2015/02/03 by Yan Xu, Xu, Yan
Mathematics · #Advanced Mathematical Identities #Fractional Differential Equations Solutions #Mathematical functions and polynomials #math.CA #math.CO #msc:11B68 #msc:33C45 #msc:41A15 #msc:42C05
paper · pdf · doi:10.48550/arxiv.1503.05387
arxiv created 2015/02/03 · arxiv updated 2015/03/19
In this paper, the structures to a family of biorthogonal polynomials that approximate to the Hermite and Generalized Laguerre polynomials are discussed respectively. Therefore, the asymptotic relation between several orthogonal polynomials and combinatorial polynomials are derived from the systems, which in turn verify the Askey scheme of hypergeometric orthogonal polynomials. As the applications of these properties, the asymptotic representations of the generalized Buchholz, Laguerre, Ultraspherical(Gegenbauer), Bernoulli, Euler, Meixner and Meixner-Pllaczekare polynomials are derived from the theorems directly. The relationship between Bernoulli and Euler polynomials are shown as a special case of the characterization theorem of the Appell sequence generated by α scaling functions.