Determinant Representations of Polynomial Sequences of Riordan Type

Abstract

In this paper, using the production matrix of a Riordan array, we obtain a recurrence relation for polynomial sequence associated with the Riordan array, and we also show that the general term for the sequence can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, a unified determinant expression for the four kinds of Chebyshev polynomials is given.

1. Introduction

The concept of a Riordan array is very useful in combinatorics. The infinite triangles of Pascal, Catalan, Motzkin, and Schröder are important and meaningful examples of Riordan array, and many others have been proposed and developed (see, e.g., [1–7]). In the recent literature, Riordan arrays have attracted the attention of various authors from many points of view and many examples and generalizations can be found (see, e.g., [8–12]).

A Riordan array denoted by (𝑔(𝑡),𝑓(𝑡)) is an infinite lower triangular matrix such that its column 𝑘  (𝑘=0,1,2,…) has generating function 𝑔(𝑡)𝑓(𝑡)^𝑘, where 𝑔(𝑡)=∑^∞_𝑛=0𝑔_𝑛𝑡^𝑛 and 𝑓(𝑡)=∑^∞_𝑛=0𝑓_𝑛𝑡^𝑛 are formal power series with 𝑔₀=1, 𝑓₀=0, and 𝑓₁≠0. That is, the general term of matrix 𝑅=(𝑔(𝑡),𝑓(𝑡)) is 𝑟_𝑛,𝑘=[𝑡^𝑛]𝑔(𝑡)𝑓(𝑡)^𝑘; here [𝑡^𝑛]ℎ(𝑡) denotes the coefficient of 𝑡^𝑛 in power series ℎ(𝑡). Given a Riordan array (𝑔(𝑡),𝑓(𝑡)) and column vector 𝐵=(𝑏₀,𝑏₁,𝑏₂,…)^𝑇, the product of (𝑔(𝑡),𝑓(𝑡)) and 𝐵 gives a column vector whose generating function is 𝑔(𝑡)𝑏(𝑓(𝑡)), where 𝑏(𝑡)=∑^∞_𝑛=0𝑏_𝑛𝑡^𝑛. If we identify a vector with its ordinary generating function, the composition rule can be rewritten as 𝑔(𝑡),𝑓(𝑡)𝑏(𝑡)=𝑔(𝑡)𝑏𝑓(𝑡). (1) This property is called the fundamental theorem for Riordan arrays and this leads to the matrix multiplication for Riordan arrays: 𝑔(𝑡),𝑓(𝑡)(ℎ(𝑡),𝑙(𝑡))=𝑔(𝑡)ℎ𝑓(𝑡),𝑙𝑓(𝑡). (2) The set of all Riordan arrays forms a group under the previos operation of a matrix multiplication. The identity element of the group is (1,𝑡). The inverse element of (𝑔(𝑡),𝑓(𝑡)) is 𝑔(𝑡),𝑓(𝑡)⁻¹=1𝑔𝑓(𝑡),𝑓(𝑡), (3) where 𝑓(𝑡) is compositional inverse of 𝑓(𝑡).

A Riordan array 𝑅=(𝑔(𝑡),𝑓(𝑡))=(𝑟_𝑛,𝑘)_{𝑛,𝑘≥0} can be characterized by two sequences 𝐴=(𝑎_𝑖)_𝑖≥0 and 𝑍=(𝑧_𝑖)_𝑖≥0 such that, for 𝑛,𝑘≥0𝑟_𝑛+1,0=𝑧₀𝑟_𝑛,0+𝑧₁𝑟_𝑛,1+𝑧₂𝑟_𝑛,2𝑟+⋯,_{𝑛+1,𝑘+1}=𝑎₀𝑟_𝑛,𝑘+𝑎₁𝑟_𝑛,𝑘+1+𝑎₂𝑟_𝑛,𝑘+2+⋯. (4) If 𝐴(𝑡) and 𝑍(𝑡) are the generating functions for the 𝐴- and 𝑍-sequences, respectively, then it follows that [9, 13] 𝑔(𝑡)=1𝑓(𝑡)(𝑡)𝑓(𝑡.1−𝑡𝑍,𝑓=𝑡𝐴) (5) If the inverse of 𝑅=(𝑔(𝑡),𝑓(𝑡)) is 𝑅⁻¹=(𝑑(𝑡),ℎ(𝑡)), then the 𝐴- and 𝑍-sequences of 𝑅 are 𝐴(𝑡)=𝑡ℎ(𝑡)(𝑡)=1,𝑍ℎ(𝑡)(1−𝑑(𝑡)). (6)

