Topic #7 -------------- OP-SF NET 8.3 ---------------- May 15, 2001 ~~~~~~~~~~~~~ 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. (2) Vadim Kuznetsov (vadim@amsta.leeds.ac.uk) wrote: 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. Readers are encouraged to subscribe and sent their questions and comments to opsftalk. (Note from Webmaster: See About OP-SF TALK.)

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