Determinant inequalities for sieved ultraspherical polynomials (Q5945168)
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scientific article; zbMATH DE number 1656127
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Determinant inequalities for sieved ultraspherical polynomials |
scientific article; zbMATH DE number 1656127 |
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Determinant inequalities for sieved ultraspherical polynomials (English)
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29 May 2002
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other special orthogonal polynomials and functions
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other functions coming from differential, difference and integral equations
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0.9112621
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0.90339506
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0.9019761
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0.90117896
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0.90045655
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0.8974586
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0.89311945
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The authors prove the so-called Turán and Turán-type inequalities for orthogonal polynomials. The paper starts proving a very general theorem: If \(\{P_n\}\) are orthogonal polynomials on \(a\leq x\leq b\) then for each \(n\) there exists \(c_n\), \(a\leq c_n\leq b\) such that NEWLINE\[NEWLINE {P_n^2(x)\over P_n^2(c_n)}-{P_{n+1}(x)\over P_{n+1}(c_n)} {P_{n-1}(x)\over P_{n-1}(c_n)}\geq 0,\qquad a\leq x\leq b. NEWLINE\]NEWLINE which is called a Turán inequality. After that, the authors prove similar inequalities for shieved ultraspherical polynomials of the second kind.
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