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.

Some results on convolved (p,q)-Fibonacci polynomials

“…As usual, we may also call them the Lucas u-polynomial sequence and Lucas v-polynomial sequence, respectively. Some special cases of these two polynomial sequences, which can be found in several related references, [1][2][3][4][5][6][7] are listed in Table 1 …”

Use Impulse Response Sequences in the Construction of Number Sequence Identities

“…A numerous examples of IRS in sets A 2 and A 3 will be giving in Section 3. In [5], Shiue and the authors presented a method of the construction of general term expressions and identities for the linear recurring sequences of order r = 2 by using the reduction order method. However, the reduction method is too complicated for the linear recurring sequences of order r > 2.…”

Some identities of Gaussian binomial coefficients

“…𝑈 𝑛 (0, 1; 𝑝 1 , 𝑝 2 ) can be represented by its Binet from 𝑈 𝑛 = (𝛼 𝑛 − 𝛽 𝑛 )/(𝛼 − 𝛽) (cf. the authors [11]), where 𝛼 and 𝛽 are two distinct roots of the (𝑈 𝑛 ) ′ 𝑠 characteristic equation 𝑥 2 − 𝑝 1 𝑥 + 𝑝 2 = 0. Throughout this paper, we always assume the characteristic equation 𝑥 2 −𝑝 1 𝑥+𝑝 2 = 0 has non-zero constant term 𝑝 2 and two distinct roots 𝛼 and 𝛽.…”