A201201 Coefficient triangle for the monic associated Laguerre-Sonin(e) polynomials of order one for parameter alpha=1.

1, -4, 1, 18, -10, 1, -96, 86, -18, 1, 600, -756, 246, -28, 1, -4320, 7092, -3168, 552, -40, 1, 35280, -71856, 41112, -9720, 1070, -54, 1, -322560, 787824, -552240, 165720, -24600, 1878, -70, 1, 3265920, -9329760, 7768080, -2835360, 531480, -54516, 3066, -88

Offset: 0

Author

Wolfdieter Lang, Dec 06 2011

Keywords

• sign
• easy
• tabl

Comments

See A199577 for general comments on associated Laguerre-Sonin(e) polynomials of order m, and the Ismail reference.

The monic row polynomials are La_n(1;1,x) = sum(a(n,k)*x^k,k=0..n), with the three term recurrence

La_n(1;1,x) = (x-2*(n+1))*La_{n-1}{1;1,x) - n*(n+1)*La_{n-2}{1;1,x), La_{-1}{1;1,x)=0, La_0(1;1,x)=1.

In the Ismail reference the non-monic associated Laguerre polynomials of order 1 appear on p. 160 in Theorem 5.6.1, eq. 5.6.11. The connection is: La_n(1;1,x)= L_n^{(alpha=1)}(x;1)*(n+1)!*(-1)^n.

The e.g.f. gLa(1;1,z,x) for La_n(1;1,x) can be obtained from the o.g.f. G(1;1,z,x) for the non-monic version L_n^{alpha=1}(x;1) by gLa(1;1,z,x)= (d/dz) (z*G(1;1,-z,x)).

G(1;1,z,x) satisfies an ordinary first order inhomogeneous differential equation:

(d/dz) G(1;1,z,x) = (3/(1-z)+(1-x)/(1-z)^2-1/(z*(1-z)^2 z)^2))* G(1;1,z,x) + 1/(z*(1-z)^2), with G(1;1,z=0,x)=1. The standard solution is:

G(1;1,z,x) = (exp(-x*z/(1-z))-1+z-x*exp(-x/(1-z))* (Ei(1,-x/(1-z))-Ei(1,-x)))/(z*(1-z)^2), with the exponential integral Ei(1,y)=int(exp(-t)/t,t=y..infty).

Examples

			The triangle begins:
n\k     0      1       2      3      4    5   6 7 ...
0:      1
1:     -4      1
2:     18    -10       1
3:    -96     86     -18      1
4:    600   -756     246    -28      1
5:  -4320   7092   -3168    552    -40    1
6:  35280 -71856   41112  -9720   1070  -54   1
7:-322560 787824 -552240 165720 -24600 1878 -70 1
...
		

References

• M. E. H. Ismail (two chapters by W. Van Assche), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005.

Crossrefs

Cf. A199577 (alpha=0 case), A201202 (row sums), A201203 (alternating row sums).

Programs

Maple

La := proc(n,x)
    option remember;
    if n= -1 then
        0;
    elif n = 0 then
        1;
    else
        (x-2*n-2)*procname(n-1,x)-n*(n+1)*procname(n-2,x) ;
    end if;
end proc:
A201201 := proc(n,k)
    coeftayl( La(n,x),x=0,k) ;
end proc:
seq(seq(A201201(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 07 2011

Mathematica

a[n_, k_] := (-1)^(n-k)*((n+1)*(n+1)!/((k+1)*(k+1)!))*Binomial[n, k]*HypergeometricPFQ[{-(n-k), k, 1}, {-(n+1), k+2}, 1]; Table[a[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

Formula

a(n,k)=[x^k] La_n(1;1,x), n>=0, k=0,...,n.

a(n,k)= (-1)^(n-k)*((n+1)*(n+1)!/((k+1)*(k+1)!))*

binomial(n,k)*hypergeom(-(n-k),k,1; -(n+1),k+2; 1), 0<=k<=n. [Ismail, p. 161, eq. (5.6.18)].

The e.g.f. gLa(1;1,z,x) for La_n(1;1,x) is exp(-x/(1+z))*x*(x-2*(1+z))*(Ei(1,-x/(1+z)) - Ei(1,-x))/(1+z)^4 + exp(x*z/(1+z))*(-x+2*(1+z))/(1+z)^4 +(1+z+x)/(1+z)^3 -2/(1+z)^2, with the exponential integral Ei.

The e.g.f. gLa(1;1,z,x) for the Euler-derivative

x*(d/dx) La_n(1;1,x) is x*exp(-x/(1+z))*(2*(1+z)-x)*

(Ei(1,-x/(1+z)) - Ei(1,-x))/(1+z)^4 + (1+z-x)*(1-exp(x*z/(1+z)))/(1+z)^3.

From this follows La_n(1;1,x) = (n+1)*La_n(1,x) -

x*(d/dx)La_n(1;0,x). For La_n(1;0,x) see A199577 where it is called La_n(1;x).