From: danloy@anma.ucl.ac.be (Bernard DANLOY)
Subject: Re: Chebyshev Polynomials
Date: Wed, 21 Feb 2001 17:00:04 +0200
Newsgroups: sci.math.research
Summary: (Derivatives of) Gegenbauer polynomials

In article <3A92A8AE.E03630A7@cl.cam.ac.uk>, Ioannis Ivrissimtzis
<ipi20@cl.cam.ac.uk> wrote :

: Hi,
: 
: Evaluating at 1 the k derivative of the n+k Chebyshev polynomial of
: second type, (using E.J.Scott, Amer. Math. Mon. 1964, 524-525) we get
: 
: $$(2^k)*(k!)*\sum_{i_1+...+i_{k+1}=n} (i_1+1)(i_2+1)...(i_{k+1})$$
: 
: Is there a simpler formula for that sum?
: 

Sure, your polynomial is nothing but a Gegenbauer polynomial of
index +0.5 and it is well known that the derivative of any such
Gegenbauer polynomial is another Gegenbauer polynomial ( more
precisely, everytime you derive, the degree is decreased by one
and the index is incremented by one ).

What you are looking for is just the value at 1 of the Gegenbauer
polynomial of degree n and index k+0.5 ( i let you adjust the
multiplicative factors which depend on the normalization choosen
for the Gegenbauer polynomials ).

Last, but not least, since 1 is an endpoint of the interval which
defines the orthogonality of the Gegenbauer polynomials, it appears
that any such polynomial has there a value which can be easily
expressed as a simple product ( you just need the 2-terms recurrence
involving " adjacent " Jacobi polynomials ).

I don't have the precise answer at hand but you should find all you
need in Szego or Abramowitz & Stegun.

B. Danloy