For an invertible lower triangular matrix 𝑅, its production matrix (also called its Stieltjes matrix; see [11, 14]) is the matrix 𝑃=𝑅⁻¹𝑅, where 𝑅 is the matrix 𝑅 with its first row removed. The production matrix 𝑃 can be characterized by the matrix equality 𝑅𝑃=𝐷𝑅, where 𝐷=(𝛿_𝑖+1,𝑗)_{𝑖,𝑗≥0} (𝛿 is the usual Kronecker delta).

Lemma 1 (see [14]). Assume that 𝑅=(𝑟_𝑛,𝑘) is an infinite lower triangular matrix with 𝑟_𝑛,𝑛≠0. Then 𝑅 is a Riordan array if and only if its production matrix 𝑃 is of the form 𝑧𝑃=⁰𝑎₀𝑧0000⋯₁𝑎₁𝑎₀𝑧000⋯₂𝑎₂𝑎₁𝑎₀𝑧00⋯₃𝑎₃𝑎₂𝑎₁𝑎₀𝑧0⋯₄𝑎₄𝑎₃𝑎₂𝑎₁𝑎₀⋯⋮⋮⋮⋮⋮⋮⋱, (7) where (𝑎₀,𝑎₁,𝑎₂,…) is the A-sequence and (𝑧₀,𝑧₁,𝑧₂,…) is the Z-sequence of the Riordan array 𝑅.

Definition 2. Let (𝑟_𝑛(𝑥))_𝑛≥0 be a sequence of polynomials where 𝑟_𝑛(𝑥) is of degree 𝑛 and 𝑟_𝑛(𝑥)=∑^𝑛_𝑘=0𝑟_𝑛,𝑘𝑥^𝑘. We say that (𝑟_𝑛(𝑥))_𝑛≥0 is a polynomial sequence of Riordan type if the coefficient matrix (𝑟_𝑛,𝑘)_{𝑛,𝑘≥0} is an element of the Riordan group; that is, there exists a Riordan array (𝑔(𝑡),𝑓(𝑡)) such that (𝑟_𝑛,𝑘)_{𝑛,𝑘≥0}=(𝑔(𝑡),𝑓(𝑡)). In this case, we say that (𝑟_𝑛(𝑥))_𝑛≥0 is the polynomial sequence associated with the Riordan array (𝑔(𝑡),𝑓(𝑡)).
Letting 𝑟_𝑛(𝑥)=∑^𝑛_𝑘=0𝑟_𝑛,𝑘𝑥^𝑘, 𝑛≥0, then in matrix form we have 𝑟_0,0𝑟000⋯_1,0𝑟_1,1𝑟00⋯_2,0𝑟_2,1𝑟_2,2𝑟0⋯_3,0𝑟_3,1𝑟_3,2𝑟_3,3⋯1𝑥𝑥⋮⋮⋮⋮⋱²𝑥³⋮=𝑟₀(𝑥)𝑟₁(𝑥)𝑟₂(𝑥)𝑟₃(𝑥)⋮. (8) Hence, by using (1), we have the following lemma.

Lemma 3. Let (𝑟_𝑛(𝑥))_𝑛≥0 be the polynomial sequence associated with a Riordan array (𝑔(𝑡),𝑓(𝑡)), and let 𝑟(𝑡,𝑥)=∑^∞_𝑛=0𝑟_𝑛(𝑥)𝑡^𝑛 be its generating function. Then 𝑟()=𝑔(𝑡)𝑡,𝑥(𝑡).1−𝑥𝑓 (9)

In [15], Luzón introduced a new notation 𝑇(𝑓∣𝑔) to represent the Riordan arrays and gave a recurrence relation for the family of polynomials associated to Riordan arrays. In recent works [16, 17], a new definition by means of a determinant form for Appell polynomials is given. Sequences of Appell polynomials are special of the Sheffer sequences [18]. In [19], the author obtains a determinant representation for the Sheffer sequence. The aim of this work is to propose a similar approach for polynomial sequences of Riordan type, which are special of the generalized Sheffer sequences [12, 18]. A determinant representation for polynomial sequences of Riordan type is obtained by using production matrix of Riordan array. In fact, we will show that the general formula for the polynomial sequences of Riordan type can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, determinant expressions for some classical polynomial sequences such as Fibonacci, Pell, and Chebyshev are derived, and a unified determinant expression for the four kinds of Chebyshev polynomials [20, 21] is established.

