On the Connection Coefficients of the Chebyshev-Boubaker Polynomials

Abstract

The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.

1. Introduction

In this paper, we will show that the usual recurrence relations that define elements of Riordan arrays can be used to give a set of recurrence relations for the connection coefficients of families of orthogonal polynomials defined by Riordan arrays. It will also be seen that we can give closed-form expressions for the connection coefficients themselves, at least so far, as it is easy to give closed-form expressions for the Riordan arrays concerned. Thus, in the special case of orthogonal polynomials that are defined by Riordan arrays, we can achieve two of the main goals involved in the study of connection coefficients for orthogonal polynomials [1–7].

If 𝑃_𝑛(𝑥) and 𝑄_𝑛(𝑥) are two families of orthogonal polynomials, the connection coefficients 𝑎_𝑛,𝑘 of 𝑃_𝑛 in terms of 𝑄_𝑛, defined by 𝑃_𝑛=^𝑛_𝑘=0𝑎_𝑛,𝑘𝑄_𝑘, (1) are often of interest. In terms of matrices, we have 𝐏=𝐀𝐐, (2) where 𝐏 is the coefficient array of the polynomials 𝑃_𝑛(𝑥), with elements 𝑝_𝑛,𝑘, and 𝐐 is the coefficient array of the polynomials 𝑄_𝑛(𝑥), with elements 𝑞_𝑛,𝑘. Thus, we have 𝑃_𝑛(𝑥)=^𝑛_𝑘=0𝑝_𝑛,𝑘𝑥^𝑘,𝑄_𝑛(𝑥)=^𝑛_𝑘=0𝑞_𝑛,𝑘𝑥^𝑘. (3) The matrices 𝐏 and 𝐐 are lower-triangular matrices. Clearly, the matrix 𝐀 of connection coefficients is then given by 𝐀=𝐏𝐐⁻¹. (4) We immediately note that 𝐀⁻¹=𝐐𝐏⁻¹, (5) so that the elements of 𝐀⁻¹ are the connection coefficients of 𝑄_𝑛 in terms of 𝑃_𝑛.

The matrix 𝐀 is lower triangular since both 𝐏 and 𝐐 are.

We will define a family of orthogonal polynomials 𝑃_𝑛(𝑥) to be of Chebyshev-Boubaker type, if the coefficient array 𝐏=(𝑝_𝑛,𝑘) is a Riordan array of the type 1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²,𝜆𝑥1+𝑎𝑥+𝑏𝑥². (6) This means that 𝑝_𝑛,𝑘=𝑥^𝑛1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²𝜆𝑥1+𝑎𝑥+𝑏𝑥^2𝑘. (7) In the case that 𝜆=1, the polynomials are monic. An array of the type 1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²,𝜆𝑥1+𝑎𝑥+𝑏𝑥² (8) will be called a Chebyshev-Boubaker array (with parameters (𝑎,𝑏,𝑢,𝑣,𝜆)). This is a generalization of the notion of a Chebyshev-Boubaker array considered in [8]. The Boubaker polynomials [8] correspond to the parameters (0,1,0,3,1).

Example 1. The Chebyshev polynomials of the second kind 𝑈_𝑛(𝑥) are defined by 𝑈_𝑛(𝑥)=^⌊𝑛/2⌋_𝑘=0(−1)^𝑘𝑘()𝑛−𝑘2𝑥^𝑛−2𝑘 (9) and have coefficient array 𝐔 given by A0531171𝐔=1+𝑥²,2𝑥1+𝑥². (10)

Example 2. We let 𝑆_𝑛(𝑥)=𝑈_𝑛(𝑥/2) be the monic Chebyshev polynomials of the second kind. The coefficient array 𝐒 of these polynomials is given by A0493101𝐒=1+𝑥²,𝑥1+𝑥². (11) This is the case 𝑢=𝑣=𝑎=0, 𝑏=1, and 𝜆=2.
The moments 𝑚_𝑛 of this family of orthogonal polynomials are the aerated Catalan numbers that begin with 1,0,1,0,2,0,5,0,14,0,42,…. (12) Their moment representation is given by 𝑚_𝑛=12𝜋²₋₂𝑥^𝑛√4−𝑥²𝑑𝑥. (13)

