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The Scientific World Journal
Volume 2013 (2013), Article ID 657806, 10 pages
http://dx.doi.org/10.1155/2013/657806
Research Article

On the Connection Coefficients of the Chebyshev-Boubaker Polynomials

School of Science, Waterford Institute of Technology, Waterford, Ireland

Received 22 April 2013; Accepted 23 June 2013

Academic Editors: A. Bekir, S. Hristova, and T.-z. Huang

Copyright © 2013 Paul Barry. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [17].

If and are two families of orthogonal polynomials, the connection coefficients of in terms of , defined by are often of interest. In terms of matrices, we have where is the coefficient array of the polynomials , with elements , and is the coefficient array of the polynomials , with elements . Thus, we have The matrices and are lower-triangular matrices. Clearly, the matrix of connection coefficients is then given by We immediately note that so that the elements of are the connection coefficients of in terms of .

The matrix is lower triangular since both and are.

We will define a family of orthogonal polynomials to be of Chebyshev-Boubaker type, if the coefficient array is a Riordan array of the type This means that In the case that , the polynomials are monic. An array of the type will be called a Chebyshev-Boubaker array (with parameters ). This is a generalization of the notion of a Chebyshev-Boubaker array considered in [8]. The Boubaker polynomials [8] correspond to the parameters .

Example 1. The Chebyshev polynomials of the second kind are defined by and have coefficient array given by A053117

Example 2. We let be the monic Chebyshev polynomials of the second kind. The coefficient array of these polynomials is given by A049310 This is the case , , and .
The moments of this family of orthogonal polynomials are the aerated Catalan numbers that begin with Their moment representation is given by

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 with suitable initial conditions. For instance, we have The recurrence coefficients and for the family of orthogonal polynomials are the entries in the production matrix (or Stieltjes matrix [9]) of the matrix . This production matrix is equal to , where the notation denotes the matrix with its first row removed. For instance, the production matrix of is given by corresponding to the three-term recurrence or Similarly, the production matrix of is given by corresponding to the three-term recurrence More generally, the production matrix of is given by corresponding to the three-term recurrence where

Example 3. The Boubaker polynomials [8, 1015] have coefficient array They can be expressed as We have where is the generating function of the Catalan numbers Hence, the moments of this family of orthogonal polynomials have g.f. . The sequence begins with and has the moment representation We have The sequence is A126984. We note that the Hankel transforms of and are both equal to . The production matrix of is given by

In this paper, we will give a closed-form expression for the connection coefficients of the Chebyshev-Boubaker polynomials with coefficient array in terms of the Chebyshev-Boubaker polynomials with coefficient array We note that the polynomials satisfy the recurrence with with a similar expression for .

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 , that is, an element of , the power series is called the ordinary generating function or g.f. of the sequence. is thus the coefficient of in this series. We denote this by . For instance, is the th Fibonacci number A000045, while is the th Catalan number A000108. We use the notation for the sequence ,  A000007. Thus, . Here, we have used the Iverson bracket notation [16], defined by if the proposition is true and if is false.

For a power series with , we define the reversion or compositional inverse of to be the power series such that .

For a lower triangular matrix , the row sums give the sequence with general term . For a sequence , the sequence of determinants is called the Hankel transform of [1727].

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 and where [29]. We often require in addition that . The associated matrix is the matrix whose th column is generated by (the first column being indexed by 0). The matrix corresponding to the pair is denoted by or . The group law is then given by The identity for this law is , and the inverse of is where is the compositional inverse of .

If is the matrix and is an integer sequence (expressed as an infinite column vector) with ordinary generating function , then the sequence has ordinary generating function . The (infinite) matrix can thus be considered to act on the ring of integer sequences by multiplication, where a sequence is regarded as a (infinite) column vector. We can extend this action to the ring of power series by

Example 4. The so-called binomial matrix is the element of the Riordan group. It has general element and hence as an array coincides with Pascal's triangle. More generally, is the element of the Riordan group, with general term . It is easy to show that the inverse of is given by .

The row sums of the matrix have generating function

Example 5. The inverse of the Riordan array is the Riordan array . This follows since the solution of the equation is given by We then have

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 (“Pascal's triangle”) is A007318.