2. Main Theorem

In this section we are going to develop our main theorem.

Theorem 4. Let 𝑅=(𝑔(𝑡),𝑓(𝑡)) be a Riordan array with the Z-sequence (𝑧_𝑖)_𝑖≥0 and the A-sequence (𝑎_𝑖)_𝑖≥0. Let (𝑝_𝑛(𝑥))_𝑛≥0 be the polynomial sequence associated with 𝑅⁻¹. Then (𝑝_𝑛(𝑥))_𝑛≥0 satisfies the recurrence relation: 𝑎₀𝑝_𝑛(𝑥)=𝑥𝑝_𝑛−1(𝑥)−𝑎₁𝑝_𝑛−1(𝑥)−𝑎₂𝑝_𝑛−2(𝑥)−⋯−𝑎_𝑛−1𝑝₁(𝑥)−𝑧_𝑛−1𝑝₀(𝑥),𝑛>1, (10) with initial condition 𝑝₀(𝑥)=1, and 𝑎₀𝑝₁(𝑥)=𝑥−𝑧₀. In general, for all 𝑛≥1, 𝑝_𝑛(𝑥) is given by the following Hessenberg determinant: 𝑝_𝑛(𝑥)=(−1)^𝑛𝑎₀^−𝑛×𝑧₀−𝑥𝑎₀𝑧000⋯00₁𝑎₁−𝑥𝑎₀𝑧00⋯00₂𝑎₂𝑎₁−𝑥𝑎₀𝑧0⋯00₃𝑎₃𝑎₂𝑎₁−𝑥𝑎₀𝑧⋯00₄𝑎₄𝑎₃𝑎₂𝑎₁𝑧−𝑥⋯00⋮⋮⋮⋮⋮⋱⋮⋮_𝑛−2𝑎_𝑛−2𝑎_𝑛−3𝑎_𝑛−4𝑎_𝑛−5⋯𝑎₁−𝑥𝑎₀𝑧_𝑛−1𝑎_𝑛−1𝑎_𝑛−2𝑎_𝑛−3𝑎_𝑛−4⋯𝑎₂𝑎₁.−𝑥 (11)

