## Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind

Citations: | 1 - 1 self |

### BibTeX

@MISC{Nkwanta_twocatalan-type,

author = {Asamoah Nkwanta and Earl R. Barnes},

title = {Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind},

year = {}

}

### OpenURL

### Abstract

Riordan matrix methods and properties of generating functions are used to prove that the entries of two Catalan-type Riordan arrays are linked to the Chebyshev polynomials of the first kind. The connections are that the rows of the arrays are used to expand the monomials (1/2)(2x) n and (1/2)(4x) n in terms of certain Chebyshev polynomials of degree n. In addition, we find new integral representations of the central binomial coefficients and Catalan numbers. 1

### Citations

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Citation Context ...nections makes use of Riordan matrix methods and properties of generating functions. As a by-product of the connections, we derive integral representations of the central binomial coefficients A00984 =-=[28]-=- and Catalan numbers A000108 [28] (see Equations (15), (16), (17), (18), (20), (21), (22), and (23)). The integrals do not seem to explicitly appear in Gradshteyn and Ryzhik [14]. They do not appear i... |

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Citation Context ... variety of applications of the Catalan numbers. In addition, the representations we give differ from the representations given by Penson and Sixdeniers [23], Sofo [29], Dana-Picard [9], [10], Aigner =-=[1]-=-, and Yuan [35], [36]. Thus, it appears that the integral representations are new. This paper is arranged as follows. The definition of a Riordan matrix is given in Section 2. We show that A and B are... |

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Citation Context ...btaining all other column entries, is a recurrence relation that defines the way entries of a Riordan matrix are computed. The formation rules of A and B are, respectively, [2, 2; 1, 2, 1] ([28]) and =-=[3, 1; 1, 2, 1]-=- [19, 20]. See the cited references for more information on the formation rules and dot diagrams of Riordan matrices. 7Arrays A and B are special cases of the generalized Riordan array of the form ( ... |

12 |
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Citation Context ...rdinary or exponential generating functions. However, the matrices presented in this paper are defined by ordinary generating functions. We note here that the proper Riordan arrays given by Sprugnoli =-=[30]-=- are what we call Riordan arrays or matrices. Two important results for multiplying Riordan matrices are now given. Theorem 2. ([27, 22]) If L = (ℓn,k) n,k∈N = (g (z),f (z)) is a Riordan matrix and h ... |

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Citation Context ...nd graph theory, combinatorial number theory, algebra, and special functions. Riordan array connections to the Chebyshev polynomials of the first kind have been given by Barry [4], Barry and Hennessy =-=[5]-=-, Luzon and Moron [17], and others. In this paper, we establish that the entries of two special Riordan matrices, we call Catalan-type Riordan arrays, are linked to the Chebyshev polynomials of the fi... |

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Citation Context ... column entries and A for obtaining all other column entries, is a recurrence relation that defines the way entries of a Riordan matrix are computed. The formation rules of A and B are, respectively, =-=[2, 2; 1, 2, 1]-=- ([28]) and [3, 1; 1, 2, 1] [19, 20]. See the cited references for more information on the formation rules and dot diagrams of Riordan matrices. 7Arrays A and B are special cases of the generalized R... |

5 |
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Citation Context ..., nor in Koshy’s book [15] that contains a wide variety of applications of the Catalan numbers. In addition, the representations we give differ from the representations given by Penson and Sixdeniers =-=[23]-=-, Sofo [29], Dana-Picard [9], [10], Aigner [1], and Yuan [35], [36]. Thus, it appears that the integral representations are new. This paper is arranged as follows. The definition of a Riordan matrix i... |

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Citation Context ... [28, 25], and E the matrix with all 1’s on and below the main diagonal and 0’s everywhere else (A000012 [28]). Note, the Chebyshev connection to B where B is a Riordan matrix is mentioned in Nkwanta =-=[21]-=- but not proved. Array A can be obtained by multiplying the matrices ⎛ ⎞ ⎛ 1 0 0 0 · · · 1 1 0 0 · · · PT = 1 2 1 0 · · · ⎜ ⎝ 1 3 3 1 · · · ⎟ ⎜ ⎠ ⎝ ... . . . . . . . . ... 1 0 0 0 · · · 1 1 0 0 · · · ... |

