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 $$ \left(\begin{matrix}n+2k\\k\end{matrix}\right) = \left(\begin{matrix}n+2k+1\\k+1\end{matrix}\right)-\left(\begin{matrix}n+2k\\k+1\end{matrix}\right), $$ ...

Solution 1:

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).

Solution 2:

$$I_n(z)=\left(\frac{z}{2}\right)^{n}\,\sum_{k=0}^\infty\frac{(-1)^k}{k! \, (k+n)!\,(2 k+n+1)}\left(\frac{z}{2}\right)^{2 k}$$ 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.