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11: 19.29 Reduction of General Elliptic Integrals

… ►and

\alpha,\beta,\gamma,\delta

is any permutation of the numbers

1,2,3,4

, then … ►

U_{\alpha\beta}=U_{\beta\alpha}=U_{\gamma\delta}=U_{\delta\gamma},

… ►Moreover, the requirement that one limit of integration be a branch point of the integrand is eliminated without doubling the number of standard integrals in the result. … ►The integrals in (19.29.4), (19.29.7), and (19.29.8) are

I(\boldsymbol{{0}})

I(\mathbf{e}_{\alpha}-\mathbf{e}_{\delta})

, and

I(\mathbf{e}_{\alpha}-\mathbf{e}_{5})

, respectively. … ►where

\alpha,\beta,\gamma

is any permutation of the numbers

1,2,3

, and …

12: 26.11 Integer Partitions: Compositions

… ►

c\left(n\right)

denotes the number of compositions of

n

, and

c_{m}\left(n\right)

is the number of compositions into exactly

m

parts.

c\left(\in\!T,n\right)

is the number of compositions of

n

with no 1’s, where again

T=\{2,3,4,\ldots\}

. … ►

26.11.2

c_{m}\left(0\right)=\delta_{0,m},

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $c_{\NVar{m}}\left(\NVar{n}\right)$ : number of compositions of $n$ into exactly $m$ parts and $m$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

… ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).

13: 31.7 Relations to Other Functions

… ►

31.7.1

{{}_{2}F_{1}}\left(\alpha,\beta;\gamma;z\right)=\mathit{H\!\ell}\left(1,\alpha% \beta;\alpha,\beta,\gamma,\delta;z\right)=\mathit{H\!\ell}\left(0,0;\alpha,% \beta,\gamma,\alpha+\beta+1-\gamma;z\right)=\mathit{H\!\ell}\left(a,a\alpha% \beta;\alpha,\beta,\gamma,\alpha+\beta+1-\gamma;z\right).

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

►Other reductions of

\mathit{H\!\ell}

to a

{{}_{2}F_{1}}

, with at least one free parameter, exist iff the pair

(a,p)

takes one of a finite number of values, where

q=\alpha\beta p

. … ►

31.7.2

\mathit{H\!\ell}\left(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1% ;z\right)={{}_{2}F_{1}}\left(\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-% z)^{2}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

►

31.7.3

\mathit{H\!\ell}\left(4,\alpha\beta;\alpha,\beta,\tfrac{1}{2},\tfrac{2}{3}(% \alpha+\beta);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta% ;\tfrac{1}{2};1-(1-z)^{2}(1-\tfrac{1}{4}z)\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

… ►

\gamma=\delta=\epsilon=\tfrac{1}{2},

…

14: 36.7 Zeros

… ►

\Delta z=\frac{9\pi}{2z_{n}^{2}},

►

\Delta x=\frac{6\pi}{z_{n}}.

… ►

36.7.6

\exp\left(-2\pi i\left(\frac{z-z_{n}}{\Delta z}+\frac{2x}{\Delta x}\right)% \right)\*{\left(2\exp\left(\frac{-6\pi ix}{\Delta x}\right)\cos\left(\frac{2% \sqrt{3}\pi y}{\Delta x}\right)+1\right)}=\sqrt{3}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $y$ : real parameter, $z$ : real parameter, $m$ : integer and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §36.7(iii), §36.7 and Ch.36

►The rings are almost circular (radii close to

(\Delta x)/9

and varying by less than 1%), and almost flat (deviating from the planes

z_{n}

by at most

(\Delta z)/36

). …,

y=0

), the number of rings in the

m

th row, measured from the origin and before the transition to hairpins, is given by …

15: Mathematical Introduction

… ►The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ► ►►►►

$\mathbb{C}$	complex plane (excluding infinity).
…
$\delta_{j,k}$ or $\delta_{jk}$	Kronecker delta: 0 if $j\neq k$ ; 1 if $j=k$ .
$\Delta$ (or $\Delta_{x}$ )	forward difference operator: $\Delta f(x)=f(x+1)-f(x)$ .
…

… ►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …

16: 34.2 Definition: $\mathit{3j}$ Symbol

… ►The corresponding projective quantum numbers

m_{1},m_{2},m_{3}

are given by … ►

See accompanying text — Figure 34.2.1: Angular momenta $j_{r}$ and projective quantum numbers $m_{r}$ , $r=1,2,3$ . Magnify

