Polynomial approximation (with decreasing error preferably) of Bessel functions

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The functions: $$J_1(a\sqrt x) = ?? $$ $$I_1(a\sqrt x) = ?? $$ s.t. $$a, \sqrt x \in R $$

See the first answer to a question I've asked previously for an example of what I'd be looking for.

Please provide a function approximating the following hypergeometric series?