Very often, orthogonal polynomials are specified in terms of the three-term recurrence that they satisfy. In the monic case, this will take the form 𝑃_𝑛(𝑥)=𝑥−𝛼_𝑛𝑃_𝑛−1(𝑥)−𝛽_𝑛𝑃_𝑛−2(𝑥), (14) with suitable initial conditions. For instance, we have 𝑆_𝑛(𝑥)=(𝑥−2)𝑆_𝑛−1(𝑥)−𝑆_𝑛−2(𝑥),𝑆₀(𝑥)=1,𝑆₁(𝑥)=𝑥. (15) The recurrence coefficients 𝛼_𝑛 and 𝛽_𝑛 for the family 𝑃_𝑛 of orthogonal polynomials are the entries in the production matrix (or Stieltjes matrix [9]) of the matrix 𝐏⁻¹. This production matrix is equal to 𝐏𝐏⁻¹, where the notation 𝐌 denotes the matrix 𝐌 with its first row removed. For instance, the production matrix of 𝐔⁻¹=(1/(1+𝑥²),2𝑥/(1+𝑥²))⁻¹ is given by 01210000⋯201201000⋯2012100⋯00201210⋯0002012⋯⋮⋮⋮⋮⋮⋮⋱, (16) corresponding to the three-term recurrence 12𝑈_𝑛(𝑥)=(𝑥−0)𝑈_𝑛−1(𝑥)−12𝑈_𝑛−2(𝑥), (17) or 𝑈_𝑛(𝑥)=2𝑥𝑈_𝑛−1(𝑥)−𝑈_𝑛−2(𝑥). (18) Similarly, the production matrix of 𝐒⁻¹=(1/(1+𝑥²),𝑥/(1+𝑥²))⁻¹ is given by ,010000⋯101000⋯010100⋯001010⋯000101⋯⋮⋮⋮⋮⋮⋮⋱ (19) corresponding to the three-term recurrence 𝑆_𝑛(𝑥)=𝑥𝑆_𝑛−1(𝑥)−𝑆_𝑛−2(𝑥). (20) More generally, the production matrix of 𝐒_𝑏⁻¹=(1/(1+𝑏𝑥²),𝑥/(1+𝑏𝑥²))⁻¹ is given by ,010000⋯𝑏01000⋯0𝑏0100⋯00𝑏010⋯000𝑏01⋯⋮⋮⋮⋮⋮⋮⋱ (21) corresponding to the three-term recurrence 𝑆_𝑛^(𝑏)(𝑥)=𝑥𝑆^(𝑏)_𝑛−1(𝑥)−𝑏𝑆^(𝑏)_𝑛−2(𝑥),𝑆₀^(𝑏)=1,𝑆₁^(𝑏)(𝑥)=𝑥, (22) where 𝑆_𝑛^(𝑏)(𝑥)=^⌊𝑛/2⌋_𝑘=0𝑘()𝑛−𝑘−𝑏^𝑘𝑥^𝑛−2𝑘. (23)

Example 3. The Boubaker polynomials [8, 10–15] have coefficient array 1+3𝑥²1+𝑥²,𝑥1+𝑥². (24) They can be expressed as 𝑃_𝑛(𝑥)=^⌊𝑛/2⌋_𝑘=0𝑘𝑛−𝑘𝑛−4𝑘()𝑛−𝑘−1^𝑘𝑥^𝑛−2𝑘. (25) We have 1+3𝑥²1+𝑥²,𝑥1+𝑥²⁻¹=1+𝑥²𝑐𝑥²²1+3𝑥²𝑐𝑥²²,𝑥𝑐𝑥²=11+2𝑥²𝑐𝑥²,𝑥𝑐𝑥², (26) where 𝑐(𝑥)=1−√1−4𝑥2𝑥 (27) is the generating function of the Catalan numbers 𝐶_𝑛=1𝑛𝑛+12𝑛. (28) Hence, the moments 𝑚_𝑛 of this family of orthogonal polynomials have g.f. 1/(1+2𝑥²c(𝑥²)). The sequence 𝑚_𝑛 begins with 1,0,−2,0,2,0,−4,0,2,0,−12,0,−12,0,−72,… (29) and has the moment representation 𝑚_𝑛1=−𝜋²₋₂𝑥^𝑛√4−𝑥²4+3𝑥²2𝑑𝑥+3−2√3𝑖^𝑛+232√3𝑖^𝑛. (30) We have 𝑚_2𝑛=^𝑛_𝑘=0𝑘+0^𝑛+𝑘𝑛+0^𝑛𝑘()2𝑛−𝑘−1𝑛−𝑘−2^𝑘=^𝑛_𝑘=02𝑘+1()𝑛+𝑘+12𝑛𝑛−𝑘−3^𝑘. (31) The sequence 𝑚_2𝑛 is A126984. We note that the Hankel transforms of 𝑚_𝑛 and 𝑚_2𝑛 are both equal to (−2)^𝑛. The production matrix of ((1+3𝑥²)/(1+𝑥²),𝑥/(1+𝑥²))⁻¹ is given by .010000⋯−201000⋯010100⋯001010⋯000101⋯⋮⋮⋮⋮⋮⋮⋱ (32)

In this paper, we will give a closed-form expression for the connection coefficients of the Chebyshev-Boubaker polynomials 𝑃_𝑛(𝑥) with coefficient array 𝐏=1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²,𝑥1+𝑎𝑥+𝑏𝑥² (33) in terms of the Chebyshev-Boubaker polynomials 𝑄_𝑛(𝑥) with coefficient array 𝐐=1+𝑠𝑥+𝑡𝑥²1+𝑐𝑥+𝑑𝑥²,𝑥1+𝑐𝑥+𝑑𝑥². (34) We note that the polynomials 𝑃_𝑛(𝑥) satisfy the recurrence 𝑃_𝑛(𝑥)=(𝑥−𝑎)𝑃_𝑛−1(𝑥)−𝑏𝑃_𝑛−2(𝑥),𝑛≥3, (35) with 𝑃₀(𝑥)=1,𝑃₁(𝑥)𝑃=𝑥+𝑢−𝑎,₂(𝑥)=𝑥²+(𝑢−2𝑎)𝑥−𝑢𝑎+𝑎²+𝑣−𝑏, (36) with a similar expression for 𝑄_𝑛(𝑥).

In the next section, we recall notational elements that we will use in the sequel, along with a brief introduction to Riordan arrays.

2. Preliminaries on Integer Sequences and Riordan Arrays

In this section, we define terms used later to discuss integer sequences, Riordan arrays, production matrices, orthogonal polynomials, and Hankel transforms. Readers familiar with these areas and the links between them may skip this section.

