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21: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

Encodings:

BibTeX

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:

Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.

External Links:

Code, Document

Referenced by:

1st item.

Encodings:

BibTeX

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Computation of modified Bessel functions and their ratios. Math. Comp. 28 (125), pp. 239–251.

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External Links:

ISSN 0025-5718, FullText, MathReview (L. Lorch), ZentralBlatt

Referenced by:

§10.74(v).

Encodings:

BibTeX

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

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22: 26.5 Lattice Paths: Catalan Numbers

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Table 26.5.1: Catalan numbers.

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$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
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6	132	13	7 42900	20	65641 20420

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26.5.6 $$C\left(n\right)\sim \frac{4^n}{\sqrt{\pi n^3}},$$ $n\to \mathrm{\infty }$,

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Symbols:

$C\left(n\right)$: Catalan number, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iv), §26.5 and Ch.26

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23: 15.10 Hypergeometric Differential Equation

… ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: … ►

15.10.19 $$w_5(z)=\frac{\mathrm{\Gamma }\left(1-c\right)\mathrm{\Gamma }\left(a-b+1\right)}{\mathrm{\Gamma }\left(a-c+1\right)\mathrm{\Gamma }\left(1-b\right)}{w}_1(z)+{\mathrm{e}}^{(c-1)\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c-1\right)\mathrm{\Gamma }\left(a-b+1\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_2(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_2(z)$: Kummer’s solution $w_2$ and $w_5(z)$: Kummer’s solution $w_5$

Permalink:

http://dlmf.nist.gov/15.10.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.20 $$w_6(z)=\frac{\mathrm{\Gamma }\left(1-c\right)\mathrm{\Gamma }\left(b-a+1\right)}{\mathrm{\Gamma }\left(b-c+1\right)\mathrm{\Gamma }\left(1-a\right)}{w}_1(z)+{\mathrm{e}}^{(c-1)\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c-1\right)\mathrm{\Gamma }\left(b-a+1\right)}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}{w}_2(z).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_2(z)$: Kummer’s solution $w_2$ and $w_6(z)$: Kummer’s solution $w_6$

Permalink:

http://dlmf.nist.gov/15.10.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.26 $$w_2(z)={\mathrm{e}}^{(1-c)\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(2-c\right)\mathrm{\Gamma }\left(b-a\right)}{\mathrm{\Gamma }\left(1-a\right)\mathrm{\Gamma }\left(b-c+1\right)}{w}_5(z)+{\mathrm{e}}^{(1-c)\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(2-c\right)\mathrm{\Gamma }\left(a-b\right)}{\mathrm{\Gamma }\left(1-b\right)\mathrm{\Gamma }\left(a-c+1\right)}{w}_6(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_2(z)$: Kummer’s solution $w_2$ , $w_5(z)$: Kummer’s solution $w_5$ and $w_6(z)$: Kummer’s solution $w_6$

Permalink:

http://dlmf.nist.gov/15.10.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.29 $$w_1(z)={\mathrm{e}}^{b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a-c+1\right)}{\mathrm{\Gamma }\left(a+b-c+1\right)\mathrm{\Gamma }\left(c-b\right)}{w}_3(z)+{\mathrm{e}}^{(b-c)\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a-c+1\right)}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(a-b+1\right)}{w}_5(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_3(z)$: Kummer’s solution $w_3$ and $w_5(z)$: Kummer’s solution $w_5$

Permalink:

http://dlmf.nist.gov/15.10.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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24: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

BibTeX

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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25: 1.11 Zeros of Polynomials

… ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. … ►

1.11.23 $$\sqrt[n]{R}\left(\cos \left(\frac{\alpha +2k\pi }{n}\right)+\mathrm{i}\sin \left(\frac{\alpha +2k\pi }{n}\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $k$: integer, $n$: nonnegative integer and $R$: radius

Permalink:

http://dlmf.nist.gov/1.11.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.11(iv), §1.11 and Ch.1

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26: 12.10 Uniform Asymptotic Expansions for Large Parameter

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12.10.16 $$\mathrm{\Gamma }\left(\frac{1}{2}+z\right)\sim \sqrt{2\pi }{\mathrm{e}}^{-z}{z}^z\sum _{s=0}^{\mathrm{\infty }}\frac{{\gamma }_s}{z^s};$$

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Defines:

${\gamma }_s$: coefficients (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $s$: nonnegative integer

Permalink:

http://dlmf.nist.gov/12.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(ii), §12.10 and Ch.12

