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MathGroup Archive 2001

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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



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