For an integer sequence 𝑎_𝑛, that is, an element of ℤ^ℕ, the power series 𝑓(𝑥)=∑^∞_𝑛=0𝑎_𝑛𝑥^𝑛 is called the ordinary generating function or g.f. of the sequence. 𝑎_𝑛 is thus the coefficient of 𝑥^𝑛 in this series. We denote this by 𝑎_𝑛=[𝑥^𝑛]𝑓(𝑥). For instance, 𝐹_𝑛=[𝑥^𝑛](𝑥/(1−𝑥−𝑥²)) is the 𝑛th Fibonacci number A000045, while 𝐶_𝑛=[𝑥^𝑛]((1−√1−4𝑥)/2𝑥) is the 𝑛th Catalan number A000108. We use the notation 0^𝑛=[𝑥^𝑛]1 for the sequence 1,0,0,0,…,  A000007. Thus, 0^𝑛=[𝑛=0]=𝛿_𝑛,0=⁰_𝑛. Here, we have used the Iverson bracket notation [16], defined by [𝒫]=1 if the proposition 𝒫 is true and [𝒫]=0 if 𝒫 is false.

For a power series 𝑓(𝑥)=∑^∞_𝑛=0𝑎_𝑛𝑥^𝑛 with 𝑓(0)=0, we define the reversion or compositional inverse of 𝑓 to be the power series 𝑓(𝑥) such that 𝑓(𝑓(𝑥))=𝑥.

For a lower triangular matrix (𝑎_𝑛,𝑘)_{𝑛,𝑘≥0}, the row sums give the sequence with general term ∑^𝑛_𝑘=0𝑎_𝑛,𝑘. For a sequence 𝑎_𝑛, the sequence of determinants |𝑎_𝑖+𝑗|_{0≤𝑖,𝑗≤𝑛} is called the Hankel transform of 𝑎_𝑛 [17–27].

The Riordan group [28, 29] is a set of infinite lower-triangular integer matrices, where each matrix is defined by a pair of generating functions 𝑔(𝑥)=1+𝑔₁𝑥+𝑔₂𝑥²+⋯ and 𝑓(𝑥)=𝑓₁𝑥+𝑓₂𝑥²+⋯ where 𝑓₁≠0 [29]. We often require in addition that 𝑓₁=1. The associated matrix is the matrix whose 𝑖th column is generated by 𝑔(𝑥)𝑓(𝑥)^𝑖 (the first column being indexed by 0). The matrix corresponding to the pair 𝑔,𝑓 is denoted by (𝑔,𝑓) or ℛ(𝑔,𝑓). The group law is then given by 𝑔,𝑓⋅(ℎ,𝑙)=𝑔,𝑓(ℎ,𝑙)=𝑔ℎ∘𝑓,𝑙∘𝑓. (37) The identity for this law is 𝐼=(1,𝑥), and the inverse of (𝑔,𝑓) is 𝑔,𝑓⁻¹=1𝑔∘𝑓,𝑓, (38) where 𝑓 is the compositional inverse of 𝑓.

If 𝐌 is the matrix (𝑔,𝑓) and 𝐚=(𝑎₀,𝑎₁,…)^𝑇 is an integer sequence (expressed as an infinite column vector) with ordinary generating function 𝒜(𝑥), then the sequence 𝐌𝐚 has ordinary generating function 𝑔(𝑥)𝒜(𝑓(𝑥)). The (infinite) matrix (𝑔,𝑓) can thus be considered to act on the ring of integer sequences ℤ^ℕ by multiplication, where a sequence is regarded as a (infinite) column vector. We can extend this action to the ring of power series ℤ[[𝑥]] by 𝑔,𝑓∶𝒜(𝑥)↦𝑔,𝑓⋅𝒜(𝑥)=𝑔(𝑥)𝒜𝑓(𝑥). (39)

Example 4. The so-called binomial matrix 𝐵 is the element (1/(1−𝑥),𝑥/(1−𝑥)) of the Riordan group. It has general element (^𝑛_𝑘) and hence as an array coincides with Pascal's triangle. More generally, 𝐵^𝑚 is the element (1/(1−𝑚𝑥),𝑥/(1−𝑚𝑥)) of the Riordan group, with general term (^𝑛_𝑘)𝑚^𝑛−𝑘. It is easy to show that the inverse 𝐵^−𝑚 of 𝐵^𝑚 is given by (1/(1+𝑚𝑥),𝑥/(1+𝑚𝑥)).

The row sums of the matrix (𝑔,𝑓) have generating function ⋅1𝑔,𝑓=𝑔(𝑥)1−𝑥(𝑥).1−𝑓 (40)

Example 5. The inverse of the Riordan array (1/(1+𝑥),𝑥/(1+𝑥)²) is the Riordan array (𝑐(𝑥),𝑥𝑐(𝑥)²). This follows since the solution of the equation 𝑧(1+𝑧)²=𝑥 (41) is given by 𝑧=𝑥𝑐(𝑥)². (42) We then have 1+𝑥𝑐(𝑥)²=𝑐(𝑥). (43)

Many interesting examples of sequences and Riordan arrays can be found in Neil Sloane's On-Line Encyclopedia of Integer Sequences (OEIS), [30, 31]. Sequences are frequently referred to by their OEIS number. For instance, the binomial matrix 𝐵 (“Pascal's triangle”) is A007318.

