  The asymptotic expansion of digamma function is a starting point for the
derivation of approximants for harmonic sums or Euler-Mascheroni constant. It
is usual to derive such approximations as values of logarithmic function, which
leads to the expansion of the exponentials of digamma function. In this paper
the asymptotic expansion of the function $\exp(p\psi(x+t))$ is derived and
analyzed in details, especially for integer values of parameter $p$. The
behavior for integer values of $p$ is proved and as a consequence a new
identity for Bernoulli polynomials. The obtained formulas are used to improve
know inequalities for Euler's constant and harmonic numbers.
