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  "body": "\n## Definition\n\nLet $\\mu, \\nu \\in \\C$ be such that neither $\\mu + \\nu$ nor $\\mu - \\nu$ is a negative odd integer, and let $z \\in \\C \\setminus (-\\infty, 0]$ with the principal branch of powers.\nThe *Lommel function* $s_{\\mu,\\nu}$ is\n$$s_{\\mu,\\nu}(z) = \\sum_{k=0}^{\\infty} \\frac{(-1)^k\\, z^{\\mu + 1 + 2k}}{\\prod_{j=0}^{k} \\left( (\\mu + 2j + 1)^2 - \\nu^2 \\right)}.$$\n\n## Remarks\n\nThe hypothesis on $\\mu \\pm \\nu$ says exactly that no factor $(\\mu + 2j + 1)^2 - \\nu^2$ vanishes.\nThe function $s_{\\mu,\\nu}$ is a particular solution of\n$$z^2 w'' + z w' + (z^2 - \\nu^2)\\, w = z^{\\mu + 1}.$$\nCheck: the operator $w \\mapsto z^2 w'' + z w' - \\nu^2 w$ multiplies $z^p$ by $p^2 - \\nu^2$, and $z^2 w$ shifts each term up by two powers.\nSo the coefficients $c_k$ of $z^{\\mu + 1 + 2k}$ must satisfy $\\left( (\\mu + 1)^2 - \\nu^2 \\right) c_0 = 1$ and $\\left( (\\mu + 2k + 1)^2 - \\nu^2 \\right) c_k = -c_{k-1}$, which the series does.\nExample: for $\\mu = 1$ and $\\nu = 0$ the first factor is $4$, so $s_{1,0}$ starts with $\\frac{z^2}{4}$.\nIn fact $s_{1,0}$ is $1 - J_0(z)$, where $J_0$ is the [[Bessel function of the first kind|CZCMRDWDYW]].\nA non-example: for $\\mu = -2$ and $\\nu = 1$ one has $\\mu + \\nu = -1$, so the factor with $j = 0$ is $(-2 + 1)^2 - 1 = 0$ and $s_{-2,1}$ is not defined by the series.\nThe second Lommel function $S_{\\mu,\\nu}$ is obtained from $s_{\\mu,\\nu}$ by adding a suitable combination of $J_\\nu$ and $Y_\\nu$ so that it has a simple asymptotic expansion at infinity.\n",
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