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Citation Context ...dition ∫ 1 0 T ∗ j (x)T ∗ k (x) √ dx = x − x2 15 ⎧ ⎨ ⎩ ( ) 2n + 1 = (n + 1)cn (17) n 0, if j ̸= k π, if j = k ̸= 0 2 π, if j = k = 0 = cn. (18)for the shifted Chebyshev polynomials of the first kind =-=[12]-=-. Recall that T ∗ 0 (x) = T ∗ 0 = 1/2. Then, from part (2) of Corollary 12, multiplying both sides of x n = (2/4 n ( ) T ∗ ( ) 2n 0 + T n ∗ ( ) 2n 1 + · · · + T n − 1 ∗ ) n (19) by T ∗ 0 (x) and the w... |

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Citation Context ...inatorial number theory, algebra, and special functions. Riordan array connections to the Chebyshev polynomials of the first kind have been given by Barry [4], Barry and Hennessy [5], Luzon and Moron =-=[17]-=-, and others. In this paper, we establish that the entries of two special Riordan matrices, we call Catalan-type Riordan arrays, are linked to the Chebyshev polynomials of the first kind. The followin... |

3 | A Riordan matrix approach to unifying a selected class of combinatorial arrays, Congressus Numerantium 160 - Nkwanta - 2003 |

3 |
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Citation Context ...E = ⎜ ⎝ 1 0 0 0 · · · 2 1 0 0 · · · 5 4 1 0 · · · 14 14 6 1 · · · . . . . ... ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ . 1 0 0 0 · · · 1 1 0 0 · · · 1 1 1 0 · · · 1 1 1 1 · · · where C denotes the Shapiro-Catalan matrix (A039598 =-=[28, 25]-=-, and E the matrix with all 1’s on and below the main diagonal and 0’s everywhere else (A000012 [28]). Note, the Chebyshev connection to B where B is a Riordan matrix is mentioned in Nkwanta [21] but ... |

2 |
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Citation Context ...at contains a wide variety of applications of the Catalan numbers. In addition, the representations we give differ from the representations given by Penson and Sixdeniers [23], Sofo [29], Dana-Picard =-=[9]-=-, [10], Aigner [1], and Yuan [35], [36]. Thus, it appears that the integral representations are new. This paper is arranged as follows. The definition of a Riordan matrix is given in Section 2. We sho... |

2 |
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Citation Context ... (1/ (n + 1)) denotes the nth Catalan number (A000108 [28]). ( ) 2n n The following lemma involving c (z) is useful for proving Proposition 6, Theorems 10 and 11, and Propositions 8 and 13. Lemma 5. (=-=[11]-=-) 1. 1/ √ 1 − 4z = c (z) / (1 − zc 2 (z)) = c (z)/(2 − c (z)) 2. zc 2 (z) = c (z) − 1. Proposition 6. ([19]), ([20]), ([28]) P ∗ T = A and C ∗ E = B. Proof. (Sketch) The Riordan pair forms of C and E ... |

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Citation Context ...s useful for proving Proposition 6, Theorems 10 and 11, and Propositions 8 and 13. Lemma 5. ([11]) 1. 1/ √ 1 − 4z = c (z) / (1 − zc 2 (z)) = c (z)/(2 − c (z)) 2. zc 2 (z) = c (z) − 1. Proposition 6. (=-=[19]-=-), ([20]), ([28]) P ∗ T = A and C ∗ E = B. Proof. (Sketch) The Riordan pair forms of C and E are C = (c 2 (z) ,zc 2 (z)) and E = (1/ (1 − z),z). Applying Theorem 3 and Lemma 5(1), then the Riordan pai... |

2 |
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Citation Context ...tations for n ≥ 1, 4n ∫ 1 (n + 1) 2x πn n+1 − xn ( ) (n + 1) 2n √ dx = = (n + 1)cn, (22) x − x2 n n − 1 and 0 4n ∫ 1 πn 0 2xn+1 − xn ( ) 1 2n √ dx = = cn. (23) x − x2 n n − 1 The Wolfram Alpha Widget =-=[34]-=- was used to confirm all of the integrals that were derived above analytically. The Wolfram definite integral calculator is a computational tool useful for experimental mathematics. It is interesting ... |

