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Generalization of Horadam’s Sequence

C.N. Phadte*, Y.S. Valaulikar

Department of Mathematics, G.V.M’s College of Commerce & Economics, Ponda, Goa 403401, India

Abstract: In this paper a new class of Fibonacci like sequence is introduced. Here we consider non-homogeneous recurrence relation to obtain generalization of Horadam’s Sequence. Some identities concerning this new sequence are obtained and proved. Some examples are given in support of the results.

Keywords: pseudo fibonacci numbers, non-homogeneous recurrence relation

Citation: C.N. Phadte, Y.S. Valaulikar, Generalization of Horadam’s Sequence. Turkish Journal of Analysis and Number Theory, 4 (4). 113-117. January 2016. doi:10.12691/tjant-4-4-5

1. Introduction

Fibonacci sequence is generalized in many ways. One of the most widely used extension is the one given by A.F. Horadam [2]. He defined the generalized sequence of numbers as follows. Let be a sequence defined by

[picture] | (1.1)

for where p, q, a and b are integers with W and .

One can see that reduces to generalized Fibonacci sequence when and , and subsequently to the Fibonacci sequence when , That is and . In a series of papers [1,3,4,5,8] various properties of the generalized sequence have been developed. Using Horadam Sequence and the concept of Pseudo Fibonacci sequence [A224508] defined earlier in [6] and studied further in [7], we now define a new extension of the sequence .

Definition 1. The generalized Pseudo Fibonacci (GPF) Sequence is defined as the sequence satisfying the following non-homogeneous recurrence relation

[picture] | (1.2)

for and with and

Here are integers and are distinct roots of characteristic equation of the corresponding homogeneous equation.

First few GPF numbers are given below:

Observe that each GPF number , consist of two parts. The first part is an expression in and , while the second is a polynomial in whose coefficients are times terms in and . This is shown in the following tables:

Table 1. First part of Gn, n≥2

Table 2. Second part of Gn, n≥2

From the above tables, we have the following relation between GPF and Horadam numbers .

Theorem 2. For , the term

of the sequence satisfy the non-homogeneous recurrence relation

Proof. Consider

That is,

[picture] | (1.3)

Now

Using equations (1.1) and (1.3), we write

Hence the theorem.

2. Some Identities for Gn

In this section, we obtain some fundamental identities for GPF sequence .

• Binet type Formula:

Let

Then the Binet form of is given by

[picture] | (2.1)

where

[picture] | (2.2)

are the distinct roots of the equation given by

[picture] | (2.3)

writing

Note that

[picture] | (2.4)

We can deduce from (2.2) that

[picture] | (2.5)

where

[picture] | (2.6)

• Generating function:

Generating function for is given by

We have the following result for sum of first GPF numbers. Proofs follow from recurrence relation and Binet formula.

Proposition 3. For

i)

ii)

Using the recurrence relation , we have the following result. The same can also be obtained by induction.

Proposition 4.

Proof. Using the recurrence relation (1.2), we have

On summing both sides of these equations, we get,

i.e.

i.e.

Hence

We now obtain the sum of the squares of GPFs and sum of the product of two consecutive GPFs.

For simplicity, let and

Let and

Further let

We have the following results:

Proposition 5. For

i)

ii)

Proof. Consider

Hence, summing up to n+1 terms on both sides, we get,

Adjusting the variables of summation and simplifying, we get

Similarly, starting with

and simplifying as above, we get

Solving these two equations for and , we get the required results.

Next result deals with sum of even and odd terms of GPF sequence. Again for simplicity, let and , E so that .

Proposition 6. The sum of the even (odd) indexed terms of is given by

(i)

and

(ii)

Provided p

Proof. From the recurrence relation (1.1), we have,

Summing up to terms, we get

which on simplifying yields,

[picture] | (2.7)

Similarly using the relation

and summing up to terms, we get,

[picture] | (2.8)

Solving (2.7 ) and (2.8), we get the required results.

We have following identity.

Proposition 7. For ,

Proof. Using Binets formula (2.1),

On simplification, we get,

Since, and ,

By letting we have the following result.

Corollary 8.

Note that the above corollary along with Proposition can be used to find obtained in Proposition .

Next we prove a version of Catalan’s Identity for GPF numbers.

Proposition 9.

where and is as defined by (2.6).

Proof. Using

From this result we immediately have a version of Cassini’s identity for GPF numbers.

Corollary 10.

Next we have an expression for in terms of binomial coefficients.

Proposition 11.

Proof. We have

Since, and , we get,

Hence,

which is the required result.

3. Examples

In this section we present some examples in support of some results obtained in Section 2.

Examples: Consider , with

Here, .

First few terms of are

(1) Verification of Proposition 5(i).

When , we have . Then

Result is verified.

(2) Verification of Proposition 5 (ii).

Here let . We have, Then

Result is verified.

(3) Verification of Proposition 7.

Let and . Then

Result is verified.

(4) Verification of Proposition 9.

Let and .

Result is verified.

(5) Verification of Proposition 11.

Let

Result is verified.

4. Conclusion

The well known Horadam sequence is generalized via non homogeneous recurrence relation to obtain a Fibonacci like sequence. All the usual identities and properties of Fibonacci like sequences are obtained for the new generalization of Horadam sequence.

References

[1]   | A. F. Horadam, “A Generalized Sequence of Numbers”, The American Mathematical Monthly, 68 No. 5,(1961), pp.455-459.
[2]   | A. F. Horadam, “Basic Properties of a certain Generalized Sequence of Numbers”, The Fibonacci Quarterly, 3, No.3(1965), pp.161-176.
[3]   | A. F. Horadam, “Generating functions for power of a certain Generalized Sequence of numbers”, Duke Math J. 32, No.3(1965), pp. 437-446.
[4]   | A. F. Horadam, “Special Properties of the Sequence Wn(a,b;p,q)” , The Fibonacci Quarterly, 5, No. 5 (1967), pp. 424-434.
[5]   | C. N. Phadte, S.P. Pethe, “Generalization of the Fibonacci Sequence”, Applications of Fibonacci Numbers,5, Kluwer Academic Pub. 1993, 465-472.
[6]   | C. N. Phadte, S. P. Pethe, “On Second Order Non-Homogeneous Recurrence Relation”, Annales Mathematicae et Informaticae vol.41 (2013) pp.205-210.
[7]   | C. N. Phadte, “Trigonometric Pseudo Fibonacci Sequence”, Notes on Number Theory and Discrete Mathematics,21 No.3, (2015) pp.70-76.
[8]   | J. E. Walton, A. F. Horadam, “Some Aspect of Fibonacci Numbers”, The Fibonacci Quarterly, 12.