Re: Chebyshev values
- To: mathgroup at smc.vnet.net
- Subject: [mg27360] Re: [mg27333] Chebyshev values
- From: Roberto Brambilla <rlbrambilla at cesi.it>
- Date: Thu, 22 Feb 2001 02:25:06 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
At 03.17 21/02/01 -0500, you wrote:
>
>Hello,
>I wonder if someone could tell me how can I find a formulae for the
>values of the derivatives of the Chebyshev polinomials in x=1.
>The table of this constants is easy obtained once you have selected the
>value of n with:
>Table[D[ChebyshevT[k,x],{x,m}]/.x->1,{k,1,n},{m,1,k}]//TableForm
>and the formulae if exist must depend on m and k.
>H. Ramos
>
>
Hi,
ans=D[ChebyshevT[k,x],{x,m}]/.x->1
ans=2^(m-1)(m-1)! k Binomial[m+k-1,k-m]
This result is obtained from Gradshteyn-Ryzhik
formula 8.949 :
D[ChebyshevT[k,x],{x,m}]=
2^(m-1)Gamma[m] k GegenbauerC[k-m,m,x]
and formula 8.937(4)
GegenbauerC[n,q,1]=Binomial[2q+n-1,n]
Bye, Roberto
Roberto Brambilla
CESI
Via Rubattino 54
20134 Milano
tel +39.2.2125.5875
fax +39.2.2125.610
rlbrambilla at cesi.it