In mathematics there are a lot of relations, one of which is equivalence ($\equiv$) and its sub-domain equality ($=$). This question doesn't concern the logical equivalence, and stated as:
Question: What is the difference between equivalence and equity within non-logical statements? Which operator to use in diffrent cases ?
A few examples to clarify the question
1) Assume we have a function $A^t(x,y)$ which gives a binomial expansion of $(x+y)^t$, gives exactly the expansion of $(x+y)^t$ for $t\geq 0$, so, since it is essentially the binomial expansion, can we write it as
$$A^t(x,y) \equiv \sum_{r} \binom{t}{r} x^{t-r} y^r ?$$
2) Assume we have a function $P^t(x,y)$ which gives a binomial expansion of $2t+1$ power and plus 1 in implicit form. For example for $t=1$ it gives
$$P^1(x,y) = -1 + x + y - (-1 + x + y) (x + y) (-1 - 3 x - 3 y + 2 (x + y))$$
the binomial expansion of third power is get only after whole result expression is exapnded, i.e
$$P^1(x,y) = 1 + x^3 + 3 x^2 y + 3 x y^2 + y^3$$
In that case, can we write it as
$$P^t(x,y) \equiv 1 + \sum_{r=0} \binom{2t+1}{r} x^{2t+1-r} y^r ?$$
The question arises since, essentially, $P^t(x,y)$ doesnt give binomial exapnsion itself immediately, but only after brackets are expanded.
PS. It is possible that the author of a question doesn't understand the concept of equivalence as a whole, and it is useless outside the math. logics, please, clarify it
Thanks in advance