On semiclassical orthogonal polynomials associated with a Freud‐type weixght

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.