For an invertible matrix , its production matrix (also called its Stieltjes matrix) [32, 33] is the matrix where is the matrix with its first row removed. A Riordan array is the inverse of the coefficient array of a family of orthogonal polynomials [3436] if and only if 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 is given by

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 be an infinite triangular matrix. Then, is a Riordan array if and only if there exist two sequences and with , such that (i), ;(ii), .

The coefficients and are called the -sequence and the -sequence of the Riordan array , respectively. Letting be the generating function of the -sequence and the generating function of the -sequence, we have

The first column of is then generated by , while the th column is generated by (taking the first column to be indexed by ).

Proposition 8. The inverse of the Chebyshev-Boubaker array has production matrix which begins with

Proof. We let . Then, We also have which proves the assertion.

3. Chebyshev-Boubaker Arrays

We can characterize the Chebyshev-Boubaker polynomials with coefficient array as follows.

Proposition 9 (see [8]). The Chebyshev-Boubaker coefficient array , is the coefficient array of the generalized Chebyshev polynomials of the second kind given by

The following variant is also true. The array is the coefficient array of the generalized Chebyshev polynomials of the second kind given by

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 one has

Proof. The proof is a straight-forward application of the multiplication rule for Riordan arrays. For instance, we have

For the monic case , we have

Corollary 11. If then

In the monic case , we then have We can characterize the Jacobi coefficients of in terms of those of as follows. We let be the production matrix of , and we let be the production matrix of . We then have the following proposition.

Proposition 12. One has,

Proof. We have , where is the shift matrix with ones on the super-diagonal and zeros elsewhere, with a similar expression for . We then have

4. The Chebyshev-Boubaker Connection Coefficients

We let For the matrix of connection coefficients , we recall that we have . Now, and have the following factorizations: We now proceed to list the elements of the factors involved. The matrix is the generalized inverse binomial matrix with general element The matrix gives the coefficients of the Chebyshev polynomials . We have The matrix is the banded matrix with the elements occupying the three diagonals from the main diagonal down (with zeros elsewhere). The elements of the matrix product are then given by Let us denote these elements by . Then, the matrix has elements given by A similar analysis of the factors in the expression for shows that the general element of is given by since we have We gather our results in the following proposition.

Proposition 13. Let be the coefficient arrays of two families of Chebyshev-Boubaker polynomials. Then, the connection coefficients for in terms of are given by where with given by the expression

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 , the production matrix of .

Proposition 14. Let be the matrix of connection coefficients that express the polynomials with coefficient array in terms of the polynomials with coefficient array . Then, one has

Proof. We have

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 and sequences of are denoted, respectively, by and , then the connection coefficients satisfy the following recurrences

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 , the coefficient array of the monic Chebyshev polynomials of the second kind. We have We will have This is the array of the Morgan-Voyce polynomials . Then, This implies that The matrix begins with The first column terms have generating function This form of g.f. suggests that this sequence may have an interesting Hankel transform. In fact, the sequence is A101499, a sequence which gives the number of peakless Motzkin paths of length in which the -steps at levels greater than level come in two colors (Emeric Deutsch). The Hankel transform of starts with This is A162547, which is a Somos- variant [27, 39] in the sense that we have We have We note that due to the combinatorial interpretation of and the positivity of the Catalan numbers, we can conclude that all the connection coefficients are positive in this case.

We now find expressions for the elements and of the production matrix of .

We use the values along with (46) to find that This implies that With these values for and , we thus obtain the following recurrences for the connection coefficients: We note that, in this case, the sequence elements and are essentially diagonal sums of generalized Narayana triangles [40].

Example 17. In this example, we let be the family of orthogonal polynomials with the Catalan numbers as moments, and we let be the family of orthogonal polynomials with the central binomial coefficients   A000984 as moments. We find that We obtain which is the partial sum matrix (the lower-triangular matrix all of whose nonzero elements are ). This corresponds to In fact, we have

Example 18. Let , the monic Chebyshev polynomials of the second kind with the aerated Catalan numbers as moments, and let be the family of orthogonal polynomials with as moments. We get The sequence is then A101500, with In this example, we have which is A094527. Hence, we have Note that this implies that

Example 19. In this final example, we let be the coefficient array of the Boubaker polynomials. Thus, We recall that the Boubaker polynomials can be expressed as We let be the coefficient array of the Morgan-Voyce polynomials Then, The connection coefficients of the Boubaker polynomials in terms of the Morgan-Voyce polynomials are the elements of the array Thus, Letting (see (85)), we obtain Again, we see that all the coefficients 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.

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