differentiation

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21: 7.7 Integral Representations

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22: 16.3 Derivatives and Contiguous Functions

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§16.3(i) Differentiation Formulas

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23: 18.19 Hahn Class: Definitions

… ►The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator $\frac{d}{dx}$ in the case of the classical OP’s is played by a suitable difference operator. … ►

Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

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24: 7.6 Series Expansions

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25: 10.38 Derivatives with Respect to Order

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26: 10.71 Integrals

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27: 18.9 Recurrence Relations and Derivatives

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§18.9(iii) Derivatives

… ►See also the differentiation formulas in (Erdélyi et al., 1953b, §10.9(15))). …

28: 19.4 Derivatives and Differential Equations

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29: 3.7 Ordinary Differential Equations

… ►By repeated differentiation of (3.7.1) all derivatives of $w(z)$ can be expressed in terms of $w(z)$ and $w^{\prime }(z)$ as follows. … ►The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. …

30: 10.40 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$, $I_{\nu }^{\prime }\left(z\right)$, and $K_{\nu }^{\prime }\left(z\right)$ for other ranges of $\mathrm{ph}z$ are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). …