Some Identities for Fibonacci and Incomplete Fibonacci p-Numbers via the Symmetric Matrix Method

M. Cetin Firengiz Department of Mathematics

Faculty of Education University of Baskent

Baglica, Ankara Turkey

 mcetin@baskent.edu.tr

Dursun Tasci and Naim Tuglu Department of Mathematics

University of Gazi Teknikokullar, 06500, Ankara

Turkey

   dtasci@gazi.edu.tr
 naimtuglu@gazi.edu.tr

Abstract

We obtain some new formulas for the Fibonacci and Lucas p-numbers, by using the symmetric infinite matrix method. We also give some results for the Fibonacci and Lucas p-numbers by means of the binomial inverse pairing.

1 Introduction

Dil and Mező [3] defined the symmetric infinite matrix method. For sequences (a_{n}) and (a^{n}), the recursive formula

a_{n}^{0} = a_{n}, a_{0}^{n} = a^{n} (n ≥ 0)
a_{n}^{k} = a_{n−1}^{k} + a_{n}^{k−1}  (n ≥ 1, k ≥ 1)  (1)

gives the associated symmetric infinite matrix [3]:

( \begin{matrix}. & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . \\ . & . & . & \begin{matrix}a_{n}^{k−1} \\ ↓ \end{matrix} & . & . & . & . \\ . & . & a_{n−1}^{k} → & a_{n}^{k} & . & . & . & . \\ . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . \end{matrix} ) .

Proposition 1. [3] If the relation (1) holds, the entry a_{n}^{k} of the corresponding symmetric infinite matrix is

a_{n}^{k} = ∑_{i=1}^{k} \binom{n + k − i − 1}{n − 1}a_{0}^{i} + ∑_{j=1}^{n} \binom{k + n − j − 1}{k − 1}a_{j}^{0}.  (2)

For two sequences (a_{n}) and (b_{n}), the well-known binomial inverse pair [9] is given by the relations

b_{n} = ∑_{k=0}^{n} \binom{n}{k}a_{k}  (3)

a_{n} = ∑_{k=0}^{n} \binom{n}{k} (−1)^{n−k} b_{k}.  (4)

Stakhov and Rozin [6] defined the Fibonacci p-numbers F_{p} (n) by the following recurrence relation for n > p

F_{p} (n) = F_{p} (n − 1) + F_{p} (n − p − 1)  (5)

with initial conditions

F_{p} (0) = 0, F_{p} (n) = 1 (n = 1, 2, …, p)

and the Lucas p-numbers L_{p} (n) by the following recurrence relation for n > p

L_{p} (n) = L_{p} (n − 1) + L_{p} (n − p − 1)  (6)

with initial conditions

L_{p} (0) = p + 1, L_{p} (n) = 1 (n = 1, 2, …, p) .

Note that for the case p = 1, the sequences of Fibonacci and Lucas p-numbers reduce to the well-known Fibonacci and Lucas sequences {F_{n}} , {L_{n}}, respectively. See [1, 5, 8] for more details about the Fibonacci and Lucas p-numbers.

Tasci and Cetin-Firengiz [7] introduced the incomplete Fibonacci and Lucas p-numbers. The incomplete Fibonacci p-numbers F_{p}^{k} (n) and the incomplete Lucas p-numbers L_{p}^{k} (n) are defined by

F_{p}^{k} (n) = ∑_{j=0}^{k} \binom{n − jp − 1}{j} (n = 1, 2, …; 0 ≤ k ≤ ⌊\frac{n−1}{p+1}⌋)

and

L_{p}^{k} (n) = ∑_{j=0}^{k} \frac{n}{n − jp}\binom{n − jp}{j} (n = 1, 2, …; 0 ≤ k ≤ ⌊\frac{n}{p+1}⌋) .