Proof. Let 𝑅=(𝑔(𝑡),𝑓(𝑡)) and 𝑅⁻¹=(𝑔(𝑡),𝑓(𝑡))⁻¹=(𝑑(𝑡),ℎ(𝑡)). Then from definition and (3), we have 𝑔(0)=1 and 𝑑(𝑡)=1/𝑔(𝑓(𝑡)). Hence 𝑑(0)=1 and 𝑝₀(𝑥)=1. Letting 𝐸=(1,𝑥,𝑥²,…)^𝑇, then 𝑅⁻¹𝐸=(𝑝₀(𝑥),𝑝₁(𝑥),𝑝₂(𝑥),…)^𝑇 and 𝐷𝐸=(𝑥,𝑥²,𝑥³,…)^𝑇, where 𝐷=(𝛿_𝑖+1,𝑗). Letting 𝑃 be the production matrix of 𝑅, then 𝑅𝑃=𝐷𝑅, and 𝑃𝑅⁻¹=𝑅⁻¹𝐷. Thus 𝑃𝑅⁻¹𝐸=𝑅⁻¹𝐷𝐸, and 𝑃𝑅⁻¹𝐸=𝑥𝑅⁻¹𝐸. In matrix form, we have 𝑧₀𝑎₀𝑧000⋯₁𝑎₁𝑎₀𝑧00⋯₂𝑎₂𝑎₁𝑎₀𝑧0⋯₃𝑎₃𝑎₂𝑎₁𝑎₀⋯𝑝⋮⋮⋮⋮⋮⋱⁰(𝑥)𝑝₁(𝑥)𝑝₂(𝑥)𝑝₃(𝑥)⋮=𝑥𝑝₀(𝑥)𝑥𝑝₁(𝑥)𝑥𝑝₂(𝑥)𝑥𝑝₃(𝑥)⋮. (12)
Using the block matrix method, we get 𝑝₀(𝑥)𝑧₀𝑧₁𝑧₂𝑧₃⋮+𝑎₀𝑎000⋯₁𝑎₀𝑎00⋯₂𝑎₁𝑎₀𝑎0⋯₃𝑎₂𝑎₁𝑎₀⋯×𝑝⋮⋮⋮⋮⋱₁(𝑥)𝑝₂(𝑥)𝑝₃(𝑥)𝑝₄(𝑥)⋮=𝑥𝑝₀(𝑥)𝑥𝑝₁(𝑥)𝑥𝑝₂(𝑥)𝑥𝑝₃(𝑥)⋮. (13)
Since 𝑥𝑝₀(𝑥)𝑥𝑝₁(𝑥)𝑥𝑝₂(𝑥)𝑥𝑝₃(𝑥)⋮=𝑥𝑝₀(𝑥)000⋮+0𝑥𝑝₁(𝑥)𝑥𝑝₂(𝑥)𝑥𝑝₃(𝑥)⋮,0𝑥𝑝₁(𝑥)𝑥𝑝₂(𝑥)𝑥𝑝₃(𝑥)⋮=0000⋯𝑝𝑥000⋯0𝑥00⋯00𝑥0⋯⋮⋮⋮⋮⋱¹(𝑥)𝑝₂(𝑥)𝑝₃(𝑥)𝑝₄(𝑥)⋮. (14) The previous matrix equation can be rewritten as 𝑎₀𝑎000⋯₁−𝑥𝑎₀𝑎00⋯₂𝑎₁−𝑥𝑎₀𝑎0⋯₃𝑎₂𝑎₁−𝑥𝑎₀⋯𝑝⋮⋮⋮⋮⋱¹(𝑥)𝑝₂(𝑥)𝑝₃(𝑥)𝑝₄(𝑥)⋮=𝑝₀(𝑥)𝑥−𝑧₀−𝑧₁−𝑧₂−𝑧₃⋮. (15) Therefore, 𝑎₀𝑝₁(𝑥)=𝑥−𝑧₀, and for 𝑛>1, we have 𝑎_𝑛−1𝑝₁(𝑥)+⋯+𝑎₂𝑝_𝑛−2(𝑥)+𝑎₁−𝑥𝑝_𝑛−1(𝑥)+𝑎₀𝑝_𝑛(𝑥)=−𝑧_𝑛−1𝑝₀(𝑥), (16) or equivalently 𝑎₀𝑝_𝑛(𝑥)=𝑥−𝑎₁𝑝_𝑛−1(𝑥)−𝑎₂𝑝_𝑛−2(𝑥)−⋯−𝑎_𝑛−1𝑝₁(𝑥)−𝑧_𝑛−1𝑝₀(𝑥). (17)
By applying the Cramer’s rule, we can work out the unknown 𝑝_𝑛(𝑥) operating with the first 𝑛 equations in (15): 𝑝_𝑛(𝑥)=𝑎₀^−𝑛𝑎₀0000⋯0𝑥−𝑧₀𝑝₀(𝑥)𝑎₁−𝑥𝑎₀000⋯0−𝑧₁𝑝₀(𝑥)𝑎₂𝑎₁−𝑥𝑎₀00⋯0−𝑧₂𝑝₀(𝑥)𝑎₃𝑎₂𝑎₁−𝑥𝑎₀0⋯0−𝑧₃𝑝₀(𝑥)𝑎₄𝑎₃𝑎₂𝑎₁−𝑥𝑎₀⋯0−𝑧₄𝑝₀(𝑥)𝑎⋮⋮⋮⋮⋮⋱⋮⋮_𝑛−2𝑎_𝑛−3𝑎_𝑛−4𝑎_𝑛−5𝑎_𝑛−6⋯𝑎₀−𝑧_𝑛−2𝑝₀(𝑥)𝑎_𝑛−1𝑎_𝑛−2𝑎_𝑛−3𝑎_𝑛−4𝑎_𝑛−5⋯𝑎₁−𝑥−𝑧_𝑛−1𝑝₀(𝑥)=𝑝₀(𝑥)𝑎₀^−𝑛𝑎₀0000⋯0𝑥−𝑧₀𝑎₁−𝑥𝑎₀000⋯0−𝑧₁𝑎₂𝑎₁−𝑥𝑎₀00⋯0−𝑧₂𝑎₃𝑎₂𝑎₁−𝑥𝑎₀0⋯0−𝑧₃𝑎₄𝑎₃𝑎₂𝑎₁−𝑥𝑎₀⋯0−𝑧₄𝑎⋮⋮⋮⋮⋮⋱⋮⋮_𝑛−2𝑎_𝑛−3𝑎_𝑛−4𝑎_𝑛−5𝑎_𝑛−6⋯𝑎₀−𝑧_𝑛−2𝑎_𝑛−1𝑎_𝑛−2𝑎_𝑛−3𝑎_𝑛−4𝑎_𝑛−5⋯𝑎₁−𝑥−𝑧_𝑛−1. (18)
After transferring the last column to the first position, an operation which introduces the factor (−1)^𝑛−1, the theorem follows.

