Lacunary ogf of Laguerre's

[Max Alekseyev]

Karol,

Look at formula (7) at http://mathworld.wolfram.com/LaguerrePolynomial.html
It implies that
sum(z^(2*n)*LaguerreL(2*n,x),n=0..infinity) = (g(x,z) + g(x,-z)) / 2 =
(exp(-xz/(1-z))/(1-z) + exp(xz/(1+z))/(1+z)) / 2
Now to get sum(t^n*LaguerreL(2*n,x),n=0..infinity) make a substitution z^2 = t.

Max

On 6/27/06, Karol PENSON <penson at lptl.jussieu.fr> wrote:
>
>
>
>    Dear Seqfans,
>  Does anyone know a closed form expression for the double-lacunary, ordinary
> generating function
>  of Laguerre polynomials, i.e. , in Maple notation, for
>
>                                   sum(t^n*LaguerreL(2*n,x),
> n=0..infinity) ?
>
>
>  Thanks ,      Karol A. Penson --
> _________________________________________________________________________
> Karol A. PENSON
> Universite Paris 6 | Internet : penson at lptl.jussieu.fr.
> Lab. Physique Theorique des |
> Liquides | http://www.lptl.jussieu.fr/users/penson
> Boite courrier 121 |
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>
>