---
tag: 7KA78K00QJ
kind: definition
name: Bessel–Clifford function
msc: [33C10]
defines: 16WR5EARVC
uses: [0H5SYBSED9, PY6K9AWR33, CZCMRDWDYW, 2ET068BZZH]
sources:
  - url: https://en.wikipedia.org/wiki/Bessel%E2%80%93Clifford_function
---

## Definition

Let $n \in \C$ and let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
The *Bessel–Clifford function* of order $n$ is $\mathcal{C}_n \colon \C \to \C$,
$$\mathcal{C}_n(z) = \sum_{k=0}^{\infty} \frac{1}{\Gamma(n + k + 1)}\, \frac{z^k}{k!},$$
where $\frac{1}{\Gamma}$ is the [[reciprocal gamma function|PY6K9AWR33]], equal to $0$ at the non-positive integers.

## 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|CZCMRDWDYW]] and the [[modified Bessel function|2ET068BZZH]], $J_n(x) = \left( \frac{x}{2} \right)^n \mathcal{C}_n\left( -\frac{x^2}{4} \right)$ 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.