Expression for series using odd terms of series representation of Bessel function of the first kind?

Question

We know:

$$_()=∑_{=0}^∞\frac{(−1)^}{Γ(++1)!}\bigr(\frac{}{2}\bigl)^{2+}\ (.1)$$ $$.. (,)∈ℕ,∈ℝ$$

Where $z>>1$

courtesy of WolframAlpha's entry on Bessel functions and that there are quite a few ways of approximating Bessel functions of the first kind (written above) asymptotically. To move on, if we take =1, then: $$_1()=∑_{=0}^∞\frac{(−1)^}{Γ(+2)!}\bigr(\frac{}{2}\bigl)^{2+1}\ (. 2)$$ But I am interested in approximations for a series composed of the odd terms of the series representation above, $(.2)$: namely $∈ℕ$.

$$???=∑_{=0}^∞\frac{(−1)^{2k+1}}{Γ(2+3)(2k+1)!}\bigr(\frac{}{2}\bigl)^{4+3}\ (. 3)$$ Someone has suggested: Mittag-Leffler functions in another question but I believe they missed the second factorial in the denominator of the series representation.

Answer 1

$$\sum_{k=0}^\infty\frac{(−1)^{2k+1}}{\Gamma(2+3)(2k+1)!}\left(\frac{}{2}\right)^{4+3}= \frac{J_1(z)-I_1(z)}{2} $$

Answer 2

Let \begin{align*} f(K,z) = \mathrm{sgn}(z)\sum_{k=0}^K \frac{−1}{(2k+2)\Gamma(2k+2)^2} \left(\frac{|z|}{2} \right)^{4k+3} \end{align*}

Your series is $$ f(\infty,z) = \frac{1}{2} (J_1(z)-I_1(z))$$ where $J$ and $I$ are the Bessel function of the first kind and the modified Bessel function of the first kind, respectively.

For real arguments, the relative error in $$ f(\max(\{10,\lfloor z/3 \rfloor\}),z) $$ is less than $0.02$ (i.e., ${}<2\%$). If you replace "$z/3$" with "$z/2$", the relative error is bounded by $4 \times 10^{-6}$.