| Title:
|
Some results on derangement polynomials (English) |
| Author:
|
Hassani, Mehdi |
| Author:
|
Moshtagh, Hossein |
| Author:
|
Ghorbani, Mohammad |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
307-313 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study moments of the difference $D_n(x)-x^n n! {\rm e}^{-1/x}$ concerning derangement polynomials $D_n(x)$. For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x>0$. For the higher moments, we obtain a multiple integral representation of the order of the moment under computation. (English) |
| Keyword:
|
derangement |
| Keyword:
|
permutation |
| Keyword:
|
integration |
| MSC:
|
05A05 |
| MSC:
|
05A16 |
| MSC:
|
26A06 |
| idZBL:
|
Zbl 07655802 |
| idMR:
|
MR4542791 |
| DOI:
|
10.14712/1213-7243.2022.025 |
| . |
| Date available:
|
2023-02-01T12:04:38Z |
| Last updated:
|
2024-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151478 |
| . |
| Reference:
|
[1] Aigner M., Ziegler G. M.: Proofs from The Book.Springer, Berlin, 2018. MR 3823190 |
| Reference:
|
[2] Askey R. A., Ismail M. E. H.: Permutation problems and special functions.Canadian. J. Math. 28 (1976), no. 4, 853–874. MR 0406808, 10.4153/CJM-1976-082-8 |
| Reference:
|
[3] Axler S.: Measure, Integration & Real Analysis.Graduate Texts in Mathematics, 282, Springer, Cham, 2020. MR 3972068 |
| Reference:
|
[4] Benyattou A.: Derangement polynomials with a complex variable.Notes Number Theory Discrete Math. 26 (2020), no. 4, 128–135. 10.7546/nntdm.2020.26.4.128-135 |
| Reference:
|
[5] Chow C.-O.: On derangement polynomials of type $B$.Sém. Lothar. Combin. 55 (2005/07), Art. B55b, 6 pages. MR 2223025 |
| Reference:
|
[6] Chow C.-O.: On derangement polynomials of type $B$. II.J. Combin. Theory Ser. A 116 (2009), no. 4, 816–830. MR 2513636, 10.1016/j.jcta.2008.11.006 |
| Reference:
|
[7] Hassani M.: Derangements and applications.J. Integer Seq. 6 (2003), no. 1, Art. 03.1.2, 8 pages. MR 1971432 |
| Reference:
|
[8] Hassani M.: Cycles in graphs and derangements.Math. Gaz. 88 (2004), no. 511, 123–126. MR 1971432, 10.1017/S0025557200174443 |
| Reference:
|
[9] Hassani M.: Enumeration by $ e$.Modern Discrete Mathematics and Analysis: Springer Optim. Appl., 131, Springer, Cham, 2018, pages 227–233. MR 3887936 |
| Reference:
|
[10] Hassani M.: Derangements and alternating sum of permutations by integration.J. Integer Seq. 23 (2020), no. 7, Art. 20.7.8, 9 pages. MR 4134234 |
| Reference:
|
[11] Hassani M.: On a difference concerning the number $ e$ and summation identities of permutations.J. Inequal. Spec. Funct. 12 (2021), no. 1, 14–22. MR 4246713 |
| Reference:
|
[12] Kayll P. M.: Integrals don't have anything to do with discrete math, do they?.Math. Mag. 84 (2011), no. 2, 108–119. MR 2793183, 10.4169/math.mag.84.2.108 |
| Reference:
|
[13] LeVeque W. J.: Topics in Number Theory. Vols. 1 and 2.Addison–Wesley Publishing, Mass, 1956. MR 0080682 |
| Reference:
|
[14] Radoux C.: Déterminant de Hankel construit sur des polynômes liés aux nombres de dérangements.European J. Combin. 12 (1991), no. 4, 327–329 (French). MR 1120419, 10.1016/S0195-6698(13)80115-1 |
| Reference:
|
[15] Radoux C.: Addition formulas for polynomials built on classical combinatorial sequences.Proc. of the 8th International Congress on Computational and Applied Mathematics, J. Comput. Appl. Math. 115 (2000), no. 1–2, 471–477. MR 1747239, 10.1016/S0377-0427(99)00120-X |
| . |
