On Pellnomial coefficients and Pell-Catalan numbers. (English) Zbl 1506.11028
Summary: In this paper, we first give the Pascal’s identity for Pellnomial coefficients and then we show that the Pellnomial coefficients are integers. We obtain that the product of \(r\) consecutive Pell numbers is divisible by the Pell analog of \(r!\). Also, we introduce the divisibility theorems between Pell numbers and Pellnomial coefficients. Furthermore, we first define Pell-Catalan numbers and then we derive two formulas for presenting Pell-Catalan numbers.
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 11Y55 | Calculation of integer sequences |
| 13A05 | Divisibility and factorizations in commutative rings |
Software:
OEISReferences:
| [1] | S. Y. Akbiyik, M. Akbiyik, and F. Yilmaz, One type of symmetric matrix with har-monic Pell entries, its inversion, permanents and some norms, Mathematics 9(5) (2021), 539, 15 pp. |
| [2] | E. Andrade, D. Carrasco-Olivera, and C. Manzaneda, On circulant like matrices prop-erties involving Horadam, Fibonacci, Jacobsthal and Pell numbers, Linear Algebra Appl. 617 (2021), 100-120. · Zbl 1464.15043 |
| [3] | N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995. · Zbl 0845.11001 |
| [4] | D. Bród, On a new one parameter generalization of Pell numbers, Ann. Math. Sil. 33 (2019), 66-76. · Zbl 1470.11022 |
| [5] | C. B. Ç imen and A.İpek, On pell quaternions and Pell-Lucas quaternions, Adv. Appl. Clifford Algebras 26 (2016), 39-51. · Zbl 1344.11022 |
| [6] | E. E. Enochs, Binomial coefficients, Bol. Asoc. Mat. Venez. 11 (2004), 17-28. · Zbl 1062.05007 |
| [7] | H. W. Gould, The bracket function and Fontené-Ward generalized binomial coeffi-cients with application to Fibonomial coefficients, Fibonacci Quart. 7 (1969), 23-40. · Zbl 0191.32702 |
| [8] | A. Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers, Organic mathematics (Burnaby, BC, 1995), 253-276, CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997. · Zbl 0903.11005 |
| [9] | R. P. Grimaldi, Fibonacci and Catalan Numbers: an Introduction, John Wiley & Sons, Hoboken, 2012. · Zbl 1242.11001 |
| [10] | V. E. Hoggatt Jr., Fibonacci numbers and generalized binomial coefficients, Fibonacci Quart. 5 (1967), 383-400. · Zbl 0157.03101 |
| [11] | A.İpek, On triangular Pell and Pell-Lucas numbers, Bull. Int. Math. Virtual Inst. 12 (2022), 429-434. |
| [12] | T. Koshy, Catalan Numbers with Applications, Oxford University Press, Oxford, 2009. · Zbl 1159.05001 |
| [13] | T. Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014. · Zbl 1330.11002 |
| [14] | E. Krot, An introduction to finite fibonomial calculus, Cent. Eur. J. Math. 2 (2004), 754-766. · Zbl 1160.11308 |
| [15] | R. Melham, Sums involving Fibonacci and Pell numbers, Portugal. Math. 56 (1999), 309-318. · Zbl 0944.11005 |
| [16] | S. Roman, An Introduction to Catalan Numbers, Springer, Cham, 2015. · Zbl 1342.05002 |
| [17] | S. F. Santana and J. L. Díaz-Barrero, Some properties of sums involving Pell numbers, Missouri J. Math. Sci. 18 (2006), 33-40. · Zbl 1137.05009 |
| [18] | J. Seibert and P. Trojovský, On some identities for the Fibonomial coefficients, Math. Slovaca 55 (2005), 9-19. · Zbl 1108.11019 |
| [19] | M. Z. Spivey, The Art of Proving Binomial Identities, CRC Press, Boca Raton, 2019. · Zbl 1419.05002 |
| [20] | R. P. Stanley, Catalan Numbers, Cambridge University Press, New York, 2015. · Zbl 1317.05010 |
| [21] | P. Trojovský, On some identities for the Fibonomial coefficients via generating func-tion, Discrete Appl. Math. 155 (2007), 2017-2024. · Zbl 1144.11018 |
| [22] | W. Zhang and T. Wang, The infinite sum of reciprocal Pell numbers, Appl. Math. Comput. 218 (2012), 6164-6167. · Zbl 1247.39001 |
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