A130757 - OEIS

These polynomials appear in the radial l=0 (s) wave functions of the isotropic three-dimensional harmonic quantum oscillator with the dimensionless variable x=(r/L)^2 with r>=0 and L^2=h/(m*f0). h is Planck's constant and m and f0 are the mass and the frequency of the oscillator.

See A099174 for relations to the Hermite polynomials and the link in A176230 for operator relations. The infinitesimal generator for this matrix contains A014105.

The row polynomials are P(n,x) = 2^n n! Lag(n,x/2,1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q, with raising operator R = -x^(-2) [x^(3/2) (1 - 2D)]^2 = 3 - x + (4x - 6) D - 4x D^2 with D = d/dx, i.e., R P(n,x) - P(n+1,x). A matrix reresentation of R acting on an o.g.f. (formal power series) is given by the transpose of the production matrix below. The diagonal corresponds to (3 + 4 xD) x^n = (3 + 4n) x^n; the upper diagonal, to -x x^n = -x^(n+1); and the lower diagonal, to (-6 - 4 xD) D x^n = -n (6 + 4(n-1)) x^(n-1), the sequence A002943. See A176230 for a similar relation.

(End)

Exponential Riordan array [1/(1-2x)^(3/2), -x/(1-2x)]. - Paul Barry, Mar 07 2017

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 775, 22.3.9.

a(n,m) = n!*(2^(n-m))*L(1/2,n,m) with L(1/2,n,m) = ((-1)^m)*binomial(n+1/2,n-m)/m!, n >= m >= 0, otherwise 0.

Let IP be the lower triangular matrix with its first subdiagonal equal to the first subdiagonal (cf. A014105) of this entry's unsigned matrix M and with all other elements equal to zero. Then IP is the infinitesimal generator of M, i.e., M = exp(IP). - Tom Copeland, Dec 12 2015

Production matrix is

3, -1;

-6, 7, -1;

0, -20, 11, -1;

0, 0, -42, 15, -1;

0, 0, 0, -72, 19, -1;

0, 0, 0, 0, -110, 23, -1;

0, 0, 0, 0, 0, -156, 27, -1;

0, 0, 0, 0, 0, 0, -210, 31, -1;

0, 0, 0, 0, 0, 0, 0, -272, 35, -1;

... (End)

[1]; [3,-1]; [15,-10,1]; [105,-105,21,-1]; [945,-1260,378,-36,1]; ...

seq(seq(n!*2^(n-m)*(-1)^m*binomial(n+1/2, n-m)/m!, m=0..n), n=0..10); # Robert Israel, Dec 25 2015

Table[n! (2^(n - m)) ((-1)^m) Binomial[n + 1/2, n - m]/m!, {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015 *)

Cf. A021009 (Coefficient table of n!*L(n, 0, x)).

Row sums (signed) give A131441. Row sums (unsigned) give A066224.

Title formula corrected by Tom Copeland, Dec 12 2015