In this paper, we derive a general formula to express the product of three theta functions as a linear combination of other products of three theta functions. Moreover, we use the main formula to deduce a general formula for the product of two theta functions. Furthermore, as applications, we extract several theorems in the theory of representation of integers as mixed ternary sums involving squares, triangular numbers, generalized pentagonal numbers and generalized octagonal numbers

Abstract Very recently, Pushpa and Vasuki (Arab. J. Math. 11, 355–378, 2022) have proved Eisenstein series identities of level 5 of weight 2 due to Ramanujan and some new Eisenstein identities for level 7 by the elementary way. In their paper, they introduced seven restricted color partition functions, namely $$P^{*}(n), M(n), T^{*}(n), L(n), K(n), A(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>P</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:ms…

Let Bs,t(n) denote the number of (s,t)-regular bipartitions of n. We prove several infinite families of congruences modulo t for Bs,t(n) where (s,t) ? {(2,7), (5?, 7), (3?, 11), (5?, 11), (3?, 17)}, ? ? 1.

See the abstract in the attached pdf.