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In this paper, the structures to a family of biorthogonal polynomials thatapproximate to the Hermite and Generalized Laguerre polynomials are discussedrespectively. Therefore, the asymptotic relation between several orthogonalpolynomials and combinatorial polynomials are derived from the systems, whichin turn verify the Askey scheme of hypergeometric orthogonal polynomials. Asthe applications of these properties, the asymptotic representations of thegeneralized Buchholz, Laguerre, Ultraspherical(Gegenbauer), Bernoulli, Euler,Meixner and Meixner-Pllaczekare polynomials are derived from the theoremsdirectly. The relationship between Bernoulli and Euler polynomials are shown asa special case of the characterization theorem of the Appell sequence generatedby $\alpha$ scaling functions.

Asymptotic properties of biorthogonal polynomials systems related to Hermite and Laguerre polynomials