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Citation Context ...tions in combinatorics and graph theory, combinatorial number theory, algebra, and special functions. Riordan array connections to the Chebyshev polynomials of the first kind have been given by Barry =-=[4]-=-, Barry and Hennessy [5], Luzon and Moron [17], and others. In this paper, we establish that the entries of two special Riordan matrices, we call Catalan-type Riordan arrays, are linked to the Chebysh... |

1 | Combinatorial trigonometry with Chebyshev polynomials - Benjamin, Ericksen, et al. |

1 |
Combinatorially composing Chebyshev polynomials, J. Statistical Planning and Inference 140 (2010) 2161–2167. 17 A. Cayley, On partitions of a polygon
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Citation Context ...ays by column vectors made up of the coefficients of the generating functions associated with the Chebyshev and shifted Chebyshev polynomials is of interest. See Shapiro [26] and Benjamin et al. [6], =-=[7]-=- for combinatorial interpretations of the Chebyshev polynomials. Finding connections of A and B to the Chebyshev polynomials of the second kind is of interest. See Barry [4], Barry and Hennessy [5], a... |

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Citation Context ...the Pascal Triangle (A007318 [28]) written in lower triangular matrix form and T the matrix (A094531 [28]) whose leftmost column entries record the count of the number of king walks down a chessboard =-=[13, 27]-=-. The two matrix products are confirmed below by Proposition 6. 6 . . . ... ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ (4) (5)Definition 4. The Catalan generating function, denoted by c (z), is defined as c (z) := ( 1 − √ 1 − 4z )... |

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Citation Context ...ons. We note here that the proper Riordan arrays given by Sprugnoli [30] are what we call Riordan arrays or matrices. Two important results for multiplying Riordan matrices are now given. Theorem 2. (=-=[27, 22]-=-) If L = (ℓn,k) n,k∈N = (g (z),f (z)) is a Riordan matrix and h (z) is the generating function of the sequence associated with the entries of the column vector h = (hk) k∈N , then the product of L and... |

1 | Derivatives of Catalan related sums - Sofo - 2009 |

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Citation Context ...on (10) is converted to the sequence {T1,T3,T5,...}. Then, N ( x, √ z ) = ∑ n≥0 T2n+1 (x)z n = x (1 − z) 1 + 2z + z 2 − 4x 2 z is the generating function of the sequence {T1,T3,T5,...}. See Sprugnoli =-=[31]-=- for more information on the technique involving √ z. The generating functions given above lead to the following theorems. Theorem 10. Consider the Catalan-type Riordan arrays A = ( 1/ √ 1 − 4z,c(z) −... |

1 |
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Citation Context ...A000108 [28] (see Equations (15), (16), (17), (18), (20), (21), (22), and (23)). The integrals do not seem to explicitly appear in Gradshteyn and Ryzhik [14]. They do not appear in Stanley’s textbook =-=[32, 33]-=- that contains a large number of combinatorial and analytical interpretations of the Catalan numbers, nor in Koshy’s book [15] that contains a wide variety of applications of the Catalan numbers. In a... |

1 |
on graphs and tensor products, Paper posted to Annoying Precision, http://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products
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Citation Context ...plications of the Catalan numbers. In addition, the representations we give differ from the representations given by Penson and Sixdeniers [23], Sofo [29], Dana-Picard [9], [10], Aigner [1], and Yuan =-=[35]-=-, [36]. Thus, it appears that the integral representations are new. This paper is arranged as follows. The definition of a Riordan matrix is given in Section 2. We show that A and B are Catalan-type R... |

1 |
The Catalan numbers, regular languages, and orthogonal polynomials, Paper posted to Annoying Precision, archived at http://qchu.wordpress.com/2009/06/07/the-catalan-numbers-regular-languages -and-orthogonal-polynomials/, June 7 2009. 2010 Mathematics Subj
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Citation Context ...ions of the Catalan numbers. In addition, the representations we give differ from the representations given by Penson and Sixdeniers [23], Sofo [29], Dana-Picard [9], [10], Aigner [1], and Yuan [35], =-=[36]-=-. Thus, it appears that the integral representations are new. This paper is arranged as follows. The definition of a Riordan matrix is given in Section 2. We show that A and B are Catalan-type Riordan... |