TAILIEUCHUNG - The generalized reciprocal super Catalan matrix

The reciprocal super Catalan matrix studied by Prodinger is further generalized, introducing two additional parameters. Explicit formulae are derived for the LU -decomposition and their inverses, as well as the Cholesky decomposition. | Turk J Math (2016) 40: 960 – 972 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article The generalized reciprocal super Catalan matrix 1 Emrah KILIC ¸ 1 , Talha ARIKAN2,∗ Department of Mathematics, TOBB Economics and Technology University, Ankara, Turkey 2 Department of Mathematics, Hacettepe University, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: The reciprocal super Catalan matrix studied by Prodinger is further generalized, introducing two additional parameters. Explicit formulae are derived for the LU -decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q -analysis and to leave the justification of the necessary identities to the q version of Zeilberger’s celebrated algorithm. Key words: Determinant, inverse matrix, LU factorization, Gaussian q -binomial coefficient, Zeilberger’s algorithm 1. Introduction As mentioned in [8], there are many combinatorial matrices defined by a given sequence {an } . One of them is known as the Hankel matrix and is defined as follows: a0 a1 a2 . a1 a2 a2 a3 a3 . a4 ··· ··· ··· . . for more details see [6]. Considering some special number sequences instead of {an } , there are many special matrices with nice algebraic properties. Moreover, some authors, such as [10], studied the Hankel matrix considering the reciprocal sequence of {an } 1 a0 1 a1 1 a1 1 a2 1 a2 1 a3 1 a2 1 a2 1 a4 . . ··· ··· ··· . . . For the sequence {ai,j } , a matrix can be defined by taking (i, j)th entries ai,j . Well-known types of these sequences typically include binomial coefficients. As examples, we give the family of Pascal matrices whose entries are defined via the usual binomial coefficients [2, 3]. The Pascal matrices are mainly two kinds: the first is the left .

• Toán học
• generalized reciprocal super Catalan matrix
• Inverse matrix
• LU factorization
• Gaussian q binomial coefficient
• Zeilberger’s algorithm