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11—20 of 71 matching pages

11: Bibliography

… ►

Y. Ameur and J. Cronvall (2023) Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials. Comm. Math. Phys. 398 (3), pp. 1291–1348.

ⓘ

External Links:: ISSN 0010-3616, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.11(iii), §8.11(iii), Erratum (V1.1.10) for Section 8.11(iii).
Encodings:: BibTeX

…

12: Bibliography T

… ►

K. Takasaki (2001) Painlevé-Calogero correspondence revisited. J. Math. Phys. 42 (3), pp. 1443–1473.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Andrew Pickering), ZentralBlatt
Referenced by:: §32.2(iii).
Encodings:: BibTeX

… ►

N. M. Temme (1985) Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

ⓘ

External Links:: ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt
Referenced by:: §2.4(i).
Encodings:: BibTeX

… ►

N. M. Temme (1996a) Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters. Methods Appl. Anal. 3 (3), pp. 335–344.

ⓘ

External Links:: ISSN 1073-2772, FullText, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §8.12, §8.2(i), §8.2(ii), §8.6(ii).
Encodings:: BibTeX

►

N. M. Temme (1997) Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.

ⓘ

External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §10.19(iii), §10.20(i), §10.74(i).
Encodings:: BibTeX

… ►

P. Terwilliger (2013) The universal Askey-Wilson algebra and DAHA of type

(C^{\vee}_{1},C_{1})

. SIGMA 9, pp. Paper 047, 40 pp..

ⓘ

External Links:: ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

…

13: 26.13 Permutations: Cycle Notation

… ►A permutation with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

can be written as a product of

a_{2}+2a_{3}+\cdots+(n-1)a_{n}=n-(a_{1}+a_{2}+\cdots+a_{n})

transpositions, and no fewer. …

14: Bibliography O

… ►

K. Okamoto (1981) On the

\tau

-function of the Painlevé equations. Phys. D 2 (3), pp. 525–535.

ⓘ

External Links:: ISSN 0167-2789, Document, MathReview (D. A. Lutz), ZentralBlatt
Referenced by:: §32.6(i), §32.6(ii).
Encodings:: BibTeX

… ►

S. Okui (1974) Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

ⓘ

External Links:: ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §10.22(vi), §10.43(vi).
Encodings:: BibTeX

… ►

A. B. Olde Daalhuis and F. W. J. Olver (1995b) On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

ⓘ

External Links:: ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.7(ii).
Encodings:: BibTeX

►

A. B. Olde Daalhuis and F. W. J. Olver (1998) On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.

ⓘ

External Links:: ISSN 1095-7200, Document, MathReview (W. Balser), ZentralBlatt
Referenced by:: §16.25, §2.7(ii), §3.7(iii).
Encodings:: BibTeX

►

A. B. Olde Daalhuis and N. M. Temme (1994) Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview, ZentralBlatt
Referenced by:: §2.4(v).
Encodings:: BibTeX

…

15: Bibliography S

… ►

K. Schulten and R. G. Gordon (1976) Recursive evaluation of

3j

- and

6j

- coefficients. Comput. Phys. Comm. 11 (2), pp. 269–278.

ⓘ

Notes:: Double-precision Fortran provided in 3 parts, $j_{1}$ -recursion for $3j$ -coefficients, $m_{2}$ -recursion for $3j$ -coefficients, and $j_{1}$ -recursion for $6j$ -coefficients
External Links:: Document, Code 1, Code 2, Code 3
Referenced by:: 8th item.
Encodings:: BibTeX

… ►

J. Segura (2011) Bounds for ratios of modified Bessel functions and associated Turán-type inequalities. J. Math. Anal. Appl. 374 (2), pp. 516–528.

ⓘ

External Links:: ISSN 0022-247X, Document, FullText, MathReview (Beyza Caliskan Aslan), ZentralBlatt
Referenced by:: §10.37, §10.37.
Encodings:: BibTeX

… ►

J. Shapiro (1970) Arbitrary

3n-j

symbols for

\mathrm{SU}(2)

. Comput. Phys. Comm. 1 (3), pp. 207–215.

ⓘ

Notes:: Single-precision Fortran
External Links:: Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

… ►

I. M. Sheffer (1939) Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

ⓘ

External Links:: ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

… ►

H. Skovgaard (1954) On inequalities of the Turán type. Math. Scand. 2, pp. 65–73.

ⓘ

External Links:: ISSN 0025-5521, FullText, MathReview (G. Szegő), ZentralBlatt
Referenced by:: §18.14(ii).
Encodings:: BibTeX

…

16: 18.38 Mathematical Applications

… ►

$3j$ and $6j$ Symbols

… ►

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

… ►Algebraic structures were built of which special representations involve Dunkl type operators. …Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. … ►Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and

q

-Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …

17: 27.14 Unrestricted Partitions

… ►For example,

p\left(5\right)=7

because there are exactly seven partitions of

5

5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1

. … ►

27.14.5

\omega(\pm k)=(3k^{2}\mp k)/2,

k=1,2,3,\dots

ⓘ

Defines:: $\omega(k)$ : pentagonal numbers (locally)
Symbols:: $k$ : positive integer
Permalink:: http://dlmf.nist.gov/27.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §27.14(ii), §27.14 and Ch.27

