How to solve the following recurrence equation?

Question

How to solve the following recurrence equation ?

$ T(a, b) = T \biggl( \frac{a}{2}, b \biggl) +T\biggl( a, \frac{b}{2} \biggl) \quad \quad \text{ if } a, b \geq 2$;

$ T(a, 1) = T \biggl( \frac{a}{2}, 1 \biggl) \quad \quad \quad \text{ if } a \geq 2$

$ T(1, b) = T \biggl( 1, \frac{b}{2} \biggl) \quad \quad \quad \text{ if } b \geq 2$;

$T(1, 1)=1$.

Where $a$ and $b$ are non negative integers in such a way that both are power of $2$ .

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I am not getting how to solve the recurrence !!! Is there any proper approach for it ?

Answer 1

Hint: Notice the similarity between the given recurrence and the definition of binomial coefficients.

Answer 2

Since $a$ and $b$ are supposed to be natural powers of $2$ consider the function $$P(m,n):=T\bigl(2^m,2^n\bigr)\qquad\bigl(m, \>n\in{\mathbb N}_{\geq0}\bigr)$$ ($P$ for Pascal) instead, and derive a recursion for $P(\cdot,\cdot)$.