Moments of the Gamma Distribution Involving Logarithmic Factors

This work presents a unified analysis of the sequence of integrals I_{n,m} = ∫_0^∞ x^n e^{-x} (ln x)^m dx, which are equal to the m-th derivatives of the Gamma function at positive integers, Γ^{(m)}(n+1).

The central result expresses the normalized integrals I_{n,m}/n! as a complete Bell polynomial in the polygamma functions (the derivatives of the log-Gamma function). This framework provides a powerful and unified way to generate closed-form expressions, recurrence relations, and exponential generating functions.

We explore a combinatorial interpretation, showing these values act as the regularized moments of the number of cycles in a random permutation. The paper also provides a complete asymptotic analysis for large n and discusses the analytic continuation of these integrals into the complex plane.

All theoretical results are supported by a fully runnable Python code that uses the mpmath library for high-precision numerical integration and evaluation, serving as a robust verification tool.

This paper serves as a definitive expository reference, synthesizing results from special functions, combinatorics, and asymptotic analysis into a single coherent narrative.