In this paper, we define a new matrix $ S_{n} $ constructed by super Catalan numbers. Also, we give Cholesky and LU-decompositions, Hermite normal form, and the determinant of the matrix $ S_{n} $. Moreover, we derive auxiliary results involving some summation formulas via the coefficients of Lucas polynomials and scaled coefficients of Chebyshev polynomials. Additionally, we give a matrix $ \acute{S}_{n} $ by modifying the matrix $ S_{n} $ to deduce a matrix identity related to matrices $ S_{n} $ and $ \acute{S}_{n}. $ By using the decomposition method, we give an application of solving a system of linear equations of order $ n $ with coefficients $ S(m, n) $ and find a general solution.
Citation: Serpil Halıcı, Zehra Betül Gür. On some combinatorial identities related to super Catalan matrix[J]. AIMS Mathematics, 2025, 10(7): 16139-16156. doi: 10.3934/math.2025723
In this paper, we define a new matrix $ S_{n} $ constructed by super Catalan numbers. Also, we give Cholesky and LU-decompositions, Hermite normal form, and the determinant of the matrix $ S_{n} $. Moreover, we derive auxiliary results involving some summation formulas via the coefficients of Lucas polynomials and scaled coefficients of Chebyshev polynomials. Additionally, we give a matrix $ \acute{S}_{n} $ by modifying the matrix $ S_{n} $ to deduce a matrix identity related to matrices $ S_{n} $ and $ \acute{S}_{n}. $ By using the decomposition method, we give an application of solving a system of linear equations of order $ n $ with coefficients $ S(m, n) $ and find a general solution.
| [1] |
I. M. Gessel, Super ballot numbers, J. Symb. Comput., 14 (1992), 179–194. https://doi.org/10.1016/0747-7171(92)90034-2 doi: 10.1016/0747-7171(92)90034-2
|
| [2] | E. A. Allen, Combinatorial interpretations of generalizations of Catalan numbers and ballot numbers, Carnegie Mellon University, 2014. |
| [3] |
E. Georgiadis, A. Munemasa, H. Tanaka, A note on super Catalan numbers, Interdiscip. Inf. Sci., 18 (2012), 23–24. https://doi.org/10.4036/iis.2012.23 doi: 10.4036/iis.2012.23
|
| [4] | K. Limanta, N. J. Wildberger, Super Catalan numbers and Fourier summation over finite fields, arXiv Preprint, 2021. https://doi.org/10.48550/arXiv.2108.10191 |
| [5] | J. C. Liu, Congruences on sums of super Catalan numbers, arXiv Preprint, 2018. https://doi.org/10.48550/arXiv.1804.09485 |
| [6] | K. von Szily, Uber die quadratsummen der binomialcoefficienten, Urgar. Ber., 12 (1893), 84–91. |
| [7] |
H. Prodinger, The reciprocal super Catalan matrix, Spec. Matrices, 3 (2015), 111–117. https://doi.org/10.1515/spma-2015-0010 doi: 10.1515/spma-2015-0010
|
| [8] |
E. Kılıç, İ. Akkuş, G. Kızılaslan, A variant of the reciprocal super Catalan matrix, Spec. Matrices, 3 (2015), 163–168. https://doi.org/10.1515/spma-2015-0014 doi: 10.1515/spma-2015-0014
|
| [9] |
E. Özkan, İ. Altun, Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials, Commun. Algebra, 47 (2019), 4020–4030. https://doi.org/10.1080/00927872.2019.1576186 doi: 10.1080/00927872.2019.1576186
|
| [10] | Z. Akyüz, S. Halıcı, On some combinatorial identities involving the terms of generalized Fibonacci and Lucas sequences, Hacet. J. Math. Stat., 4 (2013), 431–435. |
| [11] | J. C. Mason, D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, 2002. https://doi.org/10.1201/9781420036114 |
| [12] | N. J. A. Sloane, The on-line Encyclopedia of integer sequences, Available from: https://oeis.org. |
| [13] | P. A. Damianou, On the characteristic polynomial of Cartan matrices and Chebyshev polynomials, arXiv Preprint, 2011. https://doi.org/10.48550/arXiv.1110.6620 |
Serpil Halıcı, Zehra Betül Gür. On some combinatorial identities related to super Catalan matrix[J]. AIMS Mathematics, 2025, 10(7): 16139-16156. doi: 10.3934/math.2025723
DownLoad: