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  "body": "\n## Definition\n\nLet $n \\in \\C$ and let $\\Gamma$ be the [[gamma function|0H5SYBSED9]].\nThe *Bessel–Clifford function* of order $n$ is $\\mathcal{C}_n \\colon \\C \\to \\C$,\n$$\\mathcal{C}_n(z) = \\sum_{k=0}^{\\infty} \\frac{1}{\\Gamma(n + k + 1)}\\, \\frac{z^k}{k!},$$\nwhere $\\frac{1}{\\Gamma}$ is the [[reciprocal gamma function|PY6K9AWR33]], equal to $0$ at the non-positive integers.\n\n## Remarks\n\nThe series converges for every $z$, so $\\mathcal{C}_n$ is entire in $z$.\nIt 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 $(k + 1)(n + k + 1)\\, a_{k+1} = a_k$.\nExample: $\\mathcal{C}_{-1/2}(z) = \\frac{\\cosh(2\\sqrt{z})}{\\sqrt{\\pi}}$, since $k!\\, \\Gamma\\left( k + \\frac{1}{2} \\right) = \\frac{(2k)!\\sqrt{\\pi}}{4^k}$.\nFor 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 $I_n(x) = \\left( \\frac{x}{2} \\right)^n \\mathcal{C}_n\\left( \\frac{x^2}{4} \\right)$.\nA non-example: $J_0$ is not $\\mathcal{C}_0$, since $J_0(x) = \\mathcal{C}_0(-x^2/4)$ and, for instance, $\\mathcal{C}_0(1) = \\sum_k \\frac{1}{(k!)^2} \u003e 2 \u003e 1 \\ge \\abs{J_0(1)}$.\nIn hypergeometric notation, $\\mathcal{C}_n(z) = \\frac{1}{\\Gamma(n+1)}\\, {}_0F_1(; n + 1; z)$ when $n$ is not a negative integer.\n",
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