Uniform Approximations of Bernoulli and Euler Polynomials in Terms of Hyperbolic Functions.

摘要

Bernoulli and Euler polynomials are considered for large values of the order. Convergent expansions are obtained for Bn(nz + 1/2) and En(nz + 1/2) in powers of n to the minus 1 power, with coefficients being rational functions of z and hyperbolic functions of argument 1/2 z. These expansions are uniformly valid for /z plus or minus I /2 pi/ greater than 1/2pi and /z plus or minus 1 power i/pi greater than 1/pi, respectively. For real argument, accuracy of these approximations is restricted to the monotonic region.