Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
derangement; permutation; integration
derangement; permutation; integration
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.
References:
[1] Aigner M., Ziegler G. M.: Proofs from The Book. Springer, Berlin, 2018. MR 3823190
[2] Askey R. A., Ismail M. E. H.: Permutation problems and special functions. Canadian. J. Math. 28 (1976), no. 4, 853–874. DOI 10.4153/CJM-1976-082-8 | MR 0406808
[3] Axler S.: Measure, Integration & Real Analysis. Graduate Texts in Mathematics, 282, Springer, Cham, 2020. MR 3972068
[4] Benyattou A.: Derangement polynomials with a complex variable. Notes Number Theory Discrete Math. 26 (2020), no. 4, 128–135. DOI 10.7546/nntdm.2020.26.4.128-135
[5] Chow C.-O.: On derangement polynomials of type $B$. Sém. Lothar. Combin. 55 (2005/07), Art. B55b, 6 pages. MR 2223025
[6] Chow C.-O.: On derangement polynomials of type $B$. II. J. Combin. Theory Ser. A 116 (2009), no. 4, 816–830. DOI 10.1016/j.jcta.2008.11.006 | MR 2513636
[7] Hassani M.: Derangements and applications. J. Integer Seq. 6 (2003), no. 1, Art. 03.1.2, 8 pages. MR 1971432
[8] Hassani M.: Cycles in graphs and derangements. Math. Gaz. 88 (2004), no. 511, 123–126. DOI 10.1017/S0025557200174443 | MR 1971432
[9] Hassani M.: Enumeration by $ e$. Modern Discrete Mathematics and Analysis: Springer Optim. Appl., 131, Springer, Cham, 2018, pages 227–233. MR 3887936
[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
[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
[12] Kayll P. M.: Integrals don't have anything to do with discrete math, do they?. Math. Mag. 84 (2011), no. 2, 108–119. DOI 10.4169/math.mag.84.2.108 | MR 2793183
[13] LeVeque W. J.: Topics in Number Theory. Vols. 1 and 2. Addison–Wesley Publishing, Mass, 1956. MR 0080682
[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). DOI 10.1016/S0195-6698(13)80115-1 | MR 1120419
[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. DOI 10.1016/S0377-0427(99)00120-X | MR 1747239
Search
Partner of