Corollary 5. Let 𝑅=(𝑔(𝑡),𝑓(𝑡)) be a Riordan array with production matrix 𝑃. Let (𝑝_𝑛(𝑥))_𝑛≥0 be the polynomial sequence associated with 𝑅⁻¹=(𝑔(𝑡),𝑓(𝑡))⁻¹. Then 𝑝₀(𝑥)=1, and for all 𝑛≥1, 𝑝_𝑛(𝑥)=𝑎₀^−𝑛det𝑥𝐼_𝑛−𝑃_𝑛, (19) where 𝑎₀=𝑝_0,1, 𝑃_𝑛 is the principal submatrix of order 𝑛 of the production matrix 𝑃 and 𝐼_𝑛 is the identity matrix of order 𝑛.

3. Applications

A useful application of Theorem 4 is to find the determinant expression of a well-known sequence. We illustrate the ideal in the following examples. In the final paragraph, we will give a unified determinant expression for the four kinds of Chebyshev polynomials.

Example 6. Considering the Riordan array 𝒜=(1/(1+𝑟𝑡²),𝑎𝑡/(1+𝑟𝑡²)), we have (1/(1+𝑟𝑡²),𝑎𝑡/(1+𝑟𝑡²))(1/(1−𝑥𝑡))=1/(1−𝑎𝑥𝑡+𝑟𝑡²). The generating functions of the 𝐴- and 𝑍-sequences of 𝒜⁻¹ are 𝐴(𝑡)=1+𝑟𝑡²𝑎,𝑍(𝑡)=𝑟𝑡𝑎. (20)

Let (𝑝_𝑛(𝑥))_𝑛≥0 be the polynomial sequence associated with 𝒜=(1/(1+𝑟𝑡²),𝑎𝑡/(1+𝑟𝑡²)). Then (𝑝_𝑛(𝑥))_𝑛≥0 satisfies the recurrence relation: 𝑝_𝑛(𝑥)=𝑎𝑥𝑝_𝑛−1(𝑥)−𝑟𝑝_𝑛−2(𝑥),𝑛≥2, (21) with initial condition 𝑝₀(𝑥)=1, and 𝑝₁(𝑥)=𝑎𝑥. In general, 𝑝_𝑛(𝑥) is also given by the following Hessenberg determinant: 𝑝_𝑛(𝑥)=(−1)^𝑛−𝑎𝑥10000⋯0𝑟−𝑎𝑥1000⋯00𝑟−𝑎𝑥100⋯000𝑟−𝑎𝑥10⋯0000𝑟−𝑎𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−𝑎𝑥. (22) If 𝑎=1, 𝑟=−1, then 𝑝_𝑛(𝑥) becomes the Fibonacci polynomials: 𝐹_𝑛(𝑥)=(−1)^𝑛−𝑥10000⋯0−1−𝑥1000⋯00−1−𝑥100⋯000−1−𝑥10⋯0000−1−𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−𝑥. (23) If 𝑎=2, 𝑟=−1, then 𝑝_𝑛(𝑥) gives the Pell polynomials: 𝑃_𝑛(𝑥)=𝐹_𝑛()=()2𝑥−1^𝑛.−2𝑥10000⋯0−1−2𝑥1000⋯00−1−2𝑥100⋯000−1−2𝑥10⋯0000−1−2𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−2𝑥 (24) In case 𝑎=2, 𝑟=1, 𝑝_𝑛(𝑥) becomes the Chebyshev polynomials of the second kind: 𝑈_𝑛(𝑥)=(−1)^𝑛−2𝑥10000⋯01−2𝑥1000⋯001−2𝑥100⋯0001−2𝑥10⋯00001−2𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−2𝑥. (25)

