Notes on Binomial Transform of the Generalized Narayana Sequence

In this paper, we deﬁne the binomial transform of the generalized Narayana sequence and as special cases, the binomial transform of the Narayana, Narayana-Lucas, Narayana-Perrin sequences will be introduced. We investigate their properties in details.


Introduction and Preliminaries
Recently, there have been so many studies of the sequences of numbers in the literature and the sequences of numbers were widely used in many research areas, such as architecture, nature, art, physics and engineering. The sequence of Fibonacci numbers {F n } is defined by F n = F n−1 + F n−2 , n ≥ 2, F 0 = 0, F 1 = 1. and the sequence of Lucas numbers {L n } is defined by L n = L n−1 + L n−2 , n ≥ 2, L 0 = 2, L 1 = 1.
The Fibonacci numbers, Lucas numbers and their generalizations have many interesting properties and applications to almost every field. Horadam [10] defined In this paper, we introduce the binomial transform of the generalized Narayana sequence and we investigate, in detail, three special cases which we call them the binomial transform of the Narayana, Narayana-Lucas, Narayana-Perrin sequences. We investigate their properties in the next sections. In this section, we present some properties of the generalized Tribonacci sequence which is a generalization of Fibonacci numbers. The generalized Tribonacci sequence {W n (W 0 , W 1 , W 2 ; r, s, t)} n≥0 (or shortly {W n } n≥0 ) is defined as follows: W n = rW n−1 + sW n−2 + tW n−3 , W 0 = a, W 1 = b, W 2 = c, n ≥ 3 (1.1) where W 0 , W 1 , W 2 are arbitrary complex (or real) numbers and r, s, t are real numbers. This sequence has been studied by many authors, see for example [2,3,4,5,6,16,17,19,21,23,29,31]. The sequence {W n } n≥0 can be extended to negative subscripts by defining for n = 1, 2, 3, ... when t = 0. Therefore, recurrence (1.1) holds for all integer n. As {W n } is a third order recurrence sequence (difference equation), it's characteristic equation is , ω = −1 + i √ 3 2 = exp(2πi/3).
(1.3) can be written in the following form: Note that the Binet form of a sequence satisfying (1.2) for non-negative integers is valid for all integers n, for a proof of this result see [13]. This result of Howard and Saidak [13] is even true in the case of higher-order recurrence relations.
Next, we give the ordinary generating function ∞ n=0 W n x n of the sequence W n .
W n x n is the ordinary generating function of the generalized Tribonacci sequence {W n } n≥0 . Then, We next find Binet's formula of the generalized Tribonacci sequence {W n } by the use of generating function for W n .
Note that from (1.3) and (1.5) we have In this paper, we consider the case r = 1, s = 0, t = 1 and in this case we write V n = W n . So, the generalized Narayana sequence {V n } n≥0 = {V n (V 0 , V 1 , V 2 )} n≥0 is defined by the third-order recurrence relations with the initial values V 0 = c 0 , V 1 = c 1 , V 2 = c 2 not all being zero. The sequence {V n } n≥0 can be extended to negative subscripts by defining for n = 1, 2, 3, .... Therefore, recurrence (1.6) holds for all integer n.
(1.3) can be used to obtain Binet's formula of generalized Narayana numbers. Binet's formula of generalized Narayana numbers can be given as Here, α, β and γ are the roots of the cubic equation The sequences {N n } n≥0 , {U n } n≥0 and {H n } n≥0 can be extended to negative subscripts by defining for n = 1, 2, 3, ... respectively. Therefore, recurrences (1.7)-(1.9) hold for all integer n. For more details on the generalized Narayana numbers, see Soykan [28]. Note that N n is the sequence A000930 in [20] associated with the Narayana's cows sequence and the sequence A078012 in [20] associated with the expansion of (1 − x)/(1 − x − x 3 ) and U n is the sequence A001609 in [20].
Next, we give the ordinary generating function ∞ n=0 V n x n of the generalized Narayana sequence V n (see, Soykan [28] for more details.).
V n x n is the ordinary generating function of the generalized Narayana sequence {V n } n≥0 . Then, (1.10) Proof. Take r = 1, s = 0, t = 1 in Lemma 1. respectively.
2 Binomial Transform of the Generalized Narayana Sequence V n In [15, p. 137], Knuth introduced the idea of the binomial transform. Given a sequence of numbers (a n ), its binomial transform (â n ) may be defined by the rulê a n = n i=0 n i a i , with inversion a n = For more information on binomial transform, see, for example, [7,8,18,30] and references therein.
In this section, we define the binomial transform of the generalized Narayana sequence V n and as special cases the binomial transform of the Narayana, Narayana-Lucas, Narayana-Perrin sequences will be introduced.
Definition 2.1. The binomial transform of the generalized Narayana sequence V n is defined by Translated to matrix language, b n has the nice (lower-triangular matrix) form As special cases of b n = V n , the binomial transforms of the Narayana, Narayana-Lucas, Narayana-Perrin sequences are defined as follows: The binomial transform of the Narayana sequence N n is the binomial transform of the Narayana-Lucas sequence U n is the binomial transform of the Narayana-Perrin sequence H n is For n ≥ 0, the binomial transform of the generalized Narayana sequence V n satisfies the following relation: Proof. We use the following well-known identity: Note also that n + 1 0 = n 0 = 1 and n n + 1 = 0.
This completes the proof.
Remark 2.3. From the last Lemma, we see that The following theorem gives recurrent relations of the binomial transform of the generalized Narayana sequence.
Theorem 2.4. For n ≥ 0, the binomial transform of the generalized Narayana sequence V n satisfies the following recurrence relation: and taking the values n = 0, 1, 2 and then solving the system of equations The sequence {b n } n≥0 can be extended to negative subscripts by defining for n = 1, 2, 3, .... Therefore, recurrence (2.1) holds for all integer n.
Note that the recurence relation (2.1) is independent from initial values. So, and The first few terms of the binomial transform of the generalized Narayana sequence with positive subscript and negative subscript are given in the following Table 1.
The first few terms of the binomial transform numbers of the Narayana , Narayana-Lucas, Narayana-Perrin sequences with positive subscript and negative subscript are given in the following Table 2.
Here, θ 1 , θ 2 and θ 3 are the roots of the cubic equation Moreover, where Note that For all integers n, (Binet's formulas of) binomial transforms of Narayana , Narayana-Lucas, Narayana-Perrin numbers (using initial conditions in (2.2)) can be expressed using Binet's formulas as respectively.