… ►where

K=\pi\sqrt{2/3}

(Hardy and Ramanujan (1918)). … ►After decades of nearly fruitless searching for further congruences of this type, it was believed that no others existed, until it was shown in Ono (2000) that there are infinitely many. Ono proved that for every prime

q>3

there are integers

a

and

b

such that

p\left(an+b\right)\equiv 0\pmod{q}

for all

n

. …

18: 12.12 Integrals

… ►

12.12.1

\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{\mu-1}U\left(a,t\right)\,\mathrm{d}t=% \frac{\sqrt{\pi}2^{-\frac{1}{2}(\mu+a+\frac{1}{2})}\Gamma\left(\mu\right)}{% \Gamma\left(\frac{1}{2}(\mu+a+\frac{3}{2})\right)},

\Re\mu>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\Re$ : real part and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.12 and Ch.12

►

12.12.2

\int_{0}^{\infty}e^{-\frac{3}{4}t^{2}}t^{-a-\frac{3}{2}}U\left(a,t\right)\,% \mathrm{d}t=2^{\frac{1}{4}+\frac{1}{2}a}\Gamma\left(-a-\tfrac{1}{2}\right)\cos% \left((\tfrac{1}{4}a+\tfrac{1}{8})\pi\right),

\Re a<-\tfrac{1}{2}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\Re$ : real part and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §12.12 and Ch.12

►

12.12.3

\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{1}{2}}(x^{2}+t^{2})^{-1}U% \left(a,t\right)\,\mathrm{d}t=\sqrt{\pi/2}\Gamma\left(\tfrac{1}{2}-a\right)x^{% -a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}U\left(-a,x\right),

\Re a<\tfrac{1}{2},x>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\Re$ : real part, $x$ : real variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §12.12 and Ch.12

►

Nicholson-type Integral

►

12.12.4

(U\left(a,z\right))^{2}+(\overline{U}\left(a,z\right))^{2}=\frac{2^{\frac{3}{2% }}}{\pi}\Gamma\left(\tfrac{1}{2}-a\right)\int_{0}^{\infty}\frac{e^{2at+\frac{1% }{2}z^{2}\tanh t}}{\sqrt{\sinh\left(2t\right)}}\,\mathrm{d}t,

\Re a<\tfrac{1}{2}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\overline{U}\left(\NVar{a},\NVar{x}\right)$ : parabolic cylinder function, $\Re$ : real part, $z$ : complex variable and $a$ : real or complex parameter
Notes:: See Durand (1975).
Referenced by:: §12.12
Permalink:: http://dlmf.nist.gov/12.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §12.12, §12.12 and Ch.12

…

19: 10.9 Integral Representations

… ►

{H^{(1)}_{\nu}}\left(z\right)=\frac{\Gamma\left(\tfrac{1}{2}-\nu\right)(\tfrac% {1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1+i\infty}^{(1+)}e^{izt}(t^{2}-1)^{% \nu-\frac{1}{2}}\,\mathrm{d}t,

►

{H^{(2)}_{\nu}}\left(z\right)=\frac{\Gamma\left(\tfrac{1}{2}-\nu\right)(\tfrac% {1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{1-i\infty}^{(1+)}e^{-izt}(t^{2}-1)^{% \nu-\frac{1}{2}}\,\mathrm{d}t,

\nu\neq\tfrac{1}{2},\tfrac{3}{2},\dotsc,|\operatorname{ph}z|<\tfrac{1}{2}\pi

►

Mellin–Barnes Type Integrals

… ►For (10.9.22)–(10.9.25) and further integrals of this type see Paris and Kaminski (2001, pp. 114–116). … ►

Mellin–Barnes Type

…

20: 9.12 Scorer Functions

… ►

e^{\mp 2\pi i/3}\operatorname{Hi}\left(ze^{\mp 2\pi i/3}\right)

… ►

9.12.15

\operatorname{Gi}\left(z\right)=\frac{3^{-2/3}}{\pi}\*\sum_{k=0}^{\infty}\cos% \left(\frac{2k-1}{3}\pi\right)\Gamma\left(\frac{k+1}{3}\right)\frac{(3^{1/3}z)% ^{k}}{k!},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $z$ : complex variable
Proof sketch:: Substitute into (9.12.11) by means of (9.12.17), (9.4.3), (9.2.5), (9.2.6), and use (5.5.3).
Referenced by:: (9.12.16)
Permalink:: http://dlmf.nist.gov/9.12.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §9.12(vi), §9.12 and Ch.9

… ►If

\zeta=\tfrac{2}{3}z^{3/2}

\tfrac{2}{3}x^{3/2}

, and

K_{1/3}

is the modified Bessel function (§10.25(ii)), then … ►

Mellin–Barnes Type Integral

… ►where the integration contour separates the poles of

\Gamma\left(\tfrac{1}{3}+\tfrac{1}{3}t\right)

from those of

\Gamma\left(-t\right)

. …