Example 7. Considering the Riordan array ℬ=((1−𝑏𝑡²)/(1+𝑟𝑡²),𝑎𝑡/(1+𝑟𝑡²)), we have ((1−𝑏𝑡²)/(1+𝑟𝑡²),𝑎𝑡/(1+𝑟𝑡²))(1/(1−𝑥𝑡))=(1−𝑏𝑡²)/(1−𝑎𝑥𝑡+𝑟𝑡²). Then the generating functions of the 𝐴- and 𝑍-sequences of ℬ⁻¹ are 𝐴(𝑡)=1+𝑟𝑡²𝑎(𝑡)=()𝑡,𝑍𝑏+𝑟𝑎. (26)

Let (𝑝_𝑛(𝑥))_𝑛≥0 be the polynomial sequence associated with ℬ=((1−𝑏𝑡²)/(1+𝑟𝑡²),𝑎𝑡/(1+𝑟𝑡²)). Then (𝑝_𝑛(𝑥))_𝑛≥0 satisfies the recurrence relation: 𝑝_𝑛(𝑥)=𝑎𝑥𝑝_𝑛−1(𝑥)−𝑟𝑝_𝑛−2(𝑥),𝑛≥3, (27) with initial condition 𝑝₀(𝑥)=1, and 𝑝₁(𝑥)=𝑎𝑥, 𝑝₂(𝑥)=𝑎²𝑥²−𝑏−𝑟.

In general, 𝑝_𝑛(𝑥) is also given by the following Hessenberg determinant: 𝑝_𝑛(𝑥)=(−1)^𝑛−𝑎𝑥10000⋯0𝑏+𝑟−𝑎𝑥1000⋯00𝑟−𝑎𝑥100⋯000𝑟−𝑎𝑥10⋯0000𝑟−𝑎𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−𝑎𝑥. (28)

If 𝑎=2, 𝑏=𝑟=1, then 𝑝_𝑛(𝑥) become the Chebyshev polynomials of the first kind 2𝑇_𝑛(𝑥)−0^𝑛: 2𝑇_𝑛(𝑥)=(−1)^𝑛−2𝑥10000⋯02−2𝑥1000⋯001−2𝑥100⋯0001−2𝑥10⋯00001−2𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−2𝑥. (29)

In case 𝑎=3, 𝑏=0, 𝑟=2, 𝑝_𝑛(𝑥) give the Fermat polynomials (see [15]): 𝑓_𝑛(𝑥)=(−1)^𝑛−3𝑥10000⋯02−3𝑥1000⋯002−3𝑥100⋯0002−3𝑥10⋯00002−3𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−3𝑥. (30)

Example 8. Considering the Riordan array 𝒞=((1−𝑏𝑡)/(1+𝑟𝑡²),𝑎𝑡/(1+𝑟𝑡²)), we have ((1−𝑏𝑡)/(1+𝑟𝑡²),𝑎𝑡/(1+𝑟𝑡²))(1/(1−𝑥𝑡))=(1−𝑏𝑡)/(1−𝑎𝑥𝑡+𝑟𝑡²). The generating functions of the 𝐴- and 𝑍-sequences of 𝒞⁻¹ are 𝐴(𝑡)=1+𝑟𝑡²𝑎,𝑍(𝑡)=𝑏+𝑟𝑡𝑎. (31)

Let (𝑝_𝑛(𝑥))_𝑛≥0 be the polynomial sequence associated with 𝒞=((1−𝑏𝑡)/(1+𝑟𝑡²),𝑎𝑡/(1+𝑟𝑡²)). Then (𝑝_𝑛(𝑥))_𝑛≥0 satisfies the recurrence relation: 𝑝_𝑛(𝑥)=𝑎𝑥𝑝_𝑛−1(𝑥)−𝑟𝑝_𝑛−2(𝑥),𝑛≥3, (32) with initial condition 𝑝₀(𝑥)=1, and 𝑝₁(𝑥)=𝑎𝑥−𝑏, 𝑝₂(𝑥)=𝑎²𝑥²+𝑎𝑏𝑥−𝑟.