Generating Functions and Obtaining Binet Formula of Binomial Transform From Generating Function
The generating function of the binomial transform of the generalized Narayana sequence V n is a power series centered at the origin whose coefficients are the binomial transform of the generalized Narayana sequence.
Next, we give the ordinary generating function f bn (x) = Proof. Using Lemma 1.1, we obtain Note that P. Barry shows in [1] that if A(x) is the generating function of the sequence {a n }, then is the generating function of the sequence {b n } with b n = n i=0 n i a i . In our case, since we obtain The previous lemma gives the following results as particular examples.

respectively.
We next find Binet's formula of the Binomial transform of the generalized Narayana numbers {V n } by the use of generating function for b n .
Proof. By using Lemma 3.1, the proof follows from Theorem 1.2.
Note that from (2.2) and (3.2), we have Note that we can also write Next, using Theorem 3.3, we present the Binet's formulas of binomial transform of Narayana, Narayana-Lucas, Narayana-Perrin sequences.

Simson Formulas
There is a well-known Simson Identity (formula) for Fibonacci sequence {F n }, namely, F n+1 F n−1 − F 2 n = (−1) n which was derived first by R. Simson in 1753 and it is now called as Cassini Identity (formula) as well. This can be written in the form The following theorem gives generalization of this result to the generalized Narayana sequence {W n }.

Proposition 4.2. For all integers n, Simson formula of binomial transforms of generalized Narayana numbers is given as
The previous proposition gives the following results as particular examples. respectively.

Some Identities
In this section, we obtain some identities of binomial transforms of Narayana, Narayana-Lucas, Narayana-Perrin numbers. First, we can give a few basic relations between { N n } and { U n }.
Proof. Note that all the identities hold for all integers n. We prove (5.1). To show (5.1), writing and solving the system of equations we find that a = − 13 279 , b = 82 279 , c = − 104 279 . The other equalities can be proved similarly.
Note that all the identities in the above Lemma can be proved by induction as well.
Next, we present a few basic relations between { N n } and { H n }.
Now, we give a few basic relations between { U n } and { H n }.

