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  "body": "\n## Summary\n\nThe Meixner polynomials $M_n(x; \\beta, c)$ are the orthogonal polynomials of the negative binomial (Pascal) distribution on the nonnegative integers.\nThey are a discrete analogue of the [[Laguerre polynomials|WAMRHWW82R]], which arise from them in a limit.\nThey sit in the [[Askey scheme|RR2S05C09B]] at the level of the ${}_2F_1$ series, together with the Jacobi and Krawtchouk polynomials.\n\n## Explanations\n\n### L1 human\n\nRepeat a trial with success probability $c$ until a fixed number $\\beta$ of failures, and count the successes.\nSuch counts follow a negative binomial distribution, with weights proportional to $\\frac{(\\beta)_x}{x!} c^x$ on $x = 0, 1, 2, \\dots$.\nThe Meixner polynomials are the polynomials in $x$ that are perpendicular to each other for these weights.\nThey play for this distribution the role that the Laguerre polynomials play for the gamma distribution.\n",
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