Asymptotic \(\gamma\)-forms generated by multiple orthogonal polynomials. (English. Russian original) Zbl 1317.33004
Proc. Steklov Inst. Math. 272, Suppl. 2, S168-S173 (2011); translation from Sovrem. Probl. Mat. 9, 55-62 (2007).
The authors find the \(n\)-large asymptotics of the integrals
\[
q_n=\int_0^{\infty}Q_ne^{-x}\,dx
\]
and
\[
f_n=\int_0^{\infty}Q_n(x)\ln x e^{-x}\,dx
\]
involving polynomials
\[
Q_n(x)=\frac{1}{(n!)^2}\frac{e^x}{x-1}\frac{d^n}{dx^n}x^n\frac{d^n}{dx^n}(x-1)^{2n+1}x^ne^{-x}.
\]
The authors utilize an appropriate modification of the classical steepest descent method.
Reviewer: Andrei A. Kapaev (St. Petersburg)
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 30E10 | Approximation in the complex plane |
References:
| [1] | A. I. Aptekarev, A. Branquinho, and W. van Assche, ”MultipleOrthogonal Polynomials for ClassicalWeights,” Trans. Amer.Math. Soc. 355(10), 3887–3914 (2003). · Zbl 1033.33002 · doi:10.1090/S0002-9947-03-03330-0 |
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