For an invertible matrix 𝑀, its production matrix (also called its Stieltjes matrix) [32, 33] is the matrix 𝒮_𝑀=𝑀⁻¹𝑀, (44) where 𝑀 is the matrix 𝑀 with its first row removed. A Riordan array 𝑀 is the inverse of the coefficient array of a family of orthogonal polynomials [34–36] if and only if 𝒮_𝑀 is tridiagonal [8, 17]. Necessarily, the Jacobi coefficients (i.e., the coefficients of the three-term recurrence [34] that defines the polynomials) of these orthogonal polynomials are then constant.

Example 6. The production matrix of (𝑐(𝑥),𝑥𝑐(𝑥)²) is given by 110000⋯121000⋯012100⋯001210⋯000121⋯000012⋯⋮⋮⋮⋮⋮⋮⋱. (45)

An important feature of Riordan arrays is that they have a number of sequence characterizations [37, 38]. The simplest of these is as follows.

Proposition 7 (see [38], Theorems 2.1, and 2.2). Let 𝐷=[𝑑_𝑛,𝑘] be an infinite triangular matrix. Then, 𝐷 is a Riordan array if and only if there exist two sequences 𝐴=[𝛼₀,𝛼₁,𝛼₂,…] and 𝑍=[𝑧₀,𝑧₁,𝑧₂,…] with 𝛼₀≠0, 𝑧₀≠0 such that (i)𝑑_{𝑛+1,𝑘+1}=∑^∞_𝑗=0𝛼_𝑗𝑑_{𝑛,𝑘+𝑗}, (𝑘,𝑛=0,1,…);(ii)𝑑_𝑛+1,0=∑^∞_𝑗=0𝑧_𝑗𝑑_𝑛,𝑗, (𝑛=0,1,…).

The coefficients 𝛼₀,𝛼₁,𝛼₂,… and 𝑧₀,𝑧₁,𝑧₂,… are called the 𝐴-sequence and the 𝑍-sequence of the Riordan array 𝑀=(𝑔(𝑥),𝑓(𝑥)), respectively. Letting 𝐴(𝑥) be the generating function of the 𝐴-sequence and 𝑍(𝑥) the generating function of the 𝑍-sequence, we have 𝐴(𝑥)=𝑥𝑓(𝑥)(𝑥)=1,𝑍𝑓(𝑥)𝑑1−^0,0𝑔𝑓(𝑥). (46)

The first column of 𝒮_𝑀 is then generated by 𝑍(𝑥), while the 𝑘th column is generated by 𝑥^𝑘−1𝐴(𝑥) (taking the first column to be indexed by 0).

Proposition 8. The inverse of the Chebyshev-Boubaker array 𝐏=1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²,11+𝑎𝑥+𝑏𝑥²⁻¹ (47) has production matrix which begins with 𝑎−𝑢10000⋯𝑏−𝑣𝑎1000⋯0𝑏𝑎100⋯00𝑏𝑎10⋯000𝑏𝑎1⋯0000𝑏𝑎⋯⋮⋮⋮⋮⋮⋮⋱. (48)

Proof. We let 𝐌=(𝑔(𝑥),𝑓(𝑥))=𝐏⁻¹. Then, 𝐴(𝑥)=𝑥𝑓(𝑥)=𝑥𝑥/1+𝑎𝑥+𝑏𝑥²=1+𝑎𝑥+𝑏𝑥². (49) We also have 𝑍(𝑥)=1+𝑎𝑥+𝑏𝑥²𝑥1−1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²=(𝑎−𝑢)+(𝑏−𝑣)𝑥, (50) which proves the assertion.

3. Chebyshev-Boubaker Arrays

We can characterize the Chebyshev-Boubaker polynomials 𝑃_𝑛(𝑥) with coefficient array 𝐏=((1+𝑢𝑥+𝑣𝑥²)/(1+𝑎𝑥+𝑏𝑥²),𝜆𝑥/(1+𝑎𝑥+𝑏𝑥²)) as follows.

Proposition 9 (see [8]). The Chebyshev-Boubaker coefficient array ((1+𝑢𝑥+𝑣𝑥²)/(1+𝑎𝑥+𝑏𝑥²),𝜆𝑥/(1+𝑎𝑥+𝑏𝑥²)) is the coefficient array of the generalized Chebyshev polynomials 𝑃_𝑛(𝑥) of the second kind given by 𝑃_𝑛(𝑥)=√𝑏^𝑛𝑈_𝑛𝜆𝑥−𝑎2√𝑏+𝑢√𝑏^𝑛−1𝑈_𝑛−1𝜆𝑥−𝑎2√𝑏+𝑣√𝑏^𝑛−2𝑈_𝑛−2𝜆𝑥−𝑎2√𝑏. (51)

The following variant is also true. The array ((1+𝑢𝑥+𝑣𝑥²)/(1+𝜆𝑎𝑥+𝑏𝑥²),𝜆𝑥/(1+𝜆𝑎𝑥+𝑏𝑥²)) is the coefficient array of the generalized Chebyshev polynomials of the second kind 𝑃_𝑛^(𝜆)(𝑥) given by 𝑃_𝑛^(𝜆)(𝑥)=√𝑏^𝑛𝑈_𝑛𝜆(𝑥−𝑎)2√𝑏+𝑢√𝑏^𝑛−1𝑈_𝑛−1×𝜆(𝑥−𝑎)2√𝑏+𝑣√𝑏^𝑛−2𝑈_𝑛−2𝜆(𝑥−𝑎)2√𝑏. (52)

The following factorization of the Chebyshev-Boubaker array will be useful when it comes to giving a closed-form expression for the connection coefficients.

