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41: 18.17 Integrals

… ►Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively. … ►Formulas (18.17.14) and (18.17.15) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (13.6.19), special cases of (13.3.18) and (13.3.20), respectively. …

42: 1.4 Calculus of One Variable

… ►Stieltjes integrability for $f$ with respect to $\alpha$ can be defined similarly as Riemann integrability in the case that $\alpha (x)$ is differentiable with respect to $x$; a generalization follows below. …

43: 3.1 Arithmetics and Error Measures

… ►Also in this arithmetic generalized precision can be defined, which includes absolute error and relative precision (§3.1(v)) as special cases. …

44: 2.3 Integrals of a Real Variable

… ►Since $q(t)$ need not be continuous (as long as the integral converges), the case of a finite integration range is included. For an extension with more general $t$-powers see Bleistein and Handelsman (1975, §4.1). … ►In general … ►Watson’s lemma can be regarded as a special case of this result. … ►For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. …

45: 18.32 OP’s with Respect to Freud Weights

… ►Of special interest are the cases $Q(x)=x^{2m}$, $m=1,2,\mathrm{\dots }$, and the case $Q(x)=\frac{1}{4}{x}^4-t{x}^2$ ($t\in ℝ$), see §32.15. …For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q(x)=x^4$ see Bo and Wong (1999). ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999). ►Generalized Freud weights have the form …The special case $Q(x)=\frac{1}{4}{x}^4-t{x}^2$ is of particular interest, see Clarkson and Jordaan (2018). …

46: 18.23 Hahn Class: Generating Functions

… ►For the definition of generalized hypergeometric functions see §16.2. … ►

18.23.2 $${}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-x,-x+\beta +N+1}{-};-z){}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{x-N,x+\alpha +1}{-};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n{\left(\alpha +1\right)}_n}{n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

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47: 23.23 Tables

… ►2 in Abramowitz and Stegun (1964) gives values of $\mathrm{\wp }\left(z\right)$, ${\mathrm{\wp }}^{\prime }\left(z\right)$, and $\zeta \left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that ${\omega }_1=1$ and ${\omega }_3=\mathrm{i}a$ (rectangular case), or ${\omega }_1=1$ and ${\omega }_3=\frac{1}{2}+\mathrm{i}a$ (rhombic case), for $a$ = 1. …05, and in the case of $\mathrm{\wp }\left(z\right)$ the user may deduce values for complex $z$ by application of the addition theorem (23.10.1). ►Abramowitz and Stegun (1964) also includes other tables to assist the computation of the Weierstrass functions, for example, the generators as functions of the lattice invariants $g_2$ and $g_3$. …

48: 3.11 Approximation Techniques

… ►Then (in general) a better approximation to $p_n(x)$ is given by … ►For general intervals $[a,b]$ we rescale: … ►The $c_n$ in (3.11.11) can be calculated from (3.11.10), but in general it is more efficient to make use of the orthogonal property (3.11.9). Also, in cases where $f(x)$ satisfies a linear ordinary differential equation with polynomial coefficients, the expansion (3.11.11) can be substituted in the differential equation to yield a recurrence relation satisfied by the $c_n$. … ►The theory of polynomial minimax approximation given in §3.11(i) can be extended to the case when $p_n(x)$ is replaced by a rational function $R_{k,\mathrm{\ell }}(x)$. …

49: 10.16 Relations to Other Functions

… ►For these and general results when $\nu$ is half an odd integer see §§10.47(ii) and 10.49(i). … ►In all cases principal branches correspond at least when $|\mathrm{ph}z|\le \frac{1}{2}\pi$. ►

Generalized Hypergeometric Functions

►

10.16.9 $$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +1\right)}{}_0{}^{}F_1^{}(-;\nu +1;-\frac{1}{4}{z}^2).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.16

Permalink:

http://dlmf.nist.gov/10.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

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50: 2.8 Differential Equations with a Parameter

… ►There are three main cases. … ►for Case I, …for Case II, …for Case III. … ►In Case III the approximating equation is …