Can $ I_n(z) = (\frac{z}{2})^n\sum_{k=0}^\infty\frac{(-1)^k}{k!(n+k)!}\frac{1}{k+(n+k)+1}(\frac{z}{2})^{2k}$ be expressed by the Bessel function? We know that the first kind of Bessel function can be expressed by the form of Taylor series
$$
J_n(z) = (\frac{z}{2})^n\sum_{k=0}^\infty\frac{(-1)^k}{k!(n+k)!}(\frac{z}{2})^{2k}
$$
when $n\in\mathbb{N}$. I'm interested in a variant of this form, that's
$$
I_n(z) = (\frac{z}{2})^n\sum_{k=0}^\infty\frac{(-1)^k}{k!(n+k)!}\frac{1}{k+(n+k)+1}(\frac{z}{2})^{2k}.
$$
So, does this variant $I_n(z)$ can be expressed by the Bessel function?
I have some ideas:
$$
I_n(z) = (\frac{z}{2})^n\sum_{k=0}^\infty\frac{(-1)^k}{(k+n+k+1)!}\left(\begin{matrix}n+2k\\k\end{matrix}\right)(\frac{z}{2})^{2k},
$$
and then use the formula
$$
\left(\begin{matrix}n+2k\\k\end{matrix}\right) = \left(\begin{matrix}n+2k+1\\k+1\end{matrix}\right)-\left(\begin{matrix}n+2k\\k+1\end{matrix}\right),
$$
...
 A: We have
\begin{align*}
zI_{2n} (z)  & = \int_0^z {J_{2n} (t)dt}  = \int_0^z {J_0 (t)dt}  - 2\sum\limits_{k = 0}^{n - 1} {J_{2k + 1} (z)} 
\\ & = \frac{\pi }{2}z(J_0 (z){\bf H}_{ - 1} (z) - J_{ - 1} (z){\bf H}_0 (z)) - 2\sum\limits_{k = 0}^{n - 1} {J_{2k + 1} (z)} 
\\ & = \frac{\pi }{2}z(J_1 (z){\bf H}_0 (z) - J_0 (z){\bf H}_1 (z)) + zJ_0 (z) - 2\sum\limits_{k = 0}^{n - 1} {J_{2k + 1} (z)} 
\end{align*}
and
$$zI_{2n + 1} (z) = \int_0^z {J_{2n + 1} (t)dt}  = 1 - J_0 (z) - 2\sum\limits_{k = 0}^n {J_{2k} (z)} .
$$
To establish these, I used $(10.22.9)$, $(10.22.2)$, $(10.4.1)$ and $(11.4.23)$.
A: $$I_n(z)=\left(\frac{z}{2}\right)^{n}\,\sum_{k=0}^\infty\frac{(-1)^k}{k! \, (k+n)!\,(2 k+n+1)}\left(\frac{z}{2}\right)^{2 k}$$ is
$$I_n(z)= \left(\frac{z}{2}\right)^{n}\,\,\frac{\,
   _1F_2\left(\frac{n+1}{2};\frac{n+3}{2},n+1;-\frac{z^2}{4}
   \right)}{(n+1)!}$$ At least for some values of $n$, we see appearing some Bessel functions
$$I_0(z)=\frac{1}{2} \pi  \pmb{H}_0(z) J_1(z)+\frac{1}{2} (2-\pi  \pmb{H}_1(z)) J_0(z)$$
$$I_1(z)=\frac{1}{z}-\frac{J_0(z)}{z}$$
$$I_2(z)=\frac{(\pi  z \pmb{H}_0(z)-4) J_1(z)}{2 z}+\frac{1}{2} (2-\pi  \pmb{H}_1(z)) J_0(z)$$
$$I_3(z)=-\frac{2 J_1(z)}{z^2}+\frac{1}{z}-\frac{J_2(z)}{z}$$
$$I_4(z)=\frac{\left(\pi  z^3 \pmb{H}_0(z)-32\right) J_1(z)}{2 z^3}+\frac{\left(-\pi  z^2
   \pmb{H}_1(z)+2 z^2+16\right) J_0(z)}{2 z^2}$$
$$I_5(z)=-\frac{10 J_3(z)}{z^2}+\frac{\left(z^2-8\right) J_2(z)}{z^3}+\frac{1}{z}$$
The next ones are too long to be typed.
What it seems is that for odd values of $n$ only Bessel J functions appear while, for even values of $n$,they appear at the same times as Struve functions.
