Symmetric Orthogonal L-Polynomials in the Complex Plane

In [7], we have shown the relation that exists between real symmetric orthogonal polynomials and certain inverse symmetric orthogonal L-polynomials. We obtain this relation through the transformation (1.1) x ( u )   =     ( u   −   β / u ) / ( 2 α ) ,         α > 0 ,       β > 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq735.tif"/> which represents a one to one correspondence between Γ₁ = {x: −∞ < x < ∞ } and Λ₁ = {u: 0 < u < ∞ }. The principal results given in [7] can be summarized as follow: For any const > 0 , let ξ ( u )   =   c o n s t   u − 1 / 2   ζ   ( x ( u ) ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq736.tif"/>