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 \(\psi _p(r/p)+ \gamma _p\) ${\psi }_p(r/p)+{\gamma }_p$ . This research was later extended by Chatterjee and Gun in 2014, who investigated the case of \(\psi _p(r/p^n)+ \gamma _p\) ${\psi }_p(r/p^n)+{\gamma }_p$ , for any integer \(n>1\) $n>1$ . 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 \(\psi _p(r/pq)+ \gamma _p\) ${\psi }_p(r/pq)+{\gamma }_p$ , where p, q are distinct primes.