbers F_{n}^{(m)} for positive and negative m. Specifically, subtracting (43) for a given m and m → −m, we obtain

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}l (F_{n+1}^{(lm)} − F_{n+1}^{(−lm)}) = \frac{1}{2}m^{n} ((−1)^{n} − 1) n(n + 1)!  (44)

which, for m = 1, yields

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}l (F_{n+1}^{(−l)} − F_{n+1}^{(l)}) = \frac{1}{2} (1 − (−1)^{n}) n(n+1)! = \{ \begin{matrix}0, & for n even; \\ n(n + 1)!, & for n odd. \end{matrix} (45)

Furthermore, application of Proposition 4 to the family of Fibonacci sequences leads to Proposition 10. The family of Fibonacci sequences F_{n}^{(m)} obeys ∀n ∈ ℕ the following relations

∑_{l=0}^{n}(−1)^{l}\binom{n}{l}F_{n+1}^{(l+m−n+1)} = (−1)^{n}n!  (46)

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}l (F_{n+1}^{(lm)} − m^{n−1}F_{n+1}^{(l)}) = \frac{1}{2}(−1)^{n−1}m^{n−1}(1 − m)n(n + 1)!  (47)

for all m ∈ ℤ.

Proof. Both identities are a consequence of (18) and (19), using (39) and (40). Relation (47) can be directly obtained also from (43).

We note that equation (47) is a special application of (43), which for m = −1 yields

∑_{l=1}^{n}(−1)^{l}\binom{n}{l}l (F_{n+1}^{(−l)} + (−1)^{n}F_{n+1}^{(l)}) = n(n + 1)!  (48)

∀n ∈ ℕ, complementing (45) above. Interestingly, (46) allows to construct, for any given n, general recursive relations in m for generalized Fibonacci numbers F_{n}^{(m)}, namely

F_{n}^{(m+1)} = ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l + 1}F_{n}^{(m−l)} + n!,  (49)

which is valid ∀m ∈ ℤ. Restricting to m ≥ 0, the resulting sequence a_{m} = F_{2}^{(m+1)} = m^{2} + 1 obtained for n = 2 from (46), obeys

a_{0} = 1, a_{1} = 2, a_{m+1} = 2a_{m} − a_{m−1} + 2, m ≥ 1

(\underline{A002522}, E. Werley, 2011). Similarly, for n = 3, the integer sequence given by the thirdorder polynomial a_{m} = F_{3}^{(m+1)} = m^{3} + 2m (\underline{A054602}), obeys the recursive relation

a_{0} = 0, a_{1} = 3, a_{2} = 12, a_{m+1} = 3a_{m} − 3a_{m−1} + a_{m−2} + 6, m ≥ 2,

and for n = 4, the sequence a_{m} = F_{4}^{(m+1)} = m^{4} + 3m^{2} + 1 (\underline{A057721}) is subject to the recursion

a_{0} = 1, a_{1} = 5, a_{2} = 29, a_{2} = 109, a_{m+1} = 4a_{m} − 6a_{m−1} + 4a_{m−2} − a_{m−3} + 24, m ≥ 3.

Finally, Corollaries 5 and 6 provide explicit representations of generalized Fibonacci numbers in terms of other members of the Fibonacci family:

Proposition 11. Generalized Fibonacci numbers F_{n}^{(m)} obey

F_{n+1}^{(m)} = (−1)^{n}\binom{m}{n} ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l}\frac{n − l}{l − m}F_{n+1}^{(l)} + \frac{m!}{(m − n)!}  (50)

F_{n+1}^{(−m)} = (−1)^{n}\binom{m}{n} ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l}\frac{n − l}{l − m}F_{n+1}^{(−l)} + (−1)^{n} \frac{m!}{(m − n)!}  (51)

∀n, m ∈ ℕ with m ≥ n, and

F_{n−p+1}^{(m)} = (−1)^{n+1}n^{−q} ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l}l^{q}F_{n−p+1}^{(m−n+l)}  (52)

F_{n−p+1}^{(m)} = (−1)^{n+1}n^{−p} ∑_{l=0}^{n−1}(−1)^{l}\binom{n}{l}l^{p}F_{n−p+1}^{(m−n+l)} + n^{−p}n!  (53)

for all n, p ∈ ℕ with n ≥ p + 1, p ≥ 1, 0 ≤ q < p and m ∈ ℤ.

Proof. The first two identities are a direct consequence of (20) and (20), the last two can be shown with (30) and (31), using (39) and (40).

