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On semiclassical orthogonal polynomials associated with a Freud‐type weixght
Author(s) -
Wang Dan,
Zhu Mengkun,
Chen Yang
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6270
Subject(s) - mathematics , orthogonal polynomials , type (biology) , recursion (computer science) , discrete orthogonal polynomials , differential equation , orthogonality , semiclassical physics , jacobi polynomials , classical orthogonal polynomials , hierarchy , wilson polynomials , hahn polynomials , gegenbauer polynomials , monic polynomial , pure mathematics , mathematical analysis , polynomial , quantum , quantum mechanics , ecology , physics , geometry , algorithm , economics , market economy , biology
The recursion relationship:z P n ( z ) = P n + 1 ( z ) + β nP n − 1 ( z ) , n = 0 , 1 , 2 … is satisfied by all monic orthogonal polynomials in regard to an arbitrary Freud‐type weight function. In current paper, one focuses on the weight functionω ( z ) = | z | αe − z 6 + t z 2, z ∈ R , t ∈ R , α > − 1 to analyze its relativeβ nandP n ( z ) . Through above equation and orthogonality, we find thatβ n ( t ) satisfy the first discrete Painlevé equation I Hierarchy and a high‐order differential‐difference equation, respectively. Then, we find that the asymptotic value ofβ nis settled by Coulomb fluid. Additionally, we talk aboutP n ( z ) with α = 0 , including approximation forP n ( 0 ) andP n ′ ( 0 ) , and bounds forP n ( z ) as n → ∞ are settled.