Proposition 10. (The canonical factorization of Chebyshev-Boubaker arrays). Given the Chebyshev-Boubaker array 𝐏=1+𝑢𝑥+𝑣𝑥²1+𝑎𝜆𝑥+𝑏𝑥²,𝜆𝑥1+𝑎𝜆𝑥+𝑏𝑥², (53) one has 𝐏=1+𝑢𝑥+𝑣𝑥²⋅1,𝑥1+𝑏𝑥²,𝜆𝑥1+𝑏𝑥²⋅1,𝑥1+𝑎𝑥.1+𝑎𝑥 (54)

Proof. The proof is a straight-forward application of the multiplication rule for Riordan arrays. For instance, we have 11+𝑏𝑥²,𝜆𝑥1+𝑏𝑥²⋅1,𝑥1+𝑎𝑥=11+𝑎𝑥1+𝑏𝑥²11+𝑎𝜆𝑥/1+𝑏𝑥²,𝜆𝑥/1+𝑏𝑥²1+𝑎𝜆𝑥/1+𝑏𝑥²=11+𝑎𝜆𝑥+𝑏𝑥²,𝜆𝑥1+𝑎𝜆𝑥+𝑏𝑥². (55)

For the monic case 𝜆=1, we have 1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²,𝑥1+𝑎𝑥+𝑏𝑥²=1+𝑢𝑥+𝑣𝑥²⋅1,𝑥1+𝑏𝑥²,𝑥1+𝑏𝑥²⋅1,𝑥1+𝑎𝑥.1+𝑎𝑥 (56)

Corollary 11. If 𝐏=1+𝑢𝑥+𝑣𝑥²1+𝑎𝜆𝑥+𝑏𝑥²,𝜆𝑥1+𝑎𝜆𝑥+𝑏𝑥², (57) then 𝐏⁻¹=1,𝑥1−𝑎𝑥⋅𝑐1−𝑎𝑥𝑏𝑥²,1𝜆𝑥𝑐𝑏𝑥²⋅11+𝑎𝑥+𝑏𝑥²,𝑥. (58)

In the monic case 𝜆=1, we then have 1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²,𝑥1+𝑎𝑥+𝑏𝑥²⁻¹=1,𝑥1−𝑎𝑥⋅𝑐1−𝑎𝑥𝑏𝑥²,𝑥𝑐𝑏𝑥²⋅11+𝑎𝑥+𝑏𝑥²,𝑥. (59) We can characterize the Jacobi coefficients of 𝑃_𝑛(𝑥) in terms of those of 𝑄_𝑛(𝑥) as follows. We let 𝒮_𝑃=𝐏𝐏⁻¹ be the production matrix of 𝐏⁻¹, and we let 𝒮_𝑄=𝐐𝐐⁻¹ be the production matrix of 𝐐⁻¹. We then have the following proposition.

Proposition 12. One has, 𝒮_𝑃=𝐀𝒮_𝑄𝐀⁻¹. (60)

Proof. We have 𝒮_𝑃=𝐏𝐏⁻¹=𝐏𝕊𝐏⁻¹, where 𝕊 is the shift matrix with ones on the super-diagonal and zeros elsewhere, with a similar expression for 𝒮_𝑄. We then have 𝒮_𝑃=𝐏𝕊𝐏⁻¹=𝐀𝐐𝕊(𝐀𝐐)⁻¹=𝐀𝐐𝕊𝐐⁻¹𝐀⁻¹=𝐀𝒮_𝑄𝐀⁻¹. (61)