Sums of Terms with Positive Subscripts
The following proposition presents some formulas of binomial transform of generalized Narayana numbers with positive subscripts. Proposition 6.1. For n ≥ 0, we have the following formulas: Proof. Take r = 4, s = −5, t = 3 in Theorem 2.1 in [24] (or take x = 1, r = 4, s = −5, t = 3 in Theorem 2.1 in [25]). From the last proposition, we have the following corollary which gives sum formulas of binomial transform of Narayana numbers (take b n = N n with N 0 = 0, N 1 = 1, N 2 = 3).
Taking b n = U n with U 0 = 3, U 1 = 4, U 2 = 6 in the last proposition, we have the following corollary which presents sum formulas of binomial transform of Narayana-Lucas numbers. Corollary 6.3. For n ≥ 0 we have the following formulas: From the last proposition, we have the following corollary which gives sum formulas of binomial transform of Narayana-Perrin numbers (take b n = H n with H 0 = 3, H 1 = 3, H 2 = 5). Corollary 6.4. For n ≥ 0 we have the following formulas:

Sums of Terms with Negative Subscripts
The following proposition presents some formulas of binomial transform of generalized Narayana numbers with negative subscripts.
Taking b n = U n with U 0 = 3, U 1 = 4, U 2 = 6 in the last proposition, we have the following corollary which presents sum formulas of binomial transform of Narayana-Lucas numbers. Corollary 6.7. For n ≥ 1, binomial transform of Narayana-Lucas numbers have the following properties.
From the last proposition, we have the following corollary which gives sum formulas of binomial transform of Narayana-Perrin numbers (take b n = H n with H 0 = 3, H 1 = 3, H 2 = 5).

Sums of Squares of Terms with Positive Subscripts
The following proposition presents some formulas of binomial transform of generalized Narayana numbers with positive subscripts. Proposition 6.9. For n ≥ 0, we have the following formulas: Proof. Take x = 1, r = 4, s = −5, t = 3 in Theorem 4.1 in [27], see also [26]. From the last proposition, we have the following Corollary which gives sum formulas of binomial transform of Narayana numbers (take b n = N n with N 0 = 0, N 1 = 1, N 2 = 3). Corollary 6.10. For n ≥ 0, binomial transform of Narayana numbers have the following properties: Taking b n = U n with U 0 = 3, U 1 = 4, U 2 = 6 in the last Proposition, we have the following Corollary which presents sum formulas of binomial transform of Narayana-Lucas numbers. Corollary 6.11. For n ≥ 0, binomial transform of Narayana-Lucas numbers have the following properties: From the last proposition, we have the following corollary which gives sum formulas of binomial transform of Narayana-Perrin numbers (take b n = H n with H 0 = 3, H 1 = 3, H 2 = 5). Corollary 6.12. For n ≥ 0, binomial transform of Narayana-Perrin numbers have the following properties:

Matrices related with Binomial Transform of Generalized Narayana numbers
Matrix formulation of W n can be given as For matrix formulation (7.1), see [14]. In fact, Kalman gave the formula in the following form We define the square matrix A of order 3 as: and from (7.1) (or using (7.2) and induction) we have If we take b n = N n in (7.2) we have For n ≥ 0, we define By convention, we assume that Theorem 7.1. For all integers m, n ≥ 0, we have (a) B n = A n .
Proof. (a) Proof can be done by mathematical induction on n.
(c) We have i.e., C n = AC n−1 . From the last equation, using induction, we obtain C n = A n−1 C 1 . Now and similarly Some properties of matrix A n can be given as Proof. From the equation C n+m = C n B m = B m C n , we see that an element of C n+m is the product of row C n and a column B m . From the last equation, we say that an element of C n+m is the product of a row C n and column B m . We just compare the linear combination of the 2nd row and 1st column entries of the matrices C n+m and C n B m . This completes the proof.
From Corollary 6.2, we know that for n ≥ 0, So, Theorem 7.2 and Corollary 7.3 can be written in the following forms: Now, we consider non-positive subscript cases. For n ≥ 0, we define By convention, we assume that    (7.5) Remark 7.11. By induction, it can be proved that for all integers m, n ≤ 0, (7.5) holds. So, for all integers m, n, (7.5) is true.