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11: 13.16 Integral Representations

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§13.16(iii) Mellin–Barnes Integrals

►If $\frac{1}{2}+\mu -\kappa \ne 0,-1,-2,\mathrm{\dots }$, then … ►If $\frac{1}{2}\pm \mu -\kappa \ne 0,-1,-2,\mathrm{\dots }$, then …where the contour of integration separates the poles of $\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)$ from those of $\mathrm{\Gamma }\left(-\kappa -t\right)$. …where the contour of integration passes all the poles of $\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)$ on the right-hand side.

12: 19.38 Approximations

… ►Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …

13: 18.14 Inequalities

… ►For further inequalities of this type see Koornwinder et al. (2018, §1) and references given there. …

14: 19.29 Reduction of General Elliptic Integrals

… ►Basic integrals of type $I(-{𝐞}_j)$, $1\le j\le h$, are not linearly independent, nor are those of type $I({𝐞}_j)$, $1\le j\le 4$. …

15: 18.34 Bessel Polynomials

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18.34.5_5 $$\frac{2^{1-a}}{\mathrm{\Gamma }\left(1-a\right)}{\int }_0^{\mathrm{\infty }}{y}_n(x;a)y_m(x;a)x^{a-2}{\mathrm{e}}^{-2x^{-1}}dx=\frac{1-a}{1-a-2n}\frac{n!}{{\left(2-a-n\right)}_n}{\delta }_{n,m},$$ $m,n=0,1,\mathrm{\dots },N=\lceil -(1+a)/2\rceil$.

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Symbols:

$y_n(x;a)$: Bessel polynomial, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\lceil x\rceil$: ceiling of $x$ , $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $N$: positive integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Lesky (1996, p.183, case a.3))(proved)

Sources:

Romanovski (1929, Type V); Koekoek et al. (2010, (9.13.2))

Referenced by:

(18.34.7_3), §18.34(ii), §18.34(iii), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E5_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(ii), §18.34 and Ch.18

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16: 32.7 Bäcklund Transformations

… ►Again, since ${\epsilon }_j=\pm 1$, $j=1,2,3$, independently, there are eight distinct transformations of type ${𝒯}_{{\epsilon }_1,{\epsilon }_2,{\epsilon }_3}$. …

17: 14.7 Integer Degree and Order

… ►For $n=0,1,2,\mathrm{\dots }$, …where $W_{-1}(x)=0$, and for $n\ge 1$, … ►

§14.7(ii) Rodrigues-Type Formulas

►For $m=0,1,2,\mathrm{\dots }$, and $n=0,1,2,\mathrm{\dots }$, … ►When and $|h|>1$, …

18: 18.2 General Orthogonal Polynomials

… ►If these $x_k$ satisfy then Szegő type asymptotics outside $[-1,1]$ can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following). …

19: Bibliography K

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Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

Encodings:

BibTeX

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On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

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External Links:

ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. Funktsional. Anal. i Prilozhen. 23 (1), pp. 24–40 (Russian).

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Notes:

In Russian. English translation: Funct. Anal. Appl. 23(1989), no. 1, pp. 19–33

External Links:

ISSN 0374-1990, Document, MathReview (Fedor Malikov), ZentralBlatt

Referenced by:

§20.12(ii).

Encodings:

BibTeX

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20: 16.6 Transformations of Variable

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16.6.2 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,2b-a-1,2-2b+a}{b,a-b+\frac{3}{2}};\frac{z}{4})={(1-z)}^{-a}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}}{b,a-b+\frac{3}{2}};\frac{-27z}{4{(1-z)}^3}).$$

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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, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.6

Permalink:

http://dlmf.nist.gov/16.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.6, §16.6 and Ch.16

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