Askey scheme

Remarks

Example of an arrow, Jacobi to Laguerre: $\lim_{\beta \to \infty} P_n^{(\alpha, \beta)}\left( 1 - \frac{2x}{\beta} \right) = L_n^{(\alpha)}(x)$. Check for $n = 1$: $P_1^{(\alpha, \beta)}(1 - 2x/\beta) = (\alpha + 1) - (\alpha + \beta + 2) \frac{x}{\beta} \to \alpha + 1 - x = L_1^{(\alpha)}(x)$. Example 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 $1 - \frac{x}{a}$. Special cases inside a vertex, such as the Gegenbauer, Legendre and Chebyshev polynomials inside the Jacobi family, are usually not drawn as separate vertices. Non-example: the Bernoulli polynomials are not in the scheme, because they are not orthogonal for any positive measure on $\R$. Among Appell sequences, only the Hermite polynomials, up to rescaling and shifting the variable, are orthogonal. The $q$-analogue of the scheme, with the Askey–Wilson polynomials at the top, is called the $q$-Askey scheme.