Asymptotic behavior of linear forms with polynomial coefficients for some functions of Stieltjes type (Q1083583)

Notation: \(\epsilon =e^{2\pi i/r}\), r a fixed natural number \(\Delta_ j=\{t\epsilon^ j|\) \(0\leq t<\infty \}\) \[ j\in R=\{0,1,2,...,r- 1\},\quad \Delta =\cup_{R}\Delta_ j,\quad d\mu = | x|^ p e^{-| x|} | dx|,\quad x\in \Delta,\quad p>-1, \] \[ \mu_ j=\mu |_{\Delta j},\quad f_ j = \int_{\Delta_ j}\frac{d\mu_ j}{x-z},\quad \tilde f_ j(r,p;z) = - \sum^{\infty}_{j=0}e^{j^ n} \Gamma (n+j+1)e^{-n-1} \] is the asymptotic expansion of \(f_ j\) in a sector at \(\infty\). The problem is to determine \(r+1\) polynomials of degree \(\leq n\), \(T_ j(n;x)\), \(j\in R\), \(U_ n(x)\) so that the expansion at \(\infty\) of \[ \tilde L_ n(z)=\sum_{R}T_ j(n,z)\tilde f_ j(z)-U_ n(z) \] starts with at least the \((nr+r)th\) power of 1/z. The \(T_ j\) exist and are (essentially) unique and satisfy \(T_ j(n,x)=\epsilon^ j T(n,\epsilon^{-j} x)\). The orthogonality property, Rodrigues formula and generating function \(\sum^{\infty}_{n=0}T(n,x)w^ n\) are studied, as well as the asymptotic behavior of T(n,x) and \[ L_ n(z)=\sum_{R}T_ j(n,z)f_ j(z)-U_ n(z). \]