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Coefficient triangle of the Hermite-Bell polynomials for power -2.
3

%I #26 Nov 24 2017 16:32:39

%S 1,2,4,-6,8,-36,24,16,-144,300,-120,32,-480,2040,-2640,720,64,-1440,

%T 10320,-27720,25200,-5040,128,-4032,43680,-199920,383040,-262080,

%U 40320,256,-10752,163968,-1142400,3764880,-5503680,2963520,-362880

%N Coefficient triangle of the Hermite-Bell polynomials for power -2.

%C The Hermite-Bell polynomials for negative powers H(n;-r;x), n=0,1,..., r=1,2,..., and x \in C\{0} are defined by the following relation: H(n;-r;x) = x^((r+1)*n)*exp(1/x^r)*(d^n exp(-1/x^r)/dx^n). These polynomials form the natural generalization of the concept of so-called generalized Hermite-Bell polynomials given for positive integers powers by D. Dominici (see also the R. B. Paris paper). We obtain the following recurrence formula:

%C H(n+1;-r;x) = (r-(r+1)*n*x^r)*H(n;-r;x) + x^(r+1)*(dH(n;-r;x)/dx). In the sequel we deduce the following special ones: H(0;-r;x)=1, H(1;-r;x)=r, H(2;-r;x)=r^2 - r*(r+1)*x^r, H(3;-r;x)=r^3 - 3*r^2*(r+1)*x^r + r*(r+1)*(r+2)*x^(2*r), H(4;-r;x)=r^4 - 6*r^3*(r+1)*x^r + r^2*(r+1)*(7*r+11)*x^(2*r) - r*(r+1)*(r+2)*(r+3)*x^(3*r) - the general formulas are given in Witula et al.'s paper.

%C There is a connection between H(n;-1;x) and the Laguerre polynomials L(n;x;a=1), see A066667 for details.

%D R. Witula, E. Hetmaniok, D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)

%H D. Dominici, <a href="http://arxiv.org/abs/math/0606324">Asymptotic analysis of generalized Hermite polynomials</a>, arXiv:math/0606324 [math.CA], 2006

%H D. Dominici, <a href="http://dx.doi.org/10.1524/anly.2008.0911">Asymptotic analysis of generalized Hermite polynomials</a>, Analysis 28 (2008), 239-261.

%H R. B. Paris, <a href="http://dx.doi.org/10.1016/j.cam.2009.05.031">The asymptotics of the generalised Hermite-Bell polynomials</a>, J. Comput. Appl. Math. 232 (2009), 216-226.

%F H(n+1;-2;x) = (2-3*n*x^2)*H(n;-2;x) + x^3*(dH(n;-2;x)/dx), with H(1;-2;x)=2.

%e Let us put W(n;x):=H(n;-2;sqrt(x)). Then we have W(0;x)=1, W(1;x)=2, W(2;x)=4-6*x, W(3;x)=8-36*x+24*x^2, W(4;x)=16-144*x+300*x^2-120*x^3, W(5;x)=32-480*x+2040*x^2-2640*x^3+720*x^4.

%e 1;

%e 2;

%e 4, -6;

%e 8, -36, 24;

%e 16, -144, 300, -120;

%e 32, -480, 2040, -2640, 720;

%e 64, -1440, 10320, -27720, 25200, -5040;

%e 128, -4032, 43680, -199920, 383040, -262080, 40320;

%p H := proc(n,r,x)

%p local e,d ;

%p e := exp(-1/x^r) ;

%p for d from 1 to n do

%p e := diff(e,x) ;

%p end do:

%p x^((r+1)*n)*exp(1/x^r)*e ;

%p expand(%) ;

%p end proc:

%p A215216 := proc(n,k)

%p subs(x=sqrt(x),H(n,2,x)) ;

%p coeftayl( %,x=0,k) ;

%p end proc:

%p seq(seq( A215216(n,k),k=0..max(0,n-1)),n=0..6) ; # _R. J. Mathar_, Aug 07 2012

%t H[n_, r_, x_] := Module[{e, d }, e = Exp[-1/x^r]; For[d = 1, d <= n, d++, e = D[e, x]]; x^((r + 1)*n)*Exp[1/x^r]*e // Expand];

%t A215216[n_, k_] := H[n, 2, x] /. x -> Sqrt[x] // SeriesCoefficient[#, {x, 0, k}]&;

%t Table[A215216[n, k], {n, 0, 8}, {k, 0, Max[0, n - 1]}] // Flatten (* _Jean-François Alcover_, Nov 24 2017, after _R. J. Mathar_ *)

%Y Cf. A066667.

%K sign,tabf

%O 0,2

%A _Roman Witula_, Aug 06 2012