A215216 Coefficient triangle of the Hermite-Bell polynomials for power -2.

1, 2, 4, -6, 8, -36, 24, 16, -144, 300, -120, 32, -480, 2040, -2640, 720, 64, -1440, 10320, -27720, 25200, -5040, 128, -4032, 43680, -199920, 383040, -262080, 40320, 256, -10752, 163968, -1142400, 3764880, -5503680, 2963520, -362880

Offset: 0

Author

Roman Witula, Aug 06 2012

Keywords

• sign
• tabf

Comments

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:

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.

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

Examples

			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.
1;
2;
4,      -6;
8,     -36,    24;
16,   -144,   300,    -120;
32,   -480,  2040,   -2640,    720;
64,  -1440, 10320,  -27720,  25200,   -5040;
128, -4032, 43680, -199920, 383040, -262080, 40320;
		

References

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

Links

• D. Dominici, Asymptotic analysis of generalized Hermite polynomials, arXiv:math/0606324 [math.CA], 2006
• D. Dominici, Asymptotic analysis of generalized Hermite polynomials, Analysis 28 (2008), 239-261.
• R. B. Paris, The asymptotics of the generalised Hermite-Bell polynomials, J. Comput. Appl. Math. 232 (2009), 216-226.

Crossrefs

Cf. A066667.

Programs

Maple

H := proc(n,r,x)
    local e,d ;
    e := exp(-1/x^r) ;
    for d from 1 to n do
        e := diff(e,x) ;
    end do:
    x^((r+1)*n)*exp(1/x^r)*e ;
    expand(%) ;
end proc:
A215216 := proc(n,k)
    subs(x=sqrt(x),H(n,2,x)) ;
    coeftayl( %,x=0,k) ;
end proc:
seq(seq( A215216(n,k),k=0..max(0,n-1)),n=0..6) ; # R. J. Mathar, Aug 07 2012

Mathematica

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];
A215216[n_, k_] := H[n, 2, x] /. x -> Sqrt[x] // SeriesCoefficient[#, {x, 0, k}]&;
Table[A215216[n, k], {n, 0, 8}, {k, 0, Max[0, n - 1]}] // Flatten (* Jean-François Alcover, Nov 24 2017, after R. J. Mathar *)

Formula

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.