4. The Chebyshev-Boubaker Connection Coefficients

We let 𝐏=1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²,𝑥1+𝑎𝑥+𝑏𝑥²,𝐐=1+𝑠𝑥+𝑡𝑥²1+𝑐𝑥+𝑑𝑥²,𝑥1+𝑐𝑥+𝑑𝑥². (62) For the matrix of connection coefficients 𝐀, we recall that we have 𝐀=𝐏𝐐⁻¹. Now, 𝐏 and 𝐐⁻¹ have the following factorizations: 𝐏=1+𝑢𝑥+𝑣𝑥²⋅1,𝑥1+𝑏𝑥²,𝑥1+𝑏𝑥²⋅1,𝑥1+𝑎𝑥,𝐐1+𝑎𝑥₋₁=1,𝑥1−𝑐𝑥⋅𝑐1−𝑐𝑥𝑑𝑥²,𝑥𝑐𝑑𝑥²⋅11+𝑠𝑥+𝑡𝑥²,𝑥. (63) We now proceed to list the elements of the factors involved. The matrix (1/(1+𝑎𝑥),𝑥/(1+𝑎𝑥)) is the generalized inverse binomial matrix with general element 1,𝑥1+𝑎𝑥=𝑛𝑘()1+𝑎𝑥−𝑎^𝑛−𝑘_{0≤𝑛,𝑘≤∞}. (64) The matrix (1/(1+𝑏𝑥²),𝑥/(1+𝑏𝑥²)) gives the coefficients of the Chebyshev polynomials 𝑈_𝑛(√𝑏𝑥). We have 11+𝑏𝑥²,𝑥1+𝑏𝑥²=𝑛+𝑘2𝑘(−𝑏)^{(𝑛−𝑘)/2}1+(−1)^𝑛−𝑘2_{0≤𝑛,𝑘≤∞}. (65) The matrix (1+𝑢𝑥+𝑣𝑥²,𝑥) is the banded matrix with the elements (1,𝑢,𝑣) occupying the three diagonals from the main diagonal down (with zeros elsewhere). The elements of the matrix product (1+𝑢𝑥+𝑣𝑥²,𝑥)⋅(1/(1+𝑏𝑥²),𝑥/(1+𝑏𝑥²)) are then given by 𝑛+𝑘2𝑘(−𝑏)^{(𝑛−𝑘)/2}1+(−1)^𝑛−𝑘2+𝑢𝑛+𝑘−12𝑘(−𝑏)^{(𝑛−𝑘−1)/2}1−(−1)^𝑛−𝑘2+𝑣𝑛+𝑘−22𝑛−𝑘−22(−𝑏)^{(𝑛−𝑘−2)/2}1+(−1)^𝑛−𝑘2. (66) Let us denote these elements by 𝑇_𝑛,𝑘. Then, the matrix 𝐏 has elements given by 𝑝_𝑛,𝑘=^𝑛_𝑗=0𝑇_𝑛,𝑗𝑘()𝑗−𝑘−𝑎^𝑗−𝑘. (67) A similar analysis of the factors in the expression for 𝐐⁻¹ shows that the general element of 𝐐⁻¹ is given by 𝑞^∗_𝑛,𝑘=^𝑛𝑛_𝑖=0𝑗=0𝑛𝑗𝑐^𝑛−𝑗𝑖+1𝑗+1𝑗+1𝑗−𝑖2×𝑑^{(𝑗−𝑖)/2}1+(−1)^𝑗−𝑖2^{⌊(𝑖−𝑘)/2⌋}_𝑟=0𝑟×()𝑖−𝑘−𝑟−𝑠^{𝑖−𝑘−𝑟}(−𝑡)^𝑟, (68) since we have 1,𝑥1−𝑐𝑥=𝑛𝑘𝑐1−𝑐𝑥^𝑛−𝑘_{0≤𝑛,𝑘≤∞},𝑐𝑑𝑥²,𝑥𝑐𝑑𝑥²=𝑘+1𝑛+1𝑛+1𝑛−𝑘2𝑑^{(𝑛−𝑘)/2}1+(−1)^𝑛−𝑘2_{0≤𝑛,𝑘≤∞},11+𝑠𝑥+𝑡𝑥²=,𝑥_{⌊(𝑛−𝑘)/2⌋𝑗=0}𝑗()𝑛−𝑘−𝑗−𝑠^{𝑛−𝑘−𝑗}(−𝑡)^𝑗_{0≤𝑛,𝑘≤∞}. (69) We gather our results in the following proposition.

Proposition 13. Let 𝐏=1+𝑢𝑥+𝑣𝑥²1+𝑎𝑥+𝑏𝑥²,𝑥1+𝑎𝑥+𝑏𝑥²,𝐐=1+𝑠𝑥+𝑡𝑥²1+𝑐𝑥+𝑑𝑥²,𝑥1+𝑐𝑥+𝑑𝑥² (70) be the coefficient arrays of two families of Chebyshev-Boubaker polynomials. Then, the connection coefficients 𝑎_𝑛,𝑘 for 𝑃_𝑛 in terms of 𝑄_𝑛 are given by 𝑎_𝑛,𝑘=^𝑛_𝑗=0𝑝_𝑛,𝑗𝑞^∗_𝑗,𝑘, (71) where 𝑝_𝑛,𝑘=^𝑛_𝑗=0𝑇_𝑛,𝑗𝑘()𝑗−𝑘−𝑎^𝑗−𝑘, (72) with 𝑇_𝑛,𝑘 given by the expression 𝑛+𝑘2𝑘(−𝑏)^{(𝑛−𝑘)/2}1+(−1)^𝑛−𝑘2+𝑢𝑛+𝑘−12𝑘(−𝑏)^{(𝑛−𝑘−1)/2}1−(−1)^𝑛−𝑘2+𝑣𝑛+𝑘−22𝑛−𝑘−22(−𝑏)^{(𝑛−𝑘−2)/2}1+(−1)^𝑛−𝑘2,𝑞^∗_𝑛,𝑘=^𝑛𝑛_𝑖=0𝑗=0𝑛𝑗𝑐^𝑛−𝑗𝑖+1𝑗+1𝑗+1𝑗−𝑖2×𝑑^{(𝑗−𝑖)/2}1+(−1)^𝑗−𝑖2^{⌊(𝑖−𝑘)/2⌋}_𝑟=0𝑟×()𝑖−𝑘−𝑟−𝑠^{𝑖−𝑘−𝑟}(−𝑡)^𝑟. (73)

The connection coefficient elements in the Chebyshev-Boubaker class of polynomials are given by Riordan arrays. The elements of these arrays can be expressed in terms of recurrences defined by the associated production matrices. Thus, we are naturally interested in the structure of 𝒮_𝐀, the production matrix of 𝐀.

Proposition 14. Let 𝐀 be the matrix of connection coefficients that express the polynomials 𝑃_𝑛(𝑥) with coefficient array 𝐏 in terms of the polynomials 𝑄_𝑛(𝑥) with coefficient array 𝐐. Then, one has 𝒮_𝐀=𝐐𝒮_𝐏𝐐⁻¹. (74)

Proof. We have 𝒮_𝐀=𝐀⁻¹=𝕊𝐀𝐏𝐐⁻¹⁻¹𝕊𝐏𝐐⁻¹=𝐐𝐏⁻¹𝕊𝐏𝐐⁻¹=𝐐𝒮_𝐏𝐐⁻¹. (75)

Proposition 7 can now be interpreted in this context as follows, giving a set of recurrence relations for the connection coefficients.