In general, 𝑝_𝑛(𝑥) is also given by the following Hessenberg determinant: 𝑝_𝑛(𝑥)=(−1)^𝑛𝑏−𝑎𝑥10000⋯0𝑟−𝑎𝑥1000⋯00𝑟−𝑎𝑥100⋯000𝑟−𝑎𝑥10⋯0000𝑟−𝑎𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−𝑎𝑥. (33)

If 𝑎=2, 𝑏=1, 𝑟=1, then 𝑝_𝑛(𝑥) becomes the Chebyshev polynomials of the third kind 𝑉_𝑛(𝑥): 𝑉_𝑛(𝑥)=(−1)^𝑛1−2𝑥10000⋯01−2𝑥1000⋯001−2𝑥100⋯0001−2𝑥10⋯00001−2𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−2𝑥. (34)

If 𝑎=2, 𝑏=−1, 𝑟=1, then 𝑝_𝑛(𝑥) gives the Chebyshev polynomials of the fourth kind 𝑊_𝑛(𝑥): 𝑊_𝑛(𝑥)=(−1)^𝑛−1−2𝑥10000⋯01−2𝑥1000⋯001−2𝑥100⋯0001−2𝑥10⋯00001−2𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−2𝑥. (35)

Finally, considering the Riordan array 𝒟=((1−𝑏𝑡−𝑐𝑡²)/(1+𝑡²),𝑎𝑡/(1+𝑡²)), we have ((1−𝑏𝑡−𝑐𝑡²)/(1+𝑡²),𝑎𝑡/(1+𝑡²))(1/(1−𝑥𝑡))=(1−𝑏𝑡−𝑐𝑡²)/(1−𝑎𝑥𝑡+𝑡²). Then the generating functions of the 𝐴- and 𝑍-sequences of 𝒟⁻¹ are 𝐴(𝑡)=1+𝑡²𝑎(𝑡)=()𝑡,𝑍𝑏+𝑐+1𝑎. (36)

Let (𝑝_𝑛(𝑥))_𝑛≥0 be the polynomial sequence associated with 𝒟=((1−𝑏𝑡−𝑐𝑡²)/(1+𝑡²),𝑎𝑡/(1+𝑡²)). Then (𝑝_𝑛(𝑥))_𝑛≥0 satisfies the recurrence relation: 𝑝_𝑛(𝑥)=𝑎𝑥𝑝_𝑛−1(𝑥)−𝑝_𝑛−2(𝑥),𝑛≥3, (37)

with initial condition 𝑝₀(𝑥)=1, and 𝑝₁(𝑥)=𝑎𝑥−𝑏, 𝑝₂(𝑥)=𝑎𝑥𝑝₁(𝑥)−(𝑐+1)𝑝₀(𝑥)=𝑎²𝑥²−𝑎𝑏𝑥−𝑐−1. For 𝑛≥1, we have 𝑝_𝑛(𝑥)=(−1)^𝑛𝑏−𝑎𝑥10000⋯0𝑐+1−𝑎𝑥1000⋯001−𝑎𝑥100⋯0001−𝑎𝑥10⋯00001−𝑎𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−𝑎𝑥. (38)

Therefore we can give, now, the following.

Definition 9. The Chebyshev polynomial of degree 𝑛, denoted by 𝐶_𝑛(𝑥,𝑎,𝑏,𝑐), is defined by 𝐶₀()𝐶𝑥,𝑎,𝑏,𝑐=1,_𝑛()=()𝑥,𝑎,𝑏,𝑐−1^𝑛,𝑏−𝑎𝑥10000⋯0𝑐+1−𝑎𝑥1000⋯001−𝑎𝑥100⋯0001−𝑎𝑥10⋯00001−𝑎𝑥1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯−𝑎𝑥𝑛≥1, (39) where 𝐶_𝑛(𝑥,𝑎,𝑏,𝑐) is represented by a Hessenberg determinant of order 𝑛.
Note that  𝐶_𝑛(𝑥,2,0,1)=2𝑇_𝑛(𝑥), 𝐶_𝑛(𝑥,2,0,0)=𝑈_𝑛(𝑥), 𝐶_𝑛(𝑥,2,1,0)=𝑉_𝑛(𝑥), and 𝐶_𝑛(𝑥,2,−1,0)=𝑊_𝑛(𝑥). Hence, Definition 9 can be considered as a unified form for the four kinds of Chebyshev polynomials.

Acknowledgments

The authors wish to thank the editor and referee for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 11261032) and the Natural Science Foundation of Gansu Province (Grant no. 1010RJZA049).