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MathGroup Archive 2006

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Re: three parallel methods for one sequence : Hermite like recurance->4th method

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69518] Re: three parallel methods for one sequence : Hermite like recurance->4th method
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Thu, 14 Sep 2006 06:56:56 -0400 (EDT)
  • References: <ee64bu$65o$1@smc.vnet.net>

Thanks to help by Paul Abbott 
and Bob Hamlon ,
I can offer a gerenal solution to the Hermite-error function type
a[n]=a0* a[n - 1] + (b0*n +c0) a[n - 2]: a0,b0,c0 Integers
in the Complex distribution z plane:


Needs["DiscreteMath`RSolve`"];


gf[x_]=ExponentialGeneratingFunction[{a[0] == k0, a[1] == k1,
    a[n]==a0* a[n - 1] + (b0*n +c0) a[n - 2]}, a[n], n, z][[1,1]]

The Bessel type is:
a[n]=(a0*n+b0) a[n - 1] + c0 a[n - 2]: a0,b0,c0 Integers

Substitution of b0=2,c0= -4,a0=2 gets a more or less pure error function
generating function:
Clear[a]

gf1[x_]=ExponentialGeneratingFunction[{a[0]==1, a[1] == 2,
    a[n]==2*a[n-1]-(2n-4)*a[n-2]}, a[n], n, z][[1,1]]

a[0] = 1; a[1] = 2;
a[n_] := a[n] = 2*a[n - 1] - (2n - 4)*a[n - 2]
Table[a[n], {n, 0, 30}]

{1, 2, 4, 4, -8, -40, -16,
   368, 928, -3296, -21440, 16448, 461696, 561536, -9957632, -34515200,
209783296, 1455022592, -3803020288, -57076808704, 22755112960, 
2214428956672,
3518653394944, -85968709390336, -326758168158208, 3301044295639040,
22286480662872064, -120479253456207872, -1399855501381763072,
3706168683871698944, 85804245445122129920}


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