We note that Propositions 9 to 11 provide a number of identities which interlink the set of generalized Fibonacci sequences defined in (39). Specifically, the defining recursive relation in n, Equation (9), for generalized Fibonacci numbers allows to express each F_{n}^{(m)} in terms of F_{n′}^{(m)}, n′ < n for fixed m, whereas relations (42)–(43), (46)–(47) and (50)–(53) allow to express each F_{n}^{(m)} in terms of F_{n}^{(m′)}, m′ ≠ m for any given n. Combining both sets of identities, we arrive at linking all members of the family of generalized Fibonacci numbers.

5 Concluding remarks

In this contribution, we investigated general properties of number sequences generated through products of the form (7). We found several identities and recursive relations which interlink such sequences and suggest their classification in terms of families (Definition 1). Although the families studied here as examples describe different integer sequences, such as Pochhammer numbers, powers of integers or generalized Fibonacci numbers, we find that each of these families is subject to the same set of identities which, in some cases, generalize interesting relations between these known sequences.

The examples presented here constitute but a small set of potential applications. For instance, q-Pochhammer sequences and sequences produced by products of Pochhammer numbers are obtained for x_{n,l} = a^{l}, a ∈ ℝ and x_{n,l} = l^{a}, a ∈ ℤ, respectively. The general relations listed in Section 2 apply in these cases, and provide a number of identities obeyed by the corresponding sequences. By setting

x_{n,l} = −2\sqrt{q} cos (\frac{lπ}{n + 1}) ,

we obtain with (6) general Lucas sequences (\underline{A108299}), i.e.,

X_{n,m} = \frac{1}{\sqrt{q}} ∏_{l=1}^{n} (m − 2\sqrt{q} cos (\frac{lπ}{n + 1})) = L_{n+1}^{(m,q)}.

For appropriate m and q, interesting identities, such as grandma’s identity [12], signed bisections of Fibonacci sequences and relations interlinking powers of Fibonacci numbers, are obtained. The study of these relations, their potential generalization and application to other families of number sequences might provide novel and potentially useful insights into properties shared by qualitatively different number sequences.

6 Acknowledgment

Research supported in part by CNRS. The author wishes to thank L. E. Muller II, J. A. G. Willow and S. Hower for valuable comments.

References

[1] D. E. Rutherford, Some continuant determinants arising in physics and chemistry, Proc. Roy. Soc. Edinburgh Sect. A 62 (1947), 229–236.

[2] D. E. Rutherford, Some continuant determinants arising in physics and chemistry II, Proc. Roy. Soc. Edinburgh Sect. A 63 (1952), 232–241.

[3] D. Lind, Problem H-64, Fibonacci Quart. 3 (1965), 116.

[4] D. Zeitlin, Solution to problem H-64, Fibonacci Quart. 5 (1967), 74–75.

[5] L. Shapiro, Solution to problem B-742, Fibonacci Quart. 32 (1994), 470–471.

[6] R. André-Jeannin, A note on a general class of polynomials, Fibonacci Quart. 32 (1994), 445–454.

[7] R. J. Hendel and C. K. Cook, Recursive properties of trigonometric products, in G. E. Bergum, A. Philippou and A. F. Horadam, eds.,Applications of Fibonacci Numbers, Vol. 6, Springer, 1996, pp. 201–214.

[8] J. Cigler, q-Fibonacci polynomials, Fibonacci Quart. 41 (2003), 31–40.

[9] Z. H. Sun, Expansion and identities concerning Lucas sequences, Fibonacci Quart. 44 (2006), 144–152.

[10] H. W. Gould, Combinatorial Identities, Morgantown, 1972.

[11] N. J. A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org.

[12] S. Humble, Grandma’s identity, Math. Gaz. 88 (2004), 524–525.

[13] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, 1972.

[14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Elsevier, 2007.

2010 Mathematics Subject Classification: Primary 11B39; Secondary 32A05, 40B05. Keywords: product representation, Fibonacci number, Pell number, Pochhammer number, integer power, recursive identity.

(Concerned with sequences \underline{A000045}, \underline{A000129}, \underline{A000290}, \underline{A000578}, \underline{A000583}, \underline{A002378}, \underline{A002522}, \underline{A007531}, \underline{A052762}, \underline{A054602}, \underline{A057721}, and \underline{A108299}.)

Received September 1 2015; revised version received March 1 2016. Published in Journal of Integer Sequences, April 6 2016.

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