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- Title
Semi-classical Orthogonal Polynomials Associated with a Modified Gaussian Weight.
- Authors
Ding, Yadan; Min, Chao
- Abstract
We are concerned with the monic orthogonal polynomials with respect to the modified Gaussian weight w (x) = w (x ; s) : = e - N [ x 2 + s (x 6 - x 2) ] , x ∈ R with parameters N > 0 and s ∈ [ 0 , 1 ] . Using the ladder operator approach and associated compatibility conditions, we show that the recurrence coefficient β n (s) satisfies a nonlinear fourth-order difference equation, which is the second member of the discrete Painlevé I hierarchy. We find that the orthogonal polynomials satisfy a second-order ordinary differential equation, with all the coefficients expressed in terms of β n (s) . By considering the s evolution, we derive the differential-difference equation for the recurrence coefficient β n (s) . We also obtain some relations between the Hankel determinant D n (s) , the sub-leading coefficient p (n , s) of the monic orthogonal polynomials and the recurrence coefficient β n (s) . Finally, we study the large n asymptotics of the recurrence coefficient β n (s) , the sub-leading coefficient p (n , s) and the logarithmic derivative of D n (s) for fixed N > 0 . We also consider the asymptotics of β n (s) when n/N is fixed as n → ∞ .
- Subjects
ORTHOGONAL polynomials; DIFFERENCE operators; NONLINEAR difference equations; DIFFERENTIAL-difference equations; ORDINARY differential equations
- Publication
Results in Mathematics / Resultate der Mathematik, 2024, Vol 79, Issue 3, p1
- ISSN
1422-6383
- Publication type
Article
- DOI
10.1007/s00025-024-02137-z