An elementary proof of a conjecture of Saikia on congruences for t-colored overpartitions

The starting point for this work is the family of functions p¯ _-t(n) which counts the number of t-colored overpartitions of n. In recent years, several infinite families of congruences satisfied by p¯ _-t(n) for specific values of t≥ 1 have been proven. In particular, in his 2023 work, Saikia proved a number of congruence properties modulo powers of 2 for p¯ _-t(n) for t= 5 , 7 , 11 , 13 . He also included the following conjecture in that paper: Conjecture: For all n≥ 0 and primes t, we have p¯-t(8n+1)≡0(mod2),p¯-t(8n+2)≡0(mod4),p¯-t(8n+3)≡0(mod8),p¯-t(8n+4)≡0(mod2),p¯-t(8n+5)≡0(mod8),p¯-t(8n+6)≡0(mod8),p¯-t(8n+7)≡0(mod32). Using a truly elementary approach, relying on classical generating function manipulations and dissections, as well as proof by induction, we show that Saikia’s conjecture holds for all odd integers t (not necessarily prime).