| Title:
|
Closed-form expression for Hankel determinants of the Narayana polynomials (English) |
| Author:
|
Petković, Marko D. |
| Author:
|
Barry, Paul |
| Author:
|
Rajković, Predrag |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
39-57 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We considered a Hankel transform evaluation of Narayana and shifted Narayana polynomials. Those polynomials arises from Narayana numbers and have many combinatorial properties. A mainly used tool for the evaluation is the method based on orthogonal polynomials. Furthermore, we provided a Hankel transform evaluation of the linear combination of two consecutive shifted Narayana polynomials, using the same method (based on orthogonal polynomials) and previously obtained moment representation of Narayana and shifted Narayana polynomials. (English) |
| Keyword:
|
Narayana numbers |
| Keyword:
|
Hankel transform |
| Keyword:
|
orthogonal polynomials |
| MSC:
|
11B83 |
| MSC:
|
11Y55 |
| MSC:
|
33C45 |
| MSC:
|
34A25 |
| idZBL:
|
Zbl 1249.11042 |
| idMR:
|
MR2899733 |
| DOI:
|
10.1007/s10587-012-0015-8 |
| . |
| Date available:
|
2012-03-05T07:10:14Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142039 |
| . |
| Reference:
|
[1] Barry, P.: On integer-sequences-based constructions of generalized Pascal triangles.J. Integer Seq. 59 (2006), Article 06.2.4. Electronic only. MR 2217230 |
| Reference:
|
[2] Barry, P., Hennessy, A.: Notes on a family of Riordan arrays and associated integer Hankel transforms.J. Integer Seq. 12 (2009), Article ID 09.5.3. Electronic only. Zbl 1201.11034, MR 2520842 |
| Reference:
|
[3] Brändén, P.: $q$-Narayana numbers and the flag $h$-vector of $J(2 \times n)$.Discrete Math. 281 (2004), 67-81. MR 2047757, 10.1016/j.disc.2003.07.006 |
| Reference:
|
[4] Brualdi, R. A., Kirkland, S.: Aztec diamonds and digraphs, and Hankel determinants of Schröder numbers.J. Comb. Theory Ser. B 94 (2005), 334-351. Zbl 1066.05009, MR 2145518, 10.1016/j.jctb.2005.02.001 |
| Reference:
|
[5] Chamberland, M., French, C.: Generalized Catalan numbers and generalized Hankel transformations.J. Integer Seq. 10 (2007). Zbl 1116.11009, MR 2268452 |
| Reference:
|
[6] Chihara, T. S.: An Introduction to Orthogonal Polynomials.Gordon and Breach New York (1978). Zbl 0389.33008, MR 0481884 |
| Reference:
|
[7] Cvetković, A., Rajković, P. M., Ivković, M.: Catalan Numbers, the Hankel transform, and Fibonacci numbers.J. Integer Seq. 5 (2002). Zbl 1041.11014, MR 1919940 |
| Reference:
|
[8] Eğecioğlu, O., Redmond, T., Ryavec, C.: Almost product evaluation of Hankel determinants.Electron. J. Comb. 15, \#R6 (2008). Zbl 1206.05009, MR 2368911, 10.37236/730 |
| Reference:
|
[9] Eğecioğlu, O., Redmond, T., Ryavec, C.: A multilinear operator for almost product evaluation of Hankel determinants.J. Comb. Theory Ser. A 117 (2010), 77-103. Zbl 1227.05031, MR 2557881, 10.1016/j.jcta.2009.03.016 |
| Reference:
|
[10] Armas, M. Garcia, Sethuraman, B. A.: A note on the Hankel transform of the central binomial coefficients.J. Integer Seq. 11 (2008), Article ID 08.5.8. MR 2465389 |
| Reference:
|
[11] Gautschi, W.: Orthogonal polynomials: Applications and computation.In: Acta Numerica Vol. 5 A. Iserles Cambridge University Press Cambridge (1996), 45-119. Zbl 0871.65011, MR 1624591, 10.1017/S0962492900002622 |
| Reference:
|
[12] Gautschi, W.: Orthogonal Polynomials. Computation and Approximation.Oxford University Press Oxford (2004). Zbl 1130.42300, MR 2061539 |
| Reference:
|
[13] Ismail, M. E. H.: Determinants with orthogonal polynomial entries.J. Comput. Appl. Math. 178 (2005), 255-266. Zbl 1083.15011, MR 2127884, 10.1016/j.cam.2004.01.042 |
| Reference:
|
[14] Junod, A.: Hankel determinants and orthogonal polynomials.Expo. Math. 21 (2003), 63-74. Zbl 1153.15304, MR 1955218, 10.1016/S0723-0869(03)80010-5 |
| Reference:
|
[15] Krattenthaler, C.: Advanced determinant calculus: A complement.Linear Algebra Appl. 411 (2005), 68-166. Zbl 1079.05008, MR 2178686, 10.1016/j.laa.2005.06.042 |
| Reference:
|
[16] Lang, W.: On sums of powers of zeros of polynomials.J. Comput. Appl. Math. 89 (1998), 237-256. Zbl 0910.30003, MR 1626522, 10.1016/S0377-0427(97)00240-9 |
| Reference:
|
[17] Layman, J. W.: The Hankel transform and some of its properties.J. Integer Seq. 4 (2001), Article 01.1.5. Electronic only. Zbl 0978.15022, MR 1848942 |
| Reference:
|
[18] Liu, L. L., Wang, Y.: A unified approach to polynomial sequences with only real zeros.Adv. Appl. Math. 38 (2007), 542-560. Zbl 1123.05009, MR 2311051, 10.1016/j.aam.2006.02.003 |
| Reference:
|
[19] MacMahon, P. A.: Combinatorial Analysis, Vols. 1 and 2.Cambridge University Press Cambridge (1915), 1916 reprinted by Chelsea, 1960. |
| Reference:
|
[20] Mansour, T., Sun, Y.: Identities involving Narayana polynomials and Catalan numbers.Discrete Math. 309 (2009), 4079-4088. Zbl 1191.05016, MR 2537400, 10.1016/j.disc.2008.12.006 |
| Reference:
|
[21] Narayana, T. V.: Sur les treillis formés par les partitions d'un entier et leurs applications à la théorie des probabilités.C. R. Acad. Sci. 240 (1955), 1188-1189 French. Zbl 0064.12705, MR 0070648 |
| Reference:
|
[22] Rajković, P. M., Petković, M. D., Barry, P.: The Hankel transform of the sum of consecutive generalized Catalan numbers.Integral Transforms Spec. Funct. 18 (2007), 285-296. Zbl 1127.11017, MR 2319589, 10.1080/10652460601092303 |
| Reference:
|
[23] Sloane, N. J. A.: The On-Line Encyclopedia of Integer Sequences.available at http://www.research.att.com/ {njas/sequences} Zbl 1159.11327 |
| Reference:
|
[24] Sulanke, R. A.: Counting lattice paths by Narayana polynomials.Electron. J. Comb. 7 (2000), \#R40. Zbl 0953.05006, MR 1779937, 10.37236/1518 |
| Reference:
|
[25] Sulanke, R. A.: The Narayana distribution.J. Statist. Plann. Inference 101 (2002), 311-326. Zbl 1001.05009, MR 1878867, 10.1016/S0378-3758(01)00192-6 |
| Reference:
|
[26] Xin, G.: Proof of the Somos-4 Hankel determinants conjecture.Adv. Appl. Math. 42 (2009), 152-156. Zbl 1169.05304, MR 2493974, 10.1016/j.aam.2008.04.003 |
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