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12.10.33 $${𝖠}_{s+1}(\tau )=-4{\tau }^2{(\tau +1)}^2\frac{d}{d\tau }{𝖠}_s(\tau )-\frac{1}{4}{\int }_0^{\tau }\left(20u^2+20u+3\right){𝖠}_s(u)du,$$ $s=0,1,2,\mathrm{\dots }$,

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Defines:

${𝖠}_s(\tau )$: coefficients (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, $s$: nonnegative integer and $\tau$

Permalink:

http://dlmf.nist.gov/12.10.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vi), §12.10 and Ch.12

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$${𝖠}_1(\tau )=-\frac{1}{12}\tau (20{\tau }^2+30\tau +9),$$

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12.10.35 $$U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim 2{\pi }^{\frac{1}{2}}{\mu }^{\frac{1}{3}}g(\mu )\varphi (\zeta )\left(\mathrm{Ai}\left({\mu }^{\frac{4}{3}}\zeta \right)\sum _{s=0}^{\mathrm{\infty }}\frac{A_s(\zeta )}{{\mu }^{4s}}+\frac{{\mathrm{Ai}}^{\prime }\left({\mu }^{\frac{4}{3}}\zeta \right)}{{\mu }^{\frac{8}{3}}}\sum _{s=0}^{\mathrm{\infty }}\frac{B_s(\zeta )}{{\mu }^{4s}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\zeta$: change of variable, $\varphi (\zeta )$: function, $A_s(\zeta )$: coefficients, $B_s(\zeta )$: coefficients and $g(\mu )$: expansion

Referenced by:

§12.10(vii), §12.10(viii)

Permalink:

http://dlmf.nist.gov/12.10.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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12.10.41 $$t=1+w-\frac{1}{10}{w}^2+\frac{11}{350}{w}^3-\frac{823}{63000}{w}^4+\frac{\mathrm{1\hspace{0.33em}50653}}{\mathrm{242\hspace{0.33em}55000}}{w}^5+\mathrm{\cdots },$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\zeta$: change of variable

Referenced by:

§12.11(iii), §12.11(iii)

Permalink:

http://dlmf.nist.gov/12.10.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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27: 6.16 Mathematical Applications

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6.16.1

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/6.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

28: 26.3 Lattice Paths: Binomial Coefficients

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Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

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$m$	$n$
	…
6	1	6	15	20	15	6	1
…

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Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

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$m$	$n$
	…
3	1	4	10	20	35	56	84	120	165
…

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26.3.12 $$\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)\sim \frac{4^n}{\sqrt{\pi n}},$$ $n\to \mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter and $n$: nonnegative integer

Referenced by:

§26.5(iv)

Permalink:

http://dlmf.nist.gov/26.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(v), §26.3 and Ch.26

29: 12.11 Zeros

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12.11.1 $$z_{a,s}={\mathrm{e}}^{\frac{3}{4}\pi \mathrm{i}}\sqrt{2{\tau }_s}\left(1-\frac{\mathrm{i}a{\lambda }_s}{2{\tau }_s}+\frac{2a^2{\lambda }_s^2-8a^2{\lambda }_s+4a^2+3}{16{\tau }_s^2}+O\left({\lambda }_s^3{\tau }_s^{-3}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter, ${\tau }_s$ and ${\lambda }_s$

Permalink:

http://dlmf.nist.gov/12.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(ii), §12.11 and Ch.12

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12.11.2 $${\tau }_s=\left(2s+\frac{1}{2}-a\right)\pi +\mathrm{i}\ln \left({\pi }^{-\frac{1}{2}}{2}^{-a-\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)\right),$$

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Defines:

${\tau }_s$ (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $s$: nonnegative integer and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(ii), §12.11 and Ch.12

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12.11.3 $${\lambda }_s=\ln {\tau }_s-\frac{1}{2}\pi \mathrm{i}.$$

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Defines:

${\lambda }_s$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $s$: nonnegative integer and ${\tau }_s$

Permalink:

http://dlmf.nist.gov/12.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(ii), §12.11 and Ch.12

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12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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30: 20.11 Generalizations and Analogs

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20.11.1 $$G(m,n)=\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n};$$

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Defines:

$G(m,n)$: Gauss sum (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/20.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

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20.11.2 $$\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}\sum _{j=0}^{m-1}{\mathrm{e}}^{\pi \mathrm{i}{j}^{2}n/m}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}G(-n,m).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $G(m,n)$: Gauss sum

Permalink:

http://dlmf.nist.gov/20.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

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