Embedding Structures Associated with Riordan Arrays and Moment Matrices
Every ordinary Riordan array contains two naturally embedded Riordan arrays. We explore this phenomenon, and we compare it to the situation for certain moment matrices of families of orthogonal polynomials.
Riordan arrays  have been used mainly to prove combinatorial identities [2, 3]. Recently, their links to orthogonal polynomials have been investigated [4, 5], while there is a growing literature surrounding their structural properties [6–10]. In this paper we investigate an embedding structure, common to all ordinary Riordan arrays. We also look at this embedding structure in the context of moment matrices of families of orthogonal polynomials. In addition to some knowledge of Riordan arrays, we assume that the reader has a basic familiarity with the theory of orthogonal polynomials on the real line [11–13], production matrices [14, 15], and continued fractions . We will meet a number of integer sequences and integer triangles in this paper. The On-Line Encyclopedia may be consulted for many of them [17, 18]. In this paper we will understand by an ordinary Riordan array an integer number triangle whose -th element is defined by a pair of power series and over the integers with , , in the following manner: The group law for Riordan arrays is given by The identity for this law is and the inverse of is , where is the compositional inverse of . For a power series with , we define the reversion or compositional inverse of to be the power series such that . We sometimes write this as .
If a matrix is the inverse of the coefficient array of a family of orthogonal polynomials, then we will call it a moment matrix, and we will single out the first column as the moment sequence.
2. The Canonical Embedding
Let be an ordinary Riordan array , with general term
Then we observe that there are two naturally associated Riordan arrays “embedded” in the array as follows.
Beginning at the first column of , we take every second column, “raising” the columns appropriately to obtain a lower-triangular matrix . The matrix is then the Riordan array with general term given by Similarly, starting at the second column of , taking every second column and “raising” all columns appropriately to obtain a lower-triangular matrix, we obtain a matrix . This matrix is then a Riordan array, given by We have The general term of is given by
Example 1. We take the example of the binomial matrix We then have The following decomposition makes this clear: The matrices and are the coefficient arrays of the Morgan-Voyce polynomials and , respectively.
Example 2. We take the Riordan array
Then we find that
The matrix begins with
We note that the matrix is the moment array for the family of orthogonal polynomials with coefficient array given by
Denoting this family by , we have
with and . Similarly the matrix is the moment array for the family of orthogonal polynomials with coefficient array given by
Denoting this family by , we have
with and .
The inverse matrix is given by which is the Riordan array . In it we see the elements of and in staggered fashion.
3. A Counter Example
It is natural to ask the question: is a matrix that contains two embedded Riordan arrays as above itself a Riordan array? The following example shows that this is not a sufficient condition on an array to be Riordan.
Example 3. We will construct an invertible integer lower-triangular matrix which has two embedded Riordan arrays in the fashion above, but which is not itself a Riordan array. We start with the essentially two-period sequence We form the matrix The inverse of this matrix begins with where we note an alternating pattern of constant columns (with generating functions and , resp.). Removing the first row of this matrix provides us with a production matrix, which is not of the form that produces a Riordan array (after the first column, subsequent columns would be shifted versions of the second column [14, 15]). Thus the resulting matrix will not be a Riordan array. This resulting matrix begins with We now observe that for this matrix, we have We notice that the sequence has generating function given by the continued fraction and secondly that This construction is easily generalized.
4. Embedding a Riordan Array
Another natural question to ask is: if we are given a Riordan array , is it possible to embed it as above into a Riordan array ? For this, we let and seek to determine such that embeds into . For this, we need Thus we require that Since we are working in the context of integer valued Riordan arrays, we require that generates an integer sequence. We can state our result as follows.
Proposition 4. The Riordan array can be embedded in the Riordan array on condition that is the generating function of an integer sequence.
Example 5. The Riordan array can be embedded in the Riordan array For this example, the matrix begins with while begins with
5. A Cascading Decomposition
We note that we can “cascade" this embedding process, in the sense that, given a Riordan array , with embedded Riordan arrays and , we can consider decomposing and in their turns and then continue this process. For instance, we can decompose into the two matrices In their turn and can be decomposed and so on.
6. Embedding and Orthogonal Polynomials
The phenomenon of embedding as described above is not confined to Riordan arrays, as the continued fraction example above shows. To further amplify this point, we give another example involving a continued fraction. Although we take a particular case, the general case can be inferred easily from it. Thus we take the particular case of the continued fraction This continued fraction is equal to By the theory of orthogonal polynomials, the power series expressed by both continued fractions is the generating function for the moment sequence of the family of orthogonal polynomials whose moment matrix (the inverse of the coefficient array of the orthogonal polynomials) has production matrix given by This production matrix generates the moment matrix that begins with In order to produce an embedding for this matrix, we proceed as follows. We form the matrix We invert this matrix, remove the first row of the resulting matrix, and use this new matrix as a production matrix. The generated matrix then begins with The moment matrix is evidently embedded in the matrix . We can show that the corresponding matrix is the moment matrix for the family of orthogonal polynomials whose moments have generating function given by The matrix begins with and it has production matrix
The matrix can now be characterized as the coefficient array of a family of polynomials defined as follows. We let be the family of orthogonal polynomials with coefficient array , and we let be the family of orthogonal polynomials with coefficient . Then we have
In the general case of a moment sequence generated by the continued fraction the matrix will be generated by the production matrix while the matrix is generated by the production matrix
7. Embedding, Orthogonal Polynomials, and Riordan Arrays
In this section, we consider the case of two related families of orthogonal polynomials and defined by with , , and with , . We note that The coefficient array of the polynomials is then given by the Riordan array while that of is given by The moment matrix begins with while the moment matrix starts with Letting we find that the inverse of the coefficient array of the family is given by Thus the two Riordan arrays and , which are the moment arrays of the two families of orthogonal polynomials and , respectively, embed into the generalized moment array for the family of polynomials . Now the production matrix of begins with where the columns have generating functions , , respectively. We now observe that this production matrix is obtained by removing the first row of the inverse of the matrix We note finally that the sequence has generating function given by We have in fact that On the other hand, if we let but this time take and , then the matrix becomes The matrix then begins with where the moment sequence has generating function or equivalently In this case, the matrix begins with This matrix is then associated with the matrix
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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