On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

Abstract

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

“…The recurrence (2.23) in [86,2009] by Tian-Xiao He and Peter Jau-Shyong Shiue defines exactly the class of Ward-Horadam functions' second order sequences and this paper standard Jacques Binet form ( 21), ( 22) of the recurrence (3) solution for Ward-Horadam functions sequences H(x) constitutes the content of their Proposition 2.7. -as has been mentioned earlier.…”

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

“…The recurrence (2.23) in [86,2009] by Tian-Xiao He and Peter Jau-Shyong Shiue defines exactly the class of Ward-Horadam functions' second order sequences and this paper standard Jacques Binet form ( 21), ( 22) of the recurrence (3) solution for Ward-Horadam functions sequences H(x) constitutes the content of their Proposition 2.7. -as has been mentioned earlier.…”

Notes on generalized and extended Leonardo numbers

“…Theorem 3.1 (He and Shiue [7]). Let {a n } be a sequence of order 2 satisfying the linear recurrence relation (8).…”

Use Impulse Response Sequences in the Construction of Number Sequence Identities

“…[5] also shows a method of finding the expressions of the linear recurring sequences of order 2 and the interrelationship among those sequences. However, [5] also pointed out "the method presented in this paper (i.e., Proposition 3.2) cannot be extended to the higher order setting." In Section 2, we have shown our method based on the IRS can be extended to the higher setting with Proposition 3.2 as a particular case of our method.…”

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

“…The recurrence (2.23) in [86,2009] by Tian-Xiao He and Peter Jau-Shyong Shiue defines exactly the class of Ward-Horadam functions' second order sequences and this paper standard Jacques Binet form ( 21), ( 22) of the recurrence (3) solution for Ward-Horadam functions sequences H(x) constitutes the content of their Proposition 2.7. -as has been mentioned earlier.…”

Notes on generalized and extended Leonardo numbers

“…Theorem 3.1 (He and Shiue [7]). Let {a n } be a sequence of order 2 satisfying the linear recurrence relation (8).…”

Use Impulse Response Sequences in the Construction of Number Sequence Identities

“…[5] also shows a method of finding the expressions of the linear recurring sequences of order 2 and the interrelationship among those sequences. However, [5] also pointed out "the method presented in this paper (i.e., Proposition 3.2) cannot be extended to the higher order setting." In Section 2, we have shown our method based on the IRS can be extended to the higher setting with Proposition 3.2 as a particular case of our method.…”

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

“…The recurrence (2.23) in [86,2009] by Tian-Xiao He and Peter Jau-Shyong Shiue defines exactly the class of Ward-Horadam functions' second order sequences and this paper standard Jacques Binet form ( 21), ( 22) of the recurrence (3) solution for Ward-Horadam functions sequences H(x) constitutes the content of their Proposition 2.7. -as has been mentioned earlier.…”

Notes on generalized and extended Leonardo numbers

“…Theorem 3.1 (He and Shiue [7]). Let {a n } be a sequence of order 2 satisfying the linear recurrence relation (8).…”

Use Impulse Response Sequences in the Construction of Number Sequence Identities

“…[5] also shows a method of finding the expressions of the linear recurring sequences of order 2 and the interrelationship among those sequences. However, [5] also pointed out "the method presented in this paper (i.e., Proposition 3.2) cannot be extended to the higher order setting." In Section 2, we have shown our method based on the IRS can be extended to the higher setting with Proposition 3.2 as a particular case of our method.…”