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  "body": "\n## Definition\n\nThe *Askey scheme* is a directed graph whose vertices are families of hypergeometric orthogonal polynomials.\nThe vertices are grouped in levels by the type of hypergeometric series that defines them:\n\n1. ${}_4F_3$: the [[Wilson|1CPF7SWQ3S]] and [[Racah polynomials|P6JYNDX5DR]].\n2. ${}_3F_2$: the continuous dual Hahn, continuous Hahn, [[Hahn|Y1M82VZYS2]] and dual Hahn polynomials.\n3. ${}_2F_1$: the Meixner–Pollaczek, [[Jacobi|TRD7DSF2KR]], [[Meixner|CEC3DESCFT]] and [[Krawtchouk polynomials|CJGKW80KBR]].\n4. ${}_1F_1$ and ${}_2F_0$: the [[Laguerre|WAMRHWW82R]], [[Bessel|M5NXATQ48S]] and [[Charlier polynomials|6KH8KVYS3R]].\n5. ${}_2F_0$: the [[Hermite polynomials|MGACY98E9H]].\n\nThere is an arrow from a family $A$ to a family $B$ when the polynomials of $B$ are limits of rescaled polynomials of $A$.\nIn such a limit, parameters of $A$ tend to special values or to infinity.\n\n## Remarks\n\nExample of an arrow, Jacobi to Laguerre: $\\lim_{\\beta \\to \\infty} P_n^{(\\alpha, \\beta)}\\left( 1 - \\frac{2x}{\\beta} \\right) = L_n^{(\\alpha)}(x)$.\nCheck for $n = 1$: $P_1^{(\\alpha, \\beta)}(1 - 2x/\\beta) = (\\alpha + 1) - (\\alpha + \\beta + 2) \\frac{x}{\\beta} \\to \\alpha + 1 - x = L_1^{(\\alpha)}(x)$.\nExample of an arrow, Krawtchouk to Charlier: $\\lim_{N \\to \\infty} K_n\\left( x; \\frac{a}{N}, N \\right) = C_n(x; a)$; for $n = 1$ both sides equal $1 - \\frac{x}{a}$.\nSpecial cases inside a vertex, such as the Gegenbauer, Legendre and Chebyshev polynomials inside the Jacobi family, are usually not drawn as separate vertices.\nNon-example: the Bernoulli polynomials are not in the scheme, because they are not orthogonal for any positive measure on $\\R$.\nAmong Appell sequences, only the Hermite polynomials, up to rescaling and shifting the variable, are orthogonal.\nThe $q$-analogue of the scheme, with the [[Askey–Wilson polynomials|RSNZNG0WA1]] at the top, is called the $q$-Askey scheme.\n",
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