Transcendental nature of $p$-adic digamma values

Abstract

For a fixed prime $p$, Murty and Saradha (2008) studied the transcendental nature of special values of the $p$-adic digamma function, denoted as $ψ_p(r/p)+ γ_p$. This research was later extended by Chatterjee and Gun in 2014, who investigated the case of $ψ_p(r/p^n)+ γ_p$, for any integer $n>1$. In this article, we generalize their results for distinct prime powers and explore the transcendental nature of the $p$-adic digamma values, with at most one exception. Further, we investigate the multiplicative independence of cyclotomic numbers satisfying certain conditions. Using this, we prove the transcendental nature of $p$-adic digamma values corresponding to $ψ_p(r/pq)+ γ_p$, where $p, q$ are distinct primes.