Expression for ( or approximation of ) series of odd terms in series representation of Bessel function of first kind?

Question

We know:

$$J_v(z) = \sum_{k=0}^{\infty}\frac{(-1)^k}{\Gamma(k+v+1)k!}\bigr(\frac{z}{2}\bigl)^{2k+v} \ \ (Eq. 1)$$ $$s.t.\ (v,k)\in \mathbb N,z\in \mathbb R$$

courtesy of Introduction to 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 $v=1$, then: $$J_1(z) = \sum_{k=0}^{\infty}\frac{(-1)^k}{\Gamma(k+2)k!}\bigr(\frac{z}{2}\bigl)^{2k+1} \ \ \ \ (Eq.\ 2) $$ But I am interested in approximations for a series composed of the odd terms of the series representation above, $(Eq. 2)$. Namely $k \in \mathbb N_{odd}$.

Answer 1

This is a generalized hypergeometric function, with

\begin{align}\sum_{k=0}^\infty\frac{(-1)^{2k+1}}{\Gamma(2k+v+2)(2k+1)!}\left(\frac z2\right)^{4k+v+2}&=-\left(\frac z2\right)^{v+2}\sum_{k=0}^\infty\frac1{\Gamma(2k+v+2)(2k+1)!}\left(\frac z2\right)^{4k}\\&=-\left(\frac z2\right)^{v+2}{}_0F_2\left(;-\frac32,\frac{v+3}2;\frac{z^4}{64}\right)\end{align}

One can also view this as a difference of the alternating and non-alternating series given by

$$J_v(z)=\sum_{k=0}^\infty\frac{(-1)^k}{\Gamma(k+v+1)k!}\left(\frac z2\right)^{2k+v}$$

$$I_v(z)=\sum_{k=0}^\infty\frac1{\Gamma(k+v+1)k!}\left(\frac z2\right)^{2k+v}$$

$$\frac{J_v(z)-I_v(z)}2=\sum_{k=0}^\infty\frac{(-1)^{2k+1}}{\Gamma(2k+v+2)(2k+1)!}\left(\frac z2\right)^{4k+v+2}$$

where $I$ is the modified Bessel function.