We note that F ⌊\frac{n−1}{2}⌋_{1} (n) = F_{n}, L⌊\frac{n}{2}⌋_{1} (n) = L_{n} and F_{1}^{k} (n) = F_{n} (k), L_{1}^{k} (n) = L_{n} (k),

where {F_{n} (k)} and {L_{n} (k)} are the sequences of incomplete Fibonacci and Lucas numbers, respectively. The same authors [7] gave the following properties of the incomplete Fibonacci and Lucas p-numbers:

∑_{j=0}^{h} \binom{h}{j}F_{p}^{k+j} (n + p (j − 1)) = F_{p}^{k+h} (n + (p + 1) h − p)  (7)

for 0 ≤ k ≤ \frac{n−p−h−1}{p+1},

∑_{j=0}^{h} \binom{h}{j}L_{p}^{k+j} (n + p (j − 1)) = L_{p}^{k+h} (n + (p + 1) h − p)  (8)

for 0 ≤ k ≤ \frac{n−p−h}{p+1}.

In this paper, we give the generalization of some results of [3]. Some properties for the Fibonacci and Lucas p-numbers are obtained via the symmetric method. The results of incomplete Fibonacci and Lucas p-numbers are given by using binomial inverse pair as used for the Euler-Seidel matrix [2, 3].

2 Applications of symmetric infinite matrix

2.1 Applications for the Fibonacci and Lucas p-numbers

Let us consider the initial sequences a_{n}^{0} = F_{p} (n − 1) and a_{0}^{n} = F_{p} (n (p + 1) − 1) , n ≥ 1. Thus the following infinite matrix is given for the special case

( \begin{matrix}0 & F_{p} (0) & F_{p} (1) & F_{p} (2) & ⋯ \\ F_{p} (p) & F_{p} (p + 1) & F_{p} (p + 2) & F_{p} (p + 3) & ⋯ \\ F_{p} (2p + 1) & F_{p} (2p + 2) & F_{p} (2p + 3) & F_{p} (2p + 4) & ⋯ \\ F_{p} (3p + 2) & F_{p} (3p + 3) & F_{p} (3p + 4) & F_{p} (3p + 5) & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ) .

Proposition 2. The Fibonacci p-numbers satisfy the relation

∑_{i=1}^{n} F_{p} (i (p + 1) − 1) = F_{p} (n (p + 1)) .  (9)

Proof. For a_{n}^{0} = F_{p} (n − 1) and a_{0}^{n} = F_{p} (n (p + 1) − 1) , n ≥ 1. We have a_{1}^{1} = F_{p} (p + 1) , a_{1}^{2} = F_{p} (2p + 2) . Suppose that the equation holds for n > 1. Now we show that the equation holds for (n + 1) . Thus we get using (1) and (5)

a_{1}^{n+1} = a_{0}^{n+1} + a_{1}^{n}
= F_{p} ((n + 1) (p + 1) − 1) + F_{p} (n (p + 1))
= F_{p} (n (p + 1) + p) + F_{p} (n (p + 1))
= F_{p} (n (p + 1) + p + 1)
= F_{p} ((n + 1) (p + 1)) .

By considering (2), we have

a_{1}^{n} = ∑_{i=1}^{n} \binom{n − i}{0}a_{0}^{i} + ∑_{j=1}^{1} \binom{n − j}{n − 1}a_{j}^{0}

= ∑_{i=1}^{n} F_{p} (i (p + 1) − 1) + a_{1}^{0}

= ∑_{i=1}^{n} F_{p} (i (p + 1) − 1) + F_{p} (0) .

Then, we can obtain

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

Taking p = 1 in (9), we get F_{2n} = ∑_{i=1}^{n} F_{2i−1} in [4].

Stakhov and Rozin [6] gave the equation F_{p} (1)+F_{p} (2)+⋯+F_{p} (n) = F_{p} (n + p + 1)−1 for the Fibonacci p-numbers. The following proposition shows that the formula can be obtained via the symmetric method.

Proposition 3. The Fibonacci p-numbers are

∑_{j=1}^{n} F_{p} (j − 1) = F_{p} (p + n) − 1.  (10)

Proof. Let a_{n}^{0} = F_{p} (n − 1) and a_{0}^{n} = F_{p} (n (p + 1) − 1) , n ≥ 1. If we take n = 1 and n = 2, then a_{1}^{1} = F_{p} (p + 1) , a_{2}^{1} = F_{p} (p + 2) . Suppose that the equation holds for n > 1. We show that the equation holds for (n + 1). We have by (1) and (5)

a_{n+1}^{1} = a_{n}^{1} + a_{n+1}^{0}
= F_{p} (p + n) + F_{p} (n)
= F_{p} (p + n + 1) .

