Extract from OP-SF NET


Topic #7  --------------   OP-SF NET 8.3  ----------------  May 15, 2001
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From: OP-SF NET Editor (muldoon@yorku.ca)
Subject: opsftalk

Three topics were recently raised in opsftalk, the discussion group of our
Activity Group.

(1) Irine Peng asked about a special _3F_2 for which she did not have a
"closed-form"  formula, in all cases but which she wanted to show to be non-zero.
R. Vidunas showed that it is not always non-zero and Victor Adamchik showed how
to write the _3F_2 as a finite sum.

I want to ask for opinions about one strange spectral problem' where, in a
sense, the spectrum is the same as the eigenvalues.  Let us call this thing an
eigen-vector-value-problem'. Technically, one solves the following system of n
equations:
(*)         \sum_k  A_{ijk} x_k = x_i x_j,        i,j,k=1,...,n,
assuming that it is compatible. Compatibility can be rephrased as the
commutativity of the matrices A_{i..} and of the matrices A_{.j.} collected from
the tensor A_{ijk}. Anyway, it is assumed.
HOW TO SOLVE SUCH (QUADRATIC?) PROBLEMS EFFECTIVELY? ON COMPUTER?
Notice that spectrum here is the same' as eigenvalues, therefore for a given A
we have to find only the vector x.

(*) can be realized as either (inverted) linearization problem or, when
n=infinity and the operator A acts in some functional space, as a product formula
for a special function.

So, my question can also be changed to the following: How to use product formulas
for producing effective numerical methods for calculating the special function
itself?  After all (*) is a very particular spectral problem and one (maybe) can
use this fact to invent a fast numerical algorithm in order to calculate x.
Notice that x can be out of the hypergeometric class, so that a problem of its
calculation can be very non-trivial task.

(3) A question appearing in sci.math.research on the generalizing (to Jacobi
polynomials) the well-known expression for Chebyshev polynomials involving
sin((n+1)arccos x) led Tom Koornwinder to suggest a sum formula for
Gegenbauer (ultraspherical) polynomials in terms of trigonometric functions with
coefficients which can be found from known results in the literature.  A similar
result is known for q-ultraspherical polynomials.

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