Euler polynomials and continued fractions of Stieltjes-Rogers
It is well known that the Euler polynomial \(E_{2n}(x)\) with \(n\geq 0\) can be expressed as a polynomial \(H_n(u)\) where \(u=x(x-1)\). The authors extend the definition of \(H_n(u)\) to \(n<0\) in which case \(H_n(u)\) is a formal power series in \(u\). If \(H_n(u)= \sum_kh_{n,k}u^k\) then the coefficients satisfy \[ 2(k+1)(2k+1)h_{n,k+1} +(k+2)(k+1)h_{n,k+2} =2n(2n-1)h_{n-1,k} \] (with \(h_{n,0} =\delta_{n,0}\) and \(h_{n,n}=1)\) for all \(n,k\in\mathbb{Z}\). This generalises recent work of Hammersley and of Horadam which corresponds to the case \(n,k\geq 0\). A number of generating functions associated with Euler polynomials, etc., can be expressed as formal Laplace transforms of expressions involving hyperbolic functions. For example \[ \sum^\infty_{n=0} H_n(u)t^{2n+2}=t{\mathcal L} \biggl(\bigl[\cosh (x-1/2)z\bigr]/ \cosh(1/2z),t\biggr) \] and \[ \sum^\infty_{n =0} E_{2n+1}(x)t^{2n+2}={\mathcal L} \biggl(\bigl[\sinh(x-1/2)z\bigr]/ \cosh(1/2z), t\biggr) \] where \[ {\mathcal L}\bigl(f(z), t\bigr)= \int^\infty_0 f(z)e^{-z/t}dz. \] The authors deduce several elegant continued fraction expansions for such generating functions some of which were published without proof by Stieltjes in 1895 and Rogers in 1906/7.
scientific article; zbMATH DE number 1230035