Asymptotic analysis of difference equations with quadratic coefficients (Q312422)
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scientific article; zbMATH DE number 6627583
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Asymptotic analysis of difference equations with quadratic coefficients |
scientific article; zbMATH DE number 6627583 |
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Asymptotic analysis of difference equations with quadratic coefficients (English)
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15 September 2016
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asymptotic analysis
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difference equations
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associated polynomials
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orthogonal polynomials
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Askey scheme
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Meixner-Pollaczek polynomials
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Meixner polynomials
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Laguerre polynomials
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0.9228468
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0.9226989
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0.92232555
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0.9203131
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0.91293323
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0.90784407
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The author studies a class of monic polynomials satisfying the second-order difference equation with quadratic coefficients NEWLINE\[NEWLINE \pi_{n+1}(x) = (x-dn-d_0)\pi_n(x)-(an^2+bn+c)\pi_{n-1}(x), \quad \pi_0(x) = 1, \quad \pi_1(x)= x-d_0NEWLINE\]NEWLINE where the parameters \( a, b, c, d \) and \( d_0 \) are all real constants. By introducing the Plancherel-Rotach scale \( x=ny \), he obtains asymptotic formulas of the orthogonal polynomials in the outer and oscillatory regions, respectively. Following the Askey scheme he gives three typical examples, namely Meixner-Pollaczek polynomials, Meixner polynomials, and Laguerre polynomials.
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