Using (2), we can write

a_{n}^{1} = ∑_{i=1}^{1} \binom{n − i}{n − 1}a_{0}^{i} + ∑_{j=1}^{n} \binom{n − j}{0}a_{j}^{0}

= a_{0}^{1} + ∑_{j=1}^{n} a_{j}^{0}

= F_{p} (p) + ∑_{j=1}^{n} F_{p} (j − 1) ,

which completes the proof.

When p = 1 in (10), we obtain ∑_{i=1}^{n} F_{i} = F_{n+2} − 1 in [4].

In particular, let a_{n}^{0} = L_{p} (n − 1) and a_{0}^{n} = L_{p} (n (p + 1) − 1) , n ≥ 1. This case gives the following infinite matrix

( \begin{matrix}0 & L_{p} (0) & L_{p} (1) & L_{p} (2) & ⋯ \\ L_{p} (p) & L_{p} (p + 1) & L_{p} (p + 2) & L_{p} (p + 3) & ⋯ \\ L_{p} (2p + 1) & L_{p} (2p + 2) & L_{p} (2p + 3) & L_{p} (2p + 4) & ⋯ \\ L_{p} (3p + 2) & L_{p} (3p + 3) & L_{p} (3p + 4) & L_{p} (3p + 5) & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ) .

Similar results for the Lucas p-numbers can be obtained likewise. Therefore we omit the proofs of Proposition 4 and 5

Proposition 4. The Lucas p-numbers L_{p} (n) satisfy the following relation

∑_{i=1}^{n} L_{p} (i (p + 1) − 1) = L_{p} (n (p + 1)) − (p + 1) .  (11)

Proposition 5. We have

∑_{j=1}^{n} L_{p} (j − 1) = L_{p} (p + n) − 1.  (12)

If p = 1 in (11) and (12), we get the well known identities ∑_{i=1}^{n} L_{2i−1} = L_{2n} − 2 and

∑_{i=1}^{n} L_{i} = L_{n+2} − 3.

2.2 Applications for the incomplete Fibonacci and Lucas p-numbers

In this subsection, we get similar formulas for (7) and (8) by using the binomial inverse pair. Let a_{n}^{0} = F_{p}^{k+n} (t + p (n − 1)). From (3) we have

a_{0}^{n} = ∑_{j=0}^{n} \binom{n}{j}F_{p}^{k+j} (t + p (j − 1)) .

By using (7)

a_{0}^{n} = F_{p}^{k+n} (t + (p + 1) n − p) .

Therefore, the dual formula of (7) is obtained from (4)

F_{p}^{k+n} (t + p (n − 1)) = ∑_{j=0}^{n} \binom{n}{j} (−1)^{n−j} F_{p}^{k+j} (t + (p + 1) j − p)  (13)

for 0 ≤ k ≤ \frac{t−p−n−1}{p+1}. Similarly, let us take a_{n}^{0} = L_{p}^{k+n} (t + p (n − 1)) . Then (3) can be rewritten as

a_{0}^{n} = ∑_{j=0}^{n} \binom{n}{j}L_{p}^{k+j} (t + p (j − 1)) .

By (8)

a_{0}^{n} = L_{p}^{k+n} (t + (p + 1) n − p) .

Finally, using (4), we obtain the dual formula (8)

L_{p}^{k+n} (t + p (n − 1)) = ∑_{j=0}^{n} \binom{n}{j} (−1)^{n−j} L_{p}^{k+j} (t + (p + 1) j − p)  (14)

for 0 ≤ k ≤ \frac{t−p−n}{p+1}.

For p = 1 in (13) and (14), we get the properties of incomplete Fibonacci and Lucas numbers in [3].

References

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[8] N. Tuglu, E. G. Kocer, and A. Stakhov, Bivariate Fibonacci like p-polynomials, Appl. Math. Comput. 217 (2011), 10239–10246.

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

2010 Mathematics Subject Classification: Primary 11B73; Secondary 11B83. Keywords: Symmetric infinite matrix, Fibonacci p-number, Lucas p-number, incomplete Fibonacci p-number. (Concerned with sequences \underline{A000032} and \underline{A000045}.)

Received August 16 2013; revised versions received November 17 2013; December 17 2013. Published in Journal of Integer Sequences, January 6 2014.

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