Let $$h(t) = \frac{\sinh(t)}{t}.$$ Let $$ f_i(t) = h\left(\frac{t}{2^i} \right)^{2^i}, $$ where $i\ge 0$ is an integer.

Is there anyway to get a series expansion of $f_i(t)$ without replacing $i$ with a fixed integer?

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Let $$h(t) = \frac{\sinh(t)}{t}.$$ Let $$ f_i(t) = h\left(\frac{t}{2^i} \right)^{2^i}, $$ where $i\ge 0$ is an integer.

Is there anyway to get a series expansion of $f_i(t)$ without replacing $i$ with a fixed integer?

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