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On a Conjecture Concerning Monotonicity of Zeros of Ultraspherical Polynomials

✍ Scribed by Dimitar K. Dimitrov

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
235 KB
Volume
85
Category
Article
ISSN
0021-9045

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✦ Synopsis

Let C * n , n=0, 1, ..., *>&1Γ‚2 be the ultraspherical (Gegenbauer) polynomials, orthogonal on (&1, 1) with respect to the weight (1&x 2 ) *&1Γ‚2 . Denote by n, k (\*), k=1, ..., [nΓ‚2] the positive zeros of C \* n enumerated in decreasing order. The problem of finding the ``extremal'' function f for which the products f (\*) n, k (*) are increasing functions of * is of recent interest. Ismail, Letessier, and Askey conjectured that f (*)=(*+1) 1Γ‚2 is the function to solve this problem. We prove the conjecture for sufficiently large n and some related results. 1996 Academic Press, Inc. 0 Z$ n, k (*)= f $ n, k (*) n, k (\*)+ f n, k (\*) $ n, k (*), article no. 0030 88

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