Proposition 15. If the 𝐴 and 𝑍 sequences of 𝒮_𝐀 are denoted, respectively, by 𝛼_𝑛 and 𝑧_𝑛, then the connection coefficients 𝑎_𝑛,𝑘 satisfy the following recurrences 𝑎_{𝑛+1,𝑘+1}=^𝑛−𝑘_𝑗=0𝛼_𝑗𝑎_{𝑛,𝑘+𝑗},𝑎_𝑛+1,0=^𝑛_𝑗=0𝑧_𝑗𝑎_𝑛,𝑗. (76)

5. Examples

In this section, we give examples of the foregoing, using simple examples that are close to the Chebyshev polynomials of the second kind. This enables us to illustrate the theory without an unnecessarily high overhead.

Example 16. We take the example of the families of orthogonal polynomials that have the aerated Catalan and the ordinary Catalan numbers as moments, respectively. Thus, let 𝐏=𝐒, the coefficient array of the monic Chebyshev polynomials of the second kind. We have 1𝐒=1+𝑥²,𝑥1+𝑥². (77) We will have 1𝐐=,𝑥1+𝑥(1+𝑥)². (78) This is the array of the Morgan-Voyce polynomials 𝐵_𝑛(𝑥)=∑^𝑛_𝑘=0^𝑛+𝑘_2𝑘(−1)^𝑛−𝑘𝑥^𝑘. Then, 𝐀=𝐒𝐐⁻¹=11+𝑥²,𝑥1+𝑥²⋅1,𝑥1+𝑥(1+𝑥)²⁻¹=11+𝑥²,𝑥1+𝑥²⋅𝑐(𝑥),𝑥𝑐(𝑥)². (79) This implies that 1𝐀=1+𝑥²𝑐𝑥1+𝑥²,𝑥1+𝑥²𝑐𝑥1+𝑥²². (80) The matrix 𝐀 begins with 100000⋯110000⋯131000⋯375100⋯91917710⋯2557553191⋯⋮⋮⋮⋮⋮⋮⋱. (81) The first column terms 𝑎_𝑛=𝑎_𝑛,0 have generating function 11+𝑥²𝑐𝑥1+𝑥²=1+𝑥²−√1−4𝑥+2𝑥²−4𝑥³+𝑥⁴2𝑥1+𝑥². (82) This form of g.f. suggests that this sequence may have an interesting Hankel transform. In fact, the sequence 𝑎_𝑛,0 is A101499, a sequence which gives the number of peakless Motzkin paths of length 𝑛 in which the (1,0)-steps at levels greater than level 0 come in two colors (Emeric Deutsch). The Hankel transform of 𝑎_𝑛,0 starts with 1,0,−4,−16,−64,0,4096,65536,1048576,0,−1073741824,…. (83) This is A162547, which is a Somos-4 variant [27, 39] in the sense that we have 𝑎_𝑛=4𝑎_𝑛−1𝑎_𝑛−3−4𝑎²_𝑛−2𝑎_𝑛−4,(𝑛≠4𝑘+1). (84) We have 𝑎_𝑛,𝑘=^𝑛−𝑘_𝑗=0^⌊𝑗/2⌋_𝑖=0(−1)^𝑖𝑖𝐶𝑗−𝑖_𝑗−2𝑖2𝑛−𝑗−1𝑛−𝑘−𝑗^{𝑛−𝑘−𝑗}. (85) We note that due to the combinatorial interpretation of 𝑎_𝑛,0 and the positivity of the Catalan numbers, we can conclude that all the connection coefficients 𝑎_𝑛,𝑘 are positive in this case.

We now find expressions for the elements 𝛼_𝑛 and 𝑧_𝑛 of the production matrix 𝒮_𝐀 of 𝐀.

We use the values 𝑔(𝑥)=11+𝑥²𝑐𝑥1+𝑥²=1+𝑥²−√1−4𝑥+2𝑥²−4𝑥³+𝑥⁴2𝑥1+𝑥²,𝑓(𝑥)=𝑥1+𝑥²𝑐𝑥1+𝑥²²=(1−𝑥)²−√1−4𝑥+2𝑥²−4𝑥³+𝑥⁴,2𝑥 (86) along with (46) to find that 𝑍(𝑥)=√1−4𝑥+2𝑥²−4𝑥³+𝑥⁴+𝑥²−1,𝐴(𝑥)=()2𝑥1+𝑥²+√1−4𝑥+2𝑥²−4𝑥³+𝑥⁴2. (87) This implies that 𝑧_𝑛=1𝑛−^{⌊(𝑛−1)/2⌋}_𝑘=01𝑘×𝑘()𝑘+1𝑛−𝑘−2𝑛−𝑘−1−2^{𝑛−1−2𝑘},𝛼_𝑛=2𝑛−^{⌊(𝑛−2)/2⌋}_𝑘=01𝑘×𝑘()𝑘+1𝑛−𝑘−3𝑛−𝑘−2−2^{𝑛−2−2𝑘}. (88) With these values for 𝑧_𝑛 and 𝛼_𝑛, we thus obtain the following recurrences for the connection coefficients: 𝑎_{𝑛+1,𝑘+1}=^𝑛−𝑘_𝑗=0𝛼_𝑗𝑎_{𝑛,𝑘+𝑗},(),𝑎𝑘,𝑛=0,1,…_𝑛+1,0=^𝑛_𝑗=0𝑧_𝑗𝑎_𝑛,𝑗,(𝑛=0,1,…). (89) We note that, in this case, the sequence elements 𝛼_𝑛 and 𝑧_𝑛 are essentially diagonal sums of generalized Narayana triangles [40].

