Bessel–Clifford function

Remarks

The series converges for every $z$, so $\mathcal{C}_n$ is entire in $z$. It solves $z y'' + (n + 1) y' - y = 0$: with $a_k = \frac{1}{k!\, \Gamma(n + k + 1)}$, the coefficient of $z^k$ in $z y'' + (n+1) y'$ is $(k + 1)(n + k + 1)\, a_{k+1} = a_k$. Example: $\mathcal{C}_{-1/2}(z) = \frac{\cosh(2\sqrt{z})}{\sqrt{\pi}}$, since $k!\, \Gamma\left( k + \frac{1}{2} \right) = \frac{(2k)!\sqrt{\pi}}{4^k}$. For the Bessel function of the first kind and the modified Bessel function, $J_n(x) = \left( \frac{x}{2} \right)^n \mathcal{C}_n\left( -\frac{x^2}{4} \right)$ and $I_n(x) = \left( \frac{x}{2} \right)^n \mathcal{C}_n\left( \frac{x^2}{4} \right)$. A non-example: $J_0$ is not $\mathcal{C}_0$, since $J_0(x) = \mathcal{C}_0(-x^2/4)$ and, for instance, $\mathcal{C}_0(1) = \sum_k \frac{1}{(k!)^2} > 2 > 1 \ge \abs{J_0(1)}$. In hypergeometric notation, $\mathcal{C}_n(z) = \frac{1}{\Gamma(n+1)}\, {}_0F_1(; n + 1; z)$ when $n$ is not a negative integer.