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**Meixner-type results for Riordan arrays and associated integer sequences.**
*(English)*
Zbl 1201.42017

Summary: We determine which (ordinary) Riordan arrays are the coefficient arrays of a family of orthogonal polynomials. In so doing, we are led to introduce a family of polynomials, which includes the Boubaker polynomials, and a scaled version of the Chebyshev polynomials, using the techniques of Riordan arrays. We classify these polynomials in terms of the Chebyshev polynomials of the first and second kinds. We also examine the Hankel transforms of sequences associated with the inverse of the polynomial coefficient arrays, including the associated moment sequences.

### MSC:

42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |

11B83 | Special sequences and polynomials |

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |

11C20 | Matrices, determinants in number theory |

15B05 | Toeplitz, Cauchy, and related matrices |

15B36 | Matrices of integers |

### Keywords:

Chebyshev polynomials; Boubaker polynomials; integer sequence; orthogonal polynomials; Riordan array; production matrix; Hankel determinant; Hankel transform### Software:

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\textit{P. Barry} and \textit{A. Hennessy}, J. Integer Seq. 13, No. 9, Article 10.9.4, 34 p. (2010; Zbl 1201.42017)

### Online Encyclopedia of Integer Sequences:

Catalan’s triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j).Catalan triangle A009766 transposed.

Triangle of coefficients of Chebyshev’s S(n,x) := U(n,x/2) polynomials (exponents in increasing order).

Triangle read by rows of coefficients of Chebyshev’s U(n,x) polynomials (exponents in increasing order).

Triangle of coefficients of Chebyshev’s T(n,x) polynomials (powers of x in increasing order).

Catalan triangle (with 0’s) read by rows.

G.f. A(x) satisfies: A(x*G(x)) = G(x), where G(x) is the g.f. for A098614(n) = Fibonacci(n+1)*Catalan(n).

Triangle read by rows: right half of Pascal’s triangle (A007318) interspersed with 0’s.

Triangle read by rows: lower triangular matrix obtained by inverting the lower triangular matrix in A108044.

a(n) = (-1)^n*n*(n-2).