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41: 18.36 Miscellaneous Polynomials

… ►The possibility of generalization to $\alpha =-k$, for $k\in \mathbb{N}$, is implicit in the identity Szegő (1975, page 102), … ►

18.36.6 $${\int }_0^{\mathrm{\infty }}{\widehat{L}}_n^{(k)}\left(x\right){\widehat{L}}_m^{(k)}\left(x\right){\widehat{W}}_k(x)dx=\frac{(n+k)\mathrm{\Gamma }\left(n+k-1\right)}{(n-1)!}{\delta }_{n,m}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , ${\widehat{L}}_n^{(k)}\left(x\right)$: exceptional Laguerre polynomial, $!$: factorial (as in $n!$), $\int$: integral, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.36.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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18.36.7 $$T_k(y)\equiv -x{y}^{\prime \prime }+\frac{x-k}{x+k}((x+k+1)y^{\prime }-y)=(n-1)y.$$

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Symbols:

$\equiv$: equals by definition, $y$: real variable, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.36(vi)

Permalink:

http://dlmf.nist.gov/18.36.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

…

42: Mathematical Introduction

… ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). … ► ►►►

$ℂ$	complex plane (excluding infinity).
…
$\equiv$	equals by definition.
…

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$(a,b]$ or $[a,b)$	half-closed intervals.
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$𝐈$	unit matrix.
$mod$ or modulo	$m\equiv n(modp)$ means $p$ divides $m-n$, where $m$, $n$, and $p$ are positive integers with $m>n$.
…

…

43: 2.11 Remainder Terms; Stokes Phenomenon

… ►As an example consider … ►From §8.19(i) the generalized exponential integral is given by …However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … ►From (2.11.5) and the identity … ►For integrals, see Berry and Howls (1991), Howls (1992), and Paris and Kaminski (2001, Chapter 6). …

44: 16.19 Identities

§16.19 Identities

… ►

16.19.6 $${\int }_0^1{t}^{-a_0}{(1-t)}^{a_0-b_{q+1}-1}{G}_{p,q}^{m,n}(zt;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})dt=\mathrm{\Gamma }\left(a_0-b_{q+1}\right)G_{p+1,q+1}^{m,n+1}(z;\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_{q+1}}),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $dx$: differential of $x$ , $\int$: integral, $p$: nonnegative integer, $q$: nonnegative integer, $z$: 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:

§16.19

Permalink:

http://dlmf.nist.gov/16.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §16.19 and Ch.16

… ►This reference and Mathai (1993, §§2.2 and 2.4) also supply additional identities.

45: 18.33 Polynomials Orthogonal on the Unit Circle

… ►

18.33.1 $$\frac{1}{2\pi \mathrm{i}}{\int }_{|z|=1}{\varphi }_n(z)\overline{{\varphi }_m(z)}w(z)\frac{dz}{z}={\delta }_{n,m},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $w(x)$: weight function, $z$: complex variable, $m$: nonnegative integer, $n$: nonnegative integer and ${\varphi }_n(z)$: polynomials

Referenced by:

§18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(i), §18.33 and Ch.18

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18.33.17 $${\int }_{|z|=1}{\mathrm{\Phi }}_n(z)\overline{{\mathrm{\Phi }}_m(z)}d\mu (z)={\kappa }_n^{-2}{\delta }_{n,m},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Source:

Simon (2005a, (1.1.4))

Referenced by:

§18.33(vi), §18.33(vi), §18.33(vi), Erratum (V1.2.0) Section 18.33

Permalink:

http://dlmf.nist.gov/18.33.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33 and Ch.18

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18.33.20 $$\frac{1}{2\pi \mathrm{i}}{\int }_{|z|=1}{\mathrm{\Phi }}_n(z)\overline{{\mathrm{\Phi }}_m(z)}w(z)\frac{dz}{z}={\kappa }_n^{-2}{\delta }_{n,m}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $w(x)$: weight function, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.33.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33 and Ch.18

… ►

18.33.22 $$p^{\ast }(z)\equiv z^n\overline{p({\overline{z}}^{-1})}=\sum _{k=0}^n\overline{c_{n-k}}{z}^k.$$

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Symbols:

$\overline{z}$: complex conjugate, $\equiv$: equals by definition, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Source:

Simon (2005a, (1.1.7))

Permalink:

http://dlmf.nist.gov/18.33.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

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18.33.26 $${\rho }_n\equiv \sqrt{1-{|{\alpha }_n|}^2}=\frac{{\kappa }_n}{{\kappa }_{n+1}}.$$

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Symbols:

$\equiv$: equals by definition and $n$: nonnegative integer

Proved:

Simon (2005a, (1.5.12))(proved)

Source:

Simon (2005a, (1.5.20))

Permalink:

http://dlmf.nist.gov/18.33.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

…

46: 21.1 Special Notation

… ► ►►►►

$g,h$	positive integers.
…
${𝐈}_g$	$g\times g$ identity matrix.
…
${\oint }_a\omega$	line integral of the differential $\omega$ over the cycle $a$.

…

47: Bibliography G

… ►

Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii), §8.28(vi).

Encodings:

BibTeX

… ►

A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§6.14(iii), §6.17, §6.7(iv).

Encodings:

BibTeX

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A new identity for some six-$j$ symbols. J. Math. Phys. 32 (6), pp. 1430–1432.

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External Links:

ISSN 0022-2488, Document, MathReview

Referenced by:

§34.5(vi).

Encodings:

BibTeX

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Definite integrals of the complete elliptic integral $K$ . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

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A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.

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External Links:

ISSN 0044-2275, Document, MathReview

Referenced by:

§10.71(ii).

Encodings:

BibTeX

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48: Guide to Searching the DLMF

… ►

Table 1: Query Examples

► ►►►►►

Query	Matching records contain
…
int sin	the integral of the sin function
int_$^$ sin	any definite integral of sin
…
int adj sin	$\int$ immediately followed by $\sin$ without any intervening terms.
…

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… ►

Table 2: Wildcard Examples

► ►►►

Query	What it stands for
…
int_$^$ sin	any definite integral of sin.

►

… ►

Table 3: A sample of recognized symbols

► ►►►

Symbols	Comments
…
-=	For equivalence $\equiv$
…

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…

49: Bibliography

… ►

Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

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Notes:

Translated from the Russian by H. H. McFaden

External Links:

ISBN 0-8218-4524-1, MathReview, ZentralBlatt

Referenced by:

§10.22(v).

Encodings:

BibTeX

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Integrals involving Airy functions. J. Phys. A 19 (13), pp. 2663–2665.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

(9.11.11), (9.11.12), (9.11.13), (9.11.14), §9.11(iv).

Encodings:

BibTeX

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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

Encodings:

BibTeX

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$q$-identities of Auluck, Carlitz, and Rogers. Duke Math. J. 33 (3), pp. 575–581.

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External Links:

Document, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§17.9(iv).

Encodings:

BibTeX

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:

ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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50: Bibliography L

… ►

Traité des fonctions elliptiques et des intégrales Eulériennes. Huzard-Courcier, Paris.

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Notes:

vol.1, 1825; vol.2, 1826; three supplements, 1828–1832.

Referenced by:

§19.14(ii), Ch.19.

Encodings:

BibTeX

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Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.

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External Links:

ISSN 0001-8708, Document, MathReview (S. I. Gel’fand), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

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A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

… ►

Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

BibTeX

… ►

Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

BibTeX

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