Orthogonal polynomials approach to the Hankel transform of sequences based on Motzkin numbers. (English) Zbl 1387.11021
Summary: In this paper, we use a method based on orthogonal polynomials to give closed-form evaluations of the Hankel transform of sequences based on the Motzkin numbers. It includes linear combinations of consecutive two, three, and four Motzkin numbers. In some cases, we were able to derive the closed-form evaluation of the Hankel transform, while in the others we showed that the Hankel transform satisfies a particular difference equation. As a corollary, we reobtain known results and show some new results regarding the Hankel transform of Motzkin and shifted Motzkin numbers. Those evaluations also give an idea on how to apply the method based on orthogonal polynomials on the sequences having zero entries in their Hankel transform.
MSC:
| 11B83 | Special sequences and polynomials |
| 11C20 | Matrices, determinants in number theory |
| 11Y55 | Calculation of integer sequences |
Software:
OEISReferences:
| [1] | Aigner, M.: Motzkin numbers. Eur. J. Combin. 19(6), 663-675 (1998) · Zbl 0915.05004 · doi:10.1006/eujc.1998.0235 |
| [2] | Barbelo, S., Cerruti, U.: Catalan moments. Congressus Numerantium 201, 187-209 (2010) · Zbl 1203.11030 |
| [3] | Brualdi, R.A., Kirkland, S.: Aztec diamonds and digraphs, and Hankel determinants of Schröder numbers. J. Combin. Theory Ser. B 94, 334-351 (2005) · Zbl 1066.05009 · doi:10.1016/j.jctb.2005.02.001 |
| [4] | Cameron, N.T., Yip, A.C.M.: Hankel determinants of sums of consecutive Motzkin numbers. Linear Algebra Appl. 434, 712-722 (2011) · Zbl 1209.15010 · doi:10.1016/j.laa.2010.09.031 |
| [5] | Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978) · Zbl 0389.33008 |
| [6] | Cvetković, A., Rajković, P.M., Ivković, M.: Catalan numbers, the Hankel transform and Fibonacci numbers. J. Integer Seq. 5, Article 02.1.3 (2002) · Zbl 1041.11014 |
| [7] | Donaghey, R., Shapiro, L.: Motzkin numbers. J. Combin. Theory Ser. A 23, 291-301 (1977) · Zbl 0417.05007 · doi:10.1016/0097-3165(77)90020-6 |
| [8] | Eğecioğlu, O., Redmond, T., Ryavec, C.: Almost product evaluation of Hankel determinants. Electron. J. Combin. 15, #R6 (2008) · Zbl 1206.05009 |
| [9] | Eğecioğlu, O., Redmond, T., Ryavec, C.: A multilinear operator for almost product evaluation of Hankel determinants. J. Combin. Theory Ser. A 117, 77-103 (2010) · Zbl 1227.05031 · doi:10.1016/j.jcta.2009.03.016 |
| [10] | Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Clarendon Press, Oxford (2003) · Zbl 1130.42300 |
| [11] | Gessel, I.M., Viennot, G.: Binomial determinants, paths, and hook length formulae. Adv. Math. 58:3, 300-321 (1985) · Zbl 0579.05004 · doi:10.1016/0001-8708(85)90121-5 |
| [12] | Junod, A.: Hankel determinants and orthogonal polynomials. Expos. Math. 21, 63-74 (2003) · Zbl 1153.15304 · doi:10.1016/S0723-0869(03)80010-5 |
| [13] | Krattenthaler, C.: Advanced determinant calculus. Seminaire Lotharingien Combin. 42 (“The Andrews Festschrift”), Article B42q (1999) · Zbl 0923.05007 |
| [14] | Krattenthaler, C.: Advanced determinant calculus: a complement. Linear Algebra Appl. 411, 68-166 (2005) · Zbl 1079.05008 · doi:10.1016/j.laa.2005.06.042 |
| [15] | Lang, W.: On sums of powers of zeros of polynomials. J. Comput. Appl. Math. 89(2), 237-256 (1998) · Zbl 0910.30003 · doi:10.1016/S0377-0427(97)00240-9 |
| [16] | Layman, J.W.: The Hankel Transform and Some of Its Properties. J. Integer Seq. 4, Article 01.1.5 (2001) · Zbl 0978.15022 |
| [17] | Matthews, L.: Enumeration of disjoint Motzkin paths systems. Congr. Numer. 165, 213-222 (2003) · Zbl 1046.05005 |
| [18] | Petković, M.D., Rajković, P.M., Barry, P.: The Hankel transform of generalized central trinomial coefficients and related sequences. Integr. Transf. Spec. Funct. 22, 29-44 (2011) · Zbl 1228.11031 · doi:10.1080/10652469.2010.497998 |
| [19] | Rajković, P.M., Petković, M.D., Barry, P.: The Hankel transform of the sum of consecutive generalized catalan numbers. Integr. Transf. Spec. Funct. 18, 285-296 (2007) · Zbl 1127.11017 · doi:10.1080/10652460601092303 |
| [20] | Romik, D.: Some formulas for the central trinomial and Motzkin numbers. J. Integer Seq. 6, Article 03.2.4 (2003) · Zbl 1030.05002 |
| [21] | Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. available at http://www.research.att.com/ njas/sequences/ · Zbl 1274.11001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
