Re: linear secod order homogeneous differential equation recursions

• To: mathgroup at smc.vnet.net
• Subject: [mg69868] Re: [mg69852] linear secod order homogeneous differential equation recursions
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Tue, 26 Sep 2006 00:59:29 -0400 (EDT)
• References: <200609250753.DAA11496@smc.vnet.net>

```Roger Bagula wrote:
> Again I have a difficult problem.
> I have this factorial based recursion:
> a[n] = (a0*n^2 + b0*n + c0)*a[n - 2]/(n*(n - 1))
>
> a[n]*n!=Integer
> I want to get a form:
> b[n]=a[n]*n!
> I've tried your RSolve , but it doesn't give a form out I can use.
>
> M athematica:
> Clear[a, a0, b0, c0]
> 0 = 1; b0 = -2; c0 = -1;
> a[n_] := a[n] = (a0*n^2 + b0*n + c0)*a[n
> - 2]/(n*(n - 1)); a[0] = 1; a[1] = 1;
> Table[ExpandAll[a[n]*n!], {n, 0, 30}]
> {1, 1, -1, 2, -7, 28, -161,
>   952, -7567, 59024, -597793, 5784352, -71137367, 821377984, -11879940289,
> 159347328896, -2649226684447,
>    40474221539584, -760328058436289, 13032699335746048, -272957772978627751,
> 5187014335626927104, -119828462337617582689, 2500140909772178864128,
> -63149599651924466077103, 1435080882209230668009472,
> -39342200583148942366035169, 967244514609021470238384128,
> -28601779823949281100107567863, 756385210424254789726416388096,
> -23996893272293446842990249437057}
>
> Clear[a]
> a[n_] := a[n] = -(n^2 - n - 1)*a[n
> - 2]/(n*(n - 1)); a[0] = 1; a[1] = 1;
> Table[a[n]*n!, {n, 0, 30}]
> {1, 1, -1, -5,
>    11, 95, -319, -3895, 17545, 276545, -1561505, -30143405, 204557155,
> 4672227775, -37024845055, -976495604975, 8848937968145, 264630308948225,
> -2698926080284225, -90238935351344725, 1022892984427721275,
> 37810113912213439775, -471553665821179507775, -19094107525667787086375,
> 259826069867469908784025, 11437370407875004464738625,
> -168627119343987970800832225, -8017596655920378129781776125,
> 127313475104710917954628329875, 6502270887951426663253020437375,
> -110635409865993787702572018661375}
>
> Clear[a]
> a[n_] := a[n] = (n - 2)*a[n
> - 2]/(n*(n - 1)); a[0] = 1; a[1] = 1;
> Table[a[n]*n!, {n, 0, 30}]
> {1, 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0,
> 2027025, 0,
> 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0,
> 7905853580625, 0,
> 213458046676875, 0}
>
>
> I didn't find these on my own:
>  they were recursion examples in an old  differential equation text
> I had. So far none of my results as a[n]*n! haven't been in OEIS.
> These are solution to differtential equations of the type:
> y''+p(x)*y'+q(x)*y=0
> f(x)=y=Sum[a[n]*(x-x0)^n,{n,0,Infinity}]
> So in Taylor series terms I'm getting ( I think):
>  b[n]=D[f(x0),{x,n}]
> a[n]=b[n]/n!
>
> I need a gerenal form for the derivatives b[n].

Here is one example, run in a development kernel.

In[14]:= InputForm[an = a[n] /. First[RSolve[
{a[n] == -(n^2 - n - 1)*a[n- 2]/(n*(n - 1)),
a[0] ==1, a[1] == 1}, a[n], n]]]

Out[14]//InputForm=
(I^n*2^(-2 + n)*((-I)*Gamma[3/4 - Sqrt[5]/4]*Gamma[3/4 + Sqrt[5]/4] +
I*(-1)^n*Gamma[3/4 - Sqrt[5]/4]*Gamma[3/4 + Sqrt[5]/4] +
2*Gamma[5/4 - Sqrt[5]/4]*Gamma[5/4 + Sqrt[5]/4] +
2*(-1)^n*Gamma[5/4 - Sqrt[5]/4]*Gamma[5/4 + Sqrt[5]/4])*
Gamma[3/4 - Sqrt[5]/4 + n/2]*Gamma[3/4 + Sqrt[5]/4 + n/2])/
(Gamma[3/4 - Sqrt[5]/4]*Gamma[5/4 - Sqrt[5]/4]*Gamma[3/4 + Sqrt[5]/4]*
Gamma[5/4 + Sqrt[5]/4]*Gamma[1 + n])

FullSimplify with Assumptions->{n>0,Element[n,Integers]}]
gives the result below.

In[17]:= InputForm[an = FullSimplify[an,
Assumptions->{n>0,Element[n,Integers]}]]

Out[17]= ((-2)^n*Cos[(Sqrt[5]*Pi)/2]*Gamma[(3 - Sqrt[5] + 2*n)/4]*
Gamma[(3 + Sqrt[5] + 2*n)/4]*(-2*Cos[(n*Pi)/2]*Gamma[(5 - Sqrt[5])/4]*
Gamma[(5 + Sqrt[5])/4] + Gamma[(3 - Sqrt[5])/4]*Gamma[(3 + Sqrt[5])/4]*
Sin[(n*Pi)/2]))/(Pi^2*n!)

In[22]:= FullSimplify[Table[an*n!, {n,0,12}]]
Out[22]= {1, 1, -1, -5, 11, 95, -319, -3895, 17545, 276545, -1561505,
-30143405, 204557155}

Daniel Lichtblau
Wolfram Research

```

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