Similar experiments for Laguerre polynomials work as expected:
(93) -> L := [ laguerreL(0,x)*laguerreL(i,x)*exp(-x) for i in 0..6 ]
(93)
- x - x 2 - x 3 2 - x
[%e , (- x + 1)%e , (x - 4x + 2)%e , (- x + 9x - 18x + 6)%e ,
4 3 2 - x
(x - 16x + 72x - 96x + 24)%e ,
5 4 3 2 - x
(- x + 25x - 200x + 600x - 600x + 120)%e ,
6 5 4 3 2 - x
(x - 36x + 450x - 2400x + 5400x - 4320x + 720)%e ]
Type: List(Expression(Integer))
(94) -> integrate(i,x=0..%plusInfinity) for i in L
(94) [1,0,0,0,0,0,0]
Type: Tuple(Union(f1: OrderedCompletion(Expression(Integer)),f2:
List(OrderedCompletion(Expression(Integer))),fail: failed,pole: potentialPole))
(95) -> L := [ laguerreL(1,x)*laguerreL(i,x)*exp(-x) for i in 0..6 ]
(95)
- x 2 - x 3 2 - x
[(- x + 1)%e , (x - 2x + 1)%e , (- x + 5x - 6x + 2)%e ,
4 3 2 - x
(x - 10x + 27x - 24x + 6)%e ,
5 4 3 2 - x
(- x + 17x - 88x + 168x - 120x + 24)%e ,
6 5 4 3 2 - x
(x - 26x + 225x - 800x + 1200x - 720x + 120)%e ,
7 6 5 4 3 2 - x
(- x + 37x - 486x + 2850x - 7800x + 9720x - 5040x + 720)%e ]
Type: List(Expression(Integer))
(96) -> integrate(i,x=0..%plusInfinity) for i in L
(96) [0,1,0,0,0,0,0]
Type: Tuple(Union(f1: OrderedCompletion(Expression(Integer)),f2:
List(OrderedCompletion(Expression(Integer))),fail: failed,pole: potentialPole))
(97) -> L := [ laguerreL(2,x)*laguerreL(i,x)*exp(-x) for i in 0..6 ]
(97)
2 - x 3 2 - x
[(x - 4x + 2)%e , (- x + 5x - 6x + 2)%e ,
4 3 2 - x
(x - 8x + 20x - 16x + 4)%e ,
5 4 3 2 - x
(- x + 13x - 56x + 96x - 60x + 12)%e ,
6 5 4 3 2 - x
(x - 20x + 138x - 416x + 552x - 288x + 48)%e ,
7 6 5 4 3 2 - x
(- x + 29x - 302x + 1450x - 3400x + 3720x - 1680x + 240)%e ,
8 7 6 5 4 3 2
x - 40x + 596x - 4272x + 15900x - 30720x + 28800x - 11520x
+
1440
*
- x
%e
]
Type: List(Expression(Integer))
(98) -> integrate(i,x=0..%plusInfinity) for i in L
(98) [0,0,4,0,0,0,0]
Type: Tuple(Union(f1: OrderedCompletion(Expression(Integer)),f2:
List(OrderedCompletion(Expression(Integer))),fail: failed,pole: potentialPole))
BTW: Note that in the definition of laguerreL(n,x) there is a prefactor 1/n!
missing compared to [1] or [2].
[1]: http://en.wikipedia.org/wiki/Laguerre_polynomials#The_first_few_polynomials
[2]: http://mathworld.wolfram.com/LaguerrePolynomial.html
Should we fix this?
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