\(L^{p}\)-norms and information entropies of Charlier polynomials (Q1867495)

The Charlier polynomials \(C_n(x,a)\) are defined for \(a>0\) by \[ C_n(x,a)=\sum_{k=0}^n {n \choose k}(-a)^{n-k}x(x-1)\cdots(n-k+1). \] The author finds very accurate asymptotic formulas for the norms \[ \|C_n\|_p=\{\sum_{x=0}^\infty |f(x)|^p w(x) \}^{1/p}, \] where \(w(x)=a^xe^{-a}/x!\), \(0<p<\infty\). For example, when \(n \to \infty\), \[ \|C_n\|_3={(n!)^{2/3}n^{-2/9} \over (2\pi)^{1/3}3^{1/6}} \exp \left (n^{2/3}-{2\over 3}n^{1/3}+{4 \over 9}+ O(n^{-1/3}) \right). \] An asymptotic formula is also found for the entropy \(S_n({\widehat C}) =\sum_{x=0}^\infty {\widehat C}_n(x)^2\log {\widehat C}_n(x)^2 w(x)\) where \({\widehat C}_n=C_n/\sqrt{a^n n!}\), \[ S_n({\widehat C})=(n+a)\log(n/(ae))+3a+1-(1/2)\log 2\pi a+o(1), \qquad n \to \infty. \] Asymptotic formulas are also given for the values \(C_n(n\beta)\), with fixed \(\beta\).