differentiation

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31: 13.3 Recurrence Relations and Derivatives

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§13.3(ii) Differentiation Formulas

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32: 15.5 Derivatives and Contiguous Functions

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§15.5(i) Differentiation Formulas

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33: 18.3 Definitions

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With the property that ${\{p_{n+1}^{\prime }(x)\}}_{n=0}^{\mathrm{\infty }}$ is again a system of OP’s. See §18.9(iii).

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34: 36.10 Differential Equations

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35: 18.2 General Orthogonal Polynomials

… ►If the orthogonality discrete set $X$ is $\{0,1,\mathrm{\dots },N\}$ or $\{0,1,2,\mathrm{\dots }\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta }}_x$, the forward-difference operator, or by ${\nabla }_x$, the backward-difference operator; compare §18.1(i). … ►

Degree lowering and raising differentiation formulas and structure relations

►For a large class of OP’s $p_n$ there exist pairs of differentiation formulas …

36: 1.10 Functions of a Complex Variable

… ►is analytic in $D$ and its derivatives of all orders can be found by differentiating under the sign of integration. …

37: 3.3 Interpolation

… ►For theory and applications see Stenger (1993, Chapter 3).

38: 9.12 Scorer Functions

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9.12.16 $${\mathrm{Gi}}^{\prime }\left(z\right)=\frac{3^{-1/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\cos \left(\frac{2k+1}{3}\pi \right)\mathrm{\Gamma }\left(\frac{k+2}{3}\right)\frac{{(3^{1/3}z)}^k}{k!}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Differentiate (9.12.15).

Permalink:

http://dlmf.nist.gov/9.12.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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9.12.18 $${\mathrm{Hi}}^{\prime }\left(z\right)=\frac{3^{-1/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{k+2}{3}\right)\frac{{(3^{1/3}z)}^k}{k!}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Differentiate (9.12.17).

Permalink:

http://dlmf.nist.gov/9.12.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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9.12.31 $${\int }_0^z\mathrm{Hi}\left(-t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }+\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $x$: real variable, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Except for the constant term, this be verified by termwise integration of (9.12.27). To evaluate the constant term replace $z$ by $-x$ ($\le 0$) in (9.12.20) and integrate (§1.5(v)) to obtain $\pi {\int }_0^x\mathrm{Hi}\left(-t\right)dt={\int }_0^{\mathrm{\infty }}(1-{\mathrm{e}}^{-xt}){\mathrm{e}}^{-\frac{1}{3}{t}^3}{t}^{-1}dt$. Next, integrate the right-hand side of this equation by parts—integrating the factor $t^{-1}$ and differentiating the rest. As $x\to \mathrm{\infty }$ the asymptotic expansions of ${\int }_0^{\mathrm{\infty }}x{\mathrm{e}}^{-xt}{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ and ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ follow from (2.3.9). Also, ${\int }_0^{\mathrm{\infty }}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ can be found by replacing $\frac{1}{3}{t}^3$ by $t$ and referring to the first of (5.9.18). (This equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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39: 5.9 Integral Representations

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40: 9.9 Zeros

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§9.9(iii) Derivatives With Respect to $k$

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