New asymptotic expansions and Pade approximants related to the triple gamma function

In this work, our main focus is to establish asymptotic expansions for the triple gamma function in terms of the triple Bernoulli polynomials. As application, an asymptotic expansion for hyperfactorial function is also obtained. Furthermore, using these asymptotic expansions, Pade approximants related to the triple gamma function are derived as a consequence. The results obtained are new, and their importance is demonstrated by deducing several interesting remarks and corollaries.

Automated summary

In this work, we focus on establishing asymptotic expansions for the triple gamma function using the triple Bernoulli polynomials. We also obtain an asymptotic expansion for the hyperfactorial function as an application. Additionally, we derive Pade approximants related to the triple gamma function using these asymptotic expansions, and demonstrate the importance of the results by deducing interesting remarks and corollaries.