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  "body": "\n## Definition\n\nLet $\\alpha$ be a positive real number, and let $\\Lambda$ be the ring of [[symmetric functions|PAJEHWXW6M]] in variables $x = (x_1, x_2, \\dots)$ with real coefficients.\nFor a [[partition|0JV30XN73Z]] $\\lambda$ with $\\ell(\\lambda)$ parts, let $m_\\lambda$ be the monomial symmetric function and $p_\\lambda = p_{\\lambda_1} p_{\\lambda_2} \\cdots$ the product of [[power sums|0ZNXZB4HNB]].\nLet $z_\\lambda = \\prod_{i \\ge 1} i^{m_i} m_i!$, where $m_i$ is the number of parts of $\\lambda$ equal to $i$.\nDefine a bilinear form on $\\Lambda$ by $\\langle p_\\lambda, p_\\mu \\rangle_\\alpha = \\delta_{\\lambda\\mu}\\, z_\\lambda\\, \\alpha^{\\ell(\\lambda)}$, and write $\\mu \u003c \\lambda$ for the strict dominance order on partitions of the same number.\nThe *monic Jack functions* are the unique family $(P_\\lambda^{(\\alpha)})_\\lambda$ in $\\Lambda$ such that:\n\n1. $P_\\lambda^{(\\alpha)} = m_\\lambda + \\sum_{\\mu \u003c \\lambda} c_{\\lambda\\mu} m_\\mu$ for some coefficients $c_{\\lambda\\mu}$;\n2. $\\langle P_\\lambda^{(\\alpha)}, P_\\mu^{(\\alpha)} \\rangle_\\alpha = 0$ for $\\lambda \\ne \\mu$.\n\nFor a box $s$ of the Young diagram of $\\lambda$, let $a(s)$ be the number of boxes to its right.\nLet $l(s)$ be the number of boxes below $s$.\nThe *Jack function* is $J_\\lambda^{(\\alpha)} = c_\\lambda(\\alpha)\\, P_\\lambda^{(\\alpha)}$ with $c_\\lambda(\\alpha) = \\prod_{s \\in \\lambda} \\big(\\alpha\\, a(s) + l(s) + 1\\big)$.\n\n## Remarks\n\nExample: in degree $2$, $P_{(2)}^{(\\alpha)} = m_{(2)} + \\frac{2}{1+\\alpha} m_{(1,1)}$ and $P_{(1,1)}^{(\\alpha)} = m_{(1,1)}$.\nCheck: $m_{(2)} = p_2$ and $m_{(1,1)} = \\frac12 (p_1^2 - p_2)$, so $P_{(2)}^{(\\alpha)} = \\frac{\\alpha}{1+\\alpha} p_2 + \\frac{1}{1+\\alpha} p_1^2$.\nWith $\\langle p_2, p_2 \\rangle_\\alpha = 2\\alpha$ and $\\langle p_1^2, p_1^2 \\rangle_\\alpha = 2\\alpha^2$, one gets $\\langle P_{(2)}^{(\\alpha)}, m_{(1,1)} \\rangle_\\alpha = \\frac12 \\left( -\\frac{2\\alpha^2}{1+\\alpha} + \\frac{2\\alpha^2}{1+\\alpha} \\right) = 0$.\nThe normalization gives $J_{(2)}^{(\\alpha)} = (1+\\alpha) m_{(2)} + 2 m_{(1,1)}$ and $J_{(1,1)}^{(\\alpha)} = 2 m_{(1,1)}$.\n\nNon-example: the Schur function $s_{(2)} = m_{(2)} + m_{(1,1)}$ satisfies clause 1 but fails clause 2 for $\\alpha \\ne 1$, since $\\langle s_{(2)}, m_{(1,1)} \\rangle_\\alpha = \\frac{\\alpha^2 - \\alpha}{2}$.\n\nAt $\\alpha = 1$ the form is the Hall inner product, $P_\\lambda^{(1)} = s_\\lambda$, and $J_\\lambda^{(1)}$ is $s_\\lambda$ times the product of the hook lengths of $\\lambda$.\nAt $\\alpha = 2$, $J_\\lambda^{(2)}$ are the zonal polynomials.\nExistence of the family is a theorem; it is not automatic, because dominance is only a partial order.\nRestricting to $n$ variables gives the Jack polynomials $J_\\lambda^{(\\alpha)}(x_1, \\dots, x_n)$, which vanish when $\\ell(\\lambda) \u003e n$.\n",
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    "name": "Jack function",
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