E.g.f. for Hermite

[FRANCISCO SALINAS]

Karol PENSON wrote:

>Dear Seqfans,I have a question related to my investigations of some
>integer sequences:
>
>  Do you know, or do you know how to obtain in closed form the exponential
>generating function of the Hermite polynomials whose order is a multiple
>of three, i.e. I am looking for , im Maple notation:
>
>        sum(t^n*HermiteH(3*n,x)/n!,n=0..infinity)=  ????
>

Only in terms of the hypergeometric function 2F0 or the Kummer function U 
(and hence, of the Whittaker function W).

See: 'A triple lacunary generating function for Hermite polynomials' by Ira 
M. Gessel and Pallavi Jayawant, 4 Mar 2004 ( 
http://arxiv.org/abs/math.CO/0403086 ), from where the following exponential 
generating function for HermiteH(3*n,x) can be obtained (the paper gives one 
for a different normalization h(3*n,x), 
h(n,x)=(I^n/2^(n/2))*HermiteH(n,-I*x/sqrt(2)) ):

exp(1/6912*(-1+sqrt(1+48*x*t))^3*(1+3*sqrt(1+48*x*t))/(t^2)) * 
hypergeom([1/6, 5/6],[],-432*t^2/((1+48*x*t)^(3/2))) / ((1+48*x*t)^(1/4))    
  (in Maple notation)

Now, for x < 0, we have 2F0(a,b,x)= hypergeom([a, b],[],x) = (-1/x)^a 
*KummerU(a,1+a-b,-1/x), and so hypergeom([1/6, 
5/6],[],-432*t^2/((1+48*x*t)^(3/2)))  can be substituted by 
((1+48*x*t)^(3/2)/(432*t^2))^(1/6) * 
KummerU(1/6,1/3,(1+48*x*t)^(3/2)/(432*t^2)).

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