Geodesic
Unimodality polynomials and generalized Pascal triangles
Algebra and discrete mathematics, Tome 26 (2018) no. 1, pp. 1-7.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we show that if $P(x)=\sum_{k=0}^{m}a_{k}x^{k}$ is a polynomial with nondecreasing, nonnegative coefficients, then the coefficients sequence of $P(x^{s}+\cdots +x+1)$ is unimodal for each integer $s\geq 1$. This paper is an extension of Boros and Moll's result “A criterion for unimodality”, who proved that the polynomial $P(x+1)$ is unimodal.
Keywords: unimodality, log-concavity, ordinary multinomials
Mots-clés : Pascal triangle. 
@article{ADM_2018_26_1_a1,
     author = {Moussa Ahmia and Hac\`ene Belbachir},
     title = {Unimodality polynomials and generalized {Pascal} triangles},
     journal = {Algebra and discrete mathematics},
     pages = {1--7},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2018_26_1_a1/}
}
TY  - JOUR
AU  - Moussa Ahmia
AU  - Hacène Belbachir
TI  - Unimodality polynomials and generalized Pascal triangles
JO  - Algebra and discrete mathematics
PY  - 2018
SP  - 1
EP  - 7
VL  - 26
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2018_26_1_a1/
LA  - en
ID  - ADM_2018_26_1_a1
ER  - 
%0 Journal Article
%A Moussa Ahmia
%A Hacène Belbachir
%T Unimodality polynomials and generalized Pascal triangles
%J Algebra and discrete mathematics
%D 2018
%P 1-7
%V 26
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2018_26_1_a1/
%G en
%F ADM_2018_26_1_a1
Moussa Ahmia; Hacène Belbachir. Unimodality polynomials and generalized Pascal triangles. Algebra and discrete mathematics, Tome 26 (2018) no. 1, pp. 1-7. http://geodesic.mathdoc.fr/item/ADM_2018_26_1_a1/

[1] M. Ahmia, H. Belbachir, “Preserving log-concavity and generalized triangles”, Diophantine analysis and related fields 2010, Conference Proceedings, 1264, eds. K. Takao, American Institute of Physics, New York, 2010, 81–89 | MR

[2] M. Ahmia, H. Belbachir, “Preserving log-convexity for generalized Pascal triangles”, Electron. J. Combin., 19:2 (2012), 16, 6 pp. | MR

[3] H. Belbachir, “Determining the mode for convolution powers of discrete uniform distribution”, Probab. Engrg. Inform. Sci., 25:4 (2011), 469–475 | DOI | MR | Zbl

[4] H. Belbachir, S. Bouroubi, A. Khelladi, “Connection between ordinary multinomials, Fibonacci numbers, Bell polynomials and discrete uniform distribution”, Annales Mathematicae et Informaticae, 35 (2008), 21–30 | MR | Zbl

[5] H. Belbachir, L. Szalay, “Unimodal rays in the regular and generalized Pascal triangles”, J. of Integer Seq., 11 (2008), 08.2.4 | MR

[6] G. Boros, V. H. Moll, “A criterion for unimodality”, Elec. Journal of Combinatorics, 6 (1999), R10 | MR | Zbl

[7] F. Brenti, “Unimodal, log-concave and Pólya frequency sequences in combinatorics”, Mem. Amer. Math. Soc., 413 (1989) | MR | Zbl

[8] F. Brenti, “Log-concave and unimodal sequence”, Algebra, combinatorics and geometry: an update, Elec. Contemp. Math., 178, 1994, 1997, 71–84 | DOI | MR

[9] F. Brenti, “Combinatorics and total positivity”, J. Combin. Theory Ser. A, 71 (1995), 175–218 | DOI | MR | Zbl

[10] F. Brenti, “The applications of total positivity to combinatorics, and conversely, Total positivity and its applications” (Jaca 1994), Math. Appl., 359, Kluwer Acad. Publ., Dordrecht, 1996, 451–473 | MR | Zbl

[11] J. N. Darroch, “On the distribution of the number of successes in independent trials”, Ann. Math. Statist., 35 (1964), 1317–1321 | DOI | MR | Zbl

[12] S. Karlin, Total Positivity, v. I, Stanford University Press, 1968 | MR | Zbl

[13] K. V. Menon, “On the convolution of logarithmically concave sequences”, Proc. Amer. Math. Soc., 23 (1969), 439–441 | DOI | MR | Zbl

[14] N. J. A. Sloane, The online Encyclopedia of Integer sequences Published electronically at http://www.research.att.com/<nobr>$\sim$</nobr>njas/sequences

[15] R. P. Stanley, “Log-concave and unimodal sequences”, Algebra, combinatorics, and geometry, Ann. New York Acad. Sci., 576, 1989, 500–534 | DOI | MR

[16] H. S. Wilf, Generating functionology, Academic Press 1990