Example 17. In this example, we let 𝑃_𝑛 be the family of orthogonal polynomials with the Catalan numbers 𝐶_𝑛 as moments, and we let 𝑄_𝑛 be the family of orthogonal polynomials with the central binomial coefficients 𝑛2𝑛  A000984 as moments. We find that 1𝐏=1+𝑥²,𝑥1+𝑥²,𝐐=1−𝑥1+𝑥²,𝑥1+𝑥². (90) We obtain 𝐀=𝐏𝐐⁻¹=1,1−𝑥,𝑥 (91) which is the partial sum matrix (the lower-triangular matrix all of whose nonzero elements are 1). This corresponds to 𝑃_𝑛(𝑥)=^𝑛_𝑘=0𝑄_𝑘(𝑥). (92) In fact, we have 𝑄_𝑛(𝑥)=()𝑛+𝑘𝑛−𝑘−1^𝑛−𝑘−()𝑛+𝑘−1𝑛−𝑘−1−1^{𝑛−𝑘−1},𝑃_𝑛(𝑥)=()𝑛+𝑘𝑛−𝑘−1^𝑛−𝑘=()𝑛+𝑘2𝑘−1^𝑛−𝑘=^𝑛_𝑘=0𝑄_𝑘(𝑥). (93)

Example 18. Let 𝑃_𝑛(𝑥)=𝑆_𝑛(𝑥), the monic Chebyshev polynomials of the second kind with the aerated Catalan numbers as moments, and let 𝑄_𝑛(𝑥) be the family of orthogonal polynomials with 𝑛2𝑛 as moments. We get 1𝐀=1+𝑥²,𝑥1+𝑥²⋅1−𝑥,𝑥1+𝑥(1+𝑥)²⁻¹=11+𝑥²,𝑥1+𝑥²⋅1√(𝑥)1−4𝑥,𝑥𝑐²=11+𝑥²11−4𝑥/1+𝑥²,𝑥1+𝑥²𝑐𝑥1+𝑥²²=1√1−4𝑥+2𝑥²−4𝑥³+𝑥⁴,(1−𝑥)²−√1−4𝑥+2𝑥²−4𝑥³+𝑥⁴.2𝑥 (94) The sequence 𝑎_𝑛=𝑎_𝑛,0 is then A101500, with 𝑎_𝑛=^⌊𝑛/2⌋_𝑘=0𝑘2()()𝑛−𝑘𝑛−2𝑘𝑛−2𝑘−1^𝑘. (95) In this example, we have 𝐐⁻¹=1−𝑥,𝑥1+𝑥(1+𝑥)²⁻¹=2𝑛𝑛−𝑘, (96) which is A094527. Hence, we have 𝑎_𝑛,𝑘=^𝑛_𝑗=0𝑛+𝑗2𝑗(−1)^{(𝑛−𝑗)/2}1+(−1)^𝑛−𝑗22𝑗𝑗−𝑘. (97) Note that this implies that 𝑎_𝑛=^𝑛_𝑗=0𝑛+𝑗2𝑗(−1)^{(𝑛−𝑗)/2}1+(−1)^𝑛−𝑗2𝑗2𝑗. (98)

Example 19. In this final example, we let 𝐏 be the coefficient array of the Boubaker polynomials. Thus, 𝐏=1+3𝑥²1+𝑥²,𝑥1+𝑥². (99) We recall that the Boubaker polynomials can be expressed as 𝑃_𝑛(𝑥)=^⌊𝑛/2⌋_𝑘=0𝑘𝑛−𝑘𝑛−4𝑘()𝑛−𝑘−1^𝑘𝑥^𝑛−2𝑘. (100) We let 𝐐 be the coefficient array of the Morgan-Voyce polynomials 𝑄_𝑛(𝑥)=^𝑛_𝑘=0()𝑛+𝑘2𝑘−1^𝑛−𝑘𝑥^𝑘. (101) Then, 1𝐐=,𝑥1+𝑥(1+𝑥)². (102) The connection coefficients of the Boubaker polynomials in terms of the Morgan-Voyce polynomials are the elements of the array 𝐀=1+3𝑥²1+𝑥²,𝑥1+𝑥²⋅1,𝑥1+𝑥(1+𝑥)²⁻¹. (103) Thus, 𝐀=1+3𝑥²1+𝑥²−√1−4𝑥+2𝑥²−4𝑥³+𝑥⁴2𝑥1+𝑥²,(1−𝑥)²−√1−4𝑥+2𝑥²−4𝑥³+𝑥⁴.2𝑥 (104) Letting 𝑇_𝑛,𝑘=^𝑛−𝑘_𝑗=0^⌊𝑗/2⌋_𝑖=0(−1)^𝑖𝑖𝐶𝑗−𝑖_𝑗−2𝑖2𝑛−𝑗−1𝑛−𝑘−𝑗^{𝑛−𝑘−𝑗}, (105) (see (85)), we obtain 𝑎_𝑛,𝑘=𝑇_𝑛,𝑘+3𝑇_{𝑛−2,𝑘}. (106) Again, we see that all the coefficients 𝑎_𝑛,𝑘 are positive.

6. Conclusions

We have shown that in the case of orthogonal polynomials defined by Riordan arrays, we can give both closed form expressions and recurrence relations for the connection coefficients, using the machinery from the theory of Riordan arrays.