A062137 Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x).

1, 4, -1, 20, -10, 1, 120, -90, 18, -1, 840, -840, 252, -28, 1, 6720, -8400, 3360, -560, 40, -1, 60480, -90720, 45360, -10080, 1080, -54, 1, 604800, -1058400, 635040, -176400, 25200, -1890, 70, -1, 6652800, -13305600, 9313920

Offset: 0

Author

Wolfdieter Lang, Jun 19 2001

Keywords

• sign
• easy
• tabl

Comments

The row polynomials s(n,x) := n!*L(n,3,x) = Sum_{m=0..n} a(n,m)*x^m have e.g.f. exp(-z*x/(1-z))/(1-z)^4. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} |A008297(n,m)|*(-x)^m, n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).

These polynomials appear in the radial part of the l=1 (p-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.

The unsigned version of this triangle is the triangle of unsigned 2-Lah numbers A143497. - Peter Bala, Aug 25 2008

This matrix (unsigned) is embedded in that for n!*L(n,-3,-x). Introduce 0,0,0 to each unsigned row and then add 1,-2,1,4,2,1 to the beginning of the array as the first three rows to generate n!*L(n,-3,-x). - Tom Copeland, Apr 21 2014

From Wolfdieter Lang, Jul 07 2014: (Start)

The standard Rodrigues formula for the monic generalized Laguerre polynomials (also called Laguerre-Sonin polynomials) is Lm(n,alpha,x) := (-1)^n*n!*L(n,3,x) is x^(-alpha)*exp(x)*(d/dx)^n(exp(-x)*x^(n+alpha)).

Another Rodrigues type formula is Lm(n,alpha,x) = exp(x)*x^(-alpha+n+1)*(-x^2*d/dx)^n*(exp(-x)*x^(alpha+1)). This is derivable from the differential - difference relation of Lm(n,alpha,x) which yields first the creation operator formula -(x*d/dx + (-x + alpha + n + 1))*Lm(n,alpha,x) = Lm(n+1,alpha,x) or in the variable q = log(x) the operator -(d/dq + alpha + n + 1 - exp(q)).

The identity (due to Christoph Mayer) (d/dq - (d/dq)W(q))*f(q) = exp(W(q)*d/dq(exp(-W(q)*f(q)) is then iterated with W(q) = W(alpha,n,q) = exp(q) - (alpha + n + 1)*q and a further change of variables leads then to the given result. (End)

Examples

			The triangle a(n,m) begins:
n\m       0        1       2     3    4   5 ...
0:        1
1:        4       -1
2:       20      -10      1
3:      120      -90     18     -1
4:      840     -840    252    -28    1
5:     6720    -8400   3360   -560   40  -1
... Formatted by _Wolfdieter Lang_, Jul 07 2014
For more rows see the link.
n = 2: 2!*L(2,3,x) = 20 - 10*x + x^2.
		

References

• A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.

Links

• Indranil Ghosh, Rows 0..125, flattened
• Wolfdieter Lang, First eleven rows of the triangle.
• Index entries for sequences related to Laguerre polynomials

Crossrefs

For m=0..5 the (unsigned) columns give A001715, A061206, A062141-A062144. The row sums (signed) give A062146, the row sums (unsigned) give A062147.

Cf. A143497. - Peter Bala, Aug 25 2008

Cf. A062139, A105278. - Wolfdieter Lang, Jul 07 2014

Programs

Mathematica

Flatten[Table[((-1)^m)*n!*Binomial[n+3,n-m]/m!,{n,0,9},{m,0,n}]] (* Indranil Ghosh, Feb 23 2017 *)

PARI

row(n) = Vecrev(n!*pollaguerre(n, 3)); \\ Michel Marcus, Feb 06 2021

Formula

a(n, m) = ((-1)^m)*n!*binomial(n+3, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^4), m >= 0.