       Re: linear second order homogeneous differential equation recursions

• To: mathgroup at smc.vnet.net
• Subject: [mg69896] Re: linear second order homogeneous differential equation recursions
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Wed, 27 Sep 2006 06:03:51 -0400 (EDT)

```Paul Abbott wrote:

>In article <ef834t\$ggg\$1 at smc.vnet.net>,
> Roger Bagula <rlbagula at sbcglobal.net> wrote:
>
>
>
>>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!
>>
>>
>
>The solution is
>
>  a[n] n! == (a0^(n/2) (2^n C + (-2)^n C) *
>    Gamma[(b0 + 2 a0 (n + 2) - Sqrt[b0^2 - 4 a0 c0])/(4 a0)] *
>    Gamma[(b0 + 2 a0 (n + 2) + Sqrt[b0^2 - 4 a0 c0])/(4 a0)])/
>   (Gamma[(4 a0 + b0 - Sqrt[b0^2 - 4 a0 c0])/(4 a0)] *
>    Gamma[(4 a0 + b0 + Sqrt[b0^2 - 4 a0 c0])/(4 a0)])
>
>Cheers,
>Paul
>
>_______________________________________________________________________
>Paul Abbott                                      Phone:  61 8 6488 2734
>School of Physics, M013                            Fax: +61 8 6488 1014
>The University of Western Australia         (CRICOS Provider No 00126G)
>AUSTRALIA                               http://physics.uwa.edu.au/~paul
>
>
>
Paul Abbott,
Thank you for your help. Both you and Daniel Lichtblau are better at
getting solution out of Mathematica than I am;
with five conmstants  ( C and C added to {a0,b0,c0}) and  even
Not exactly a simple Binet type...
seemed related to Hypergeometrics which your solution seems to confirm.
Is there any way to convert it in Mathematica to an hypergeometric form?
I had further generalized the form by
f[n]-> Binet of the Fibonacci :
f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == a[ n - 1] + a[n
- 2], a == 0, a == 1}, a[n], n][] // FullSimplify]
Clear[a]
a[n_] := a[n] = f[n]*a[n
- 2]/(n*(n - 1)); a = 1; a = 1;
Table[ExpandAll[a[n]*n!], {n, 0, 30}]
{1, 1, 1, 2, 3, 10, 24, 130, 504, 4420, 27720, 393380, 3991680, 91657540,
1504863360, 55911099400, 1485300136320, 89290025741800, 3838015552250880,
373321597626465800, 25964175210977203200, 4086378207619294646800,
459851507161617245875200, 117103340295746126693347600,
21322394684069868456741273600, 8785678105688353155168403690000,
2588389457883293541569193426124800, 1725665322163094950031867515982420000,
822618641999347403739646931950148812800,
887387152950606153059937200876123854180000,
684451614889137013807535833259801818202112000}

That idea was inspired by the Fibonacci Polynomials and the Bernoulli
/Polynomials/ constants
which I found:
g[n_]=Gamma[n+1]*BernoulliB[n-1]...= (n!*Bernoulli[n-1])
Table[g[n],{n,1,30}]
{1, -1, 1, 0, -4, 0, 120, 0, -12096, 0, 3024000, 0, -1576143360, 0,
1525620096000, 0, -2522591034163200,
0, 6686974460694528000, 0, -27033456071346536448000, 0,
160078872315904478576640000, 0, -1342964491649083924630732800000, 0,
15522270327163593186886877184000000, 0,
-241364461951740682229320388129587200000, 0}
Also:
h[n_] = (n + 1)*n!*BernoulliB[n];
Table[h[n], {n, 0, 30}]

I figure this is an even wider range of solutions that I had first thought,
with a wide range of applications in
different areas.
Roger Bagula

```

• Prev by Date: Re: Re: why does not the Mathematica kernel seem to 'multi-task' between computations in different windows?
• Next by Date: Re: strange random behavior ?
• Previous by thread: "the sum of squares removed by fitting..."?
• Next by thread: Re: linear second order homogeneous differential equation recursions