Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions

Bernoulli and Euler polynomials are considered for large values of the order. Convergent expansions are obtained for $B_n(nz+1/2)$ and $E_n(nz+1/2)$ in powers of $n^{-1$, with coefficients being rational functions of $z$ and hyperbolic functions of argument $1/2z$. These expansions are uniformly valid for $vert zpm i/2pivert>1/2pi$ and $vert zpm i/pivert>1/pi$, respectively. For real argument, accuracy of these approximations is restricted to the monotonic region. The range of validity of the uniformity parameter $z$ is enlarged, respectively, to regions of the form $vert zpm i/2(m+1)pivert>1/2(m+1)pi$ and $vert zpm i/(2m+1)pivert>1/(2m+1)pi$, $m=1,2,3,..$, by adding certain combinations of incomplete gamma functions to those uniform expansions. In addition, the convergence of these improved expansions is stronger and also for real argument the accuracy of these improved approximations is better in the oscillation region.