Transcendental nature of p-adic digamma values

For a fixed prime p , Murty and Saradha (Acta Arith 133:349–362, 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 / p q ) + γ p , where p , q are distinct primes.