Asymptotic expansions of exponentials of digamma function and identity for Bernoulli polynomials

TL;DR

This paper derives detailed asymptotic expansions for the exponential of the digamma function, especially for integer parameters, leading to new identities for Bernoulli polynomials and improved bounds for Euler's constant and harmonic numbers.

Contribution

It introduces a novel asymptotic expansion for exp(pψ(x+t)) and establishes new identities for Bernoulli polynomials, enhancing approximation methods for harmonic sums.

Findings

Derived asymptotic expansion for exp(pψ(x+t))

Proved behavior for integer p values

Improved inequalities for Euler's constant and harmonic numbers

Abstract

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.