… ►

34.2.4

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{1}-j_{2}-m_{3}}}\Delta(j_{1}j_{2}j_{3% })\left((j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!% (j_{3}-m_{3})!\right)^{\frac{1}{2}}\*\sum_{s}\frac{(-1)^{s}}{s!(j_{1}+j_{2}-j_% {3}-s)!(j_{1}-m_{1}-s)!(j_{2}+m_{2}-s)!(j_{3}-j_{2}+m_{1}+s)!(j_{3}-j_{1}-m_{2% }+s)!},

ⓘ

Defines:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol
Symbols:: $!$ : factorial (as in $n!$ ), $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $\Delta(j_{1}j_{2}j_{3})$ : product and $s$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

… ►

34.2.5

\Delta(j_{1}j_{2}j_{3})=\left(\frac{(j_{1}+j_{2}-j_{3})!(j_{1}-j_{2}+j_{3})!(-% j_{1}+j_{2}+j_{3})!}{(j_{1}+j_{2}+j_{3}+1)!}\right)^{\frac{1}{2}},

ⓘ

Defines:: $\Delta(j_{1}j_{2}j_{3})$ : product (locally)
Symbols:: $!$ : factorial (as in $n!$ ) and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

… ►

34.2.6

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_% {3})!(j_{2}+j_{3}-m_{1})!}{\Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left% (\frac{(j_{1}+m_{1})!(j_{3}-m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})% !(j_{3}+m_{3})!}\right)^{\frac{1}{2}}\*{{{}_{3}F_{2}}\left(-j_{1}-j_{2}-j_{3}-% 1,-j_{1}+m_{1},-j_{3}-m_{3};-j_{1}-j_{2}-m_{3},-j_{2}-j_{3}+m_{1};1\right)},

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $!$ : factorial (as in $n!$ ), $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

…

17: 31.3 Basic Solutions

… ►

\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)

denotes the solution of (31.2.1) that corresponds to the exponent

0

z=0

and assumes the value

1

there. If the other exponent is not a positive integer, that is, if

\gamma\neq 0,-1,-2,\dots

, then from §2.7(i) it follows that

\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)

exists, is analytic in the disk

|z|<1

, and has the Maclaurin expansion … ►Solutions of (31.2.1) corresponding to the exponents

0

and

1-\delta

z=1

are respectively, … ►

31.3.7

(1-z)^{1-\delta}\*\mathit{H\!\ell}\left(1-a,((1-a)\gamma+\epsilon)(1-\delta)+% \alpha\beta-q;\alpha+1-\delta,\beta+1-\delta,2-\delta,\gamma;1-z\right).

ⓘ

Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

… ►For example,

\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)

is equal to …

18: 2.11 Remainder Terms; Stokes Phenomenon

… ►uniformly when

\theta\in[-\pi+\delta,\pi-\delta]

(

\delta>0

) and

|\alpha|

is bounded. … ►One is uniformly valid for

-\pi+\delta\leq\operatorname{ph}z\leq 3\pi-\delta

with bounded

|\alpha|

, and achieves uniform exponential improvement throughout

0\leq\operatorname{ph}z\leq\pi

: … ►For the sector

-3\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta

the conjugate result applies. … ►We now compute the forward differences

\Delta^{j}

j=0,1,2,\dots

, of the moduli of the rounded values of the first 6 neglected terms: … ►Similar improvements are achievable by Aitken’s

\Delta^{2}

-process, Wynn’s

\epsilon

-algorithm, and other acceleration transformations. …

19: 18.26 Wilson Class: Continued

… ►

18.26.4_1

R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)=Q_{y}\left(n;\gamma,% \delta,N\right),

ⓘ

Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer
Referenced by:: §18.26(i), Erratum (V1.2.0) Section 18.26
Permalink:: http://dlmf.nist.gov/18.26.E4_1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

… ►

18.26.4_2

R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,\delta\right)=R_{y}\left(n% (n+\alpha+\beta+1);\gamma,\delta,\alpha,\beta\right).

ⓘ

Symbols:: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $y$ : real variable, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer
Referenced by:: §18.26(i), Erratum (V1.2.0) Section 18.26
Permalink:: http://dlmf.nist.gov/18.26.E4_2
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

… ►For comments on the use of the forward-difference operator

\Delta_{x}

, the backward-difference operator

\nabla_{x}

, and the central-difference operator

\delta_{x}

, see §18.2(ii). … ►

18.26.16

\frac{\Delta_{y}\left(R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,% \delta\right)\right)}{\Delta_{y}\left(y(y+\gamma+\delta+1)\right)}=\frac{n(n+% \alpha+\beta+1)}{(\alpha+1)(\beta+\delta+1)(\gamma+1)}\*R_{n-1}\left(y(y+% \gamma+\delta+2);\alpha+1,\beta+1,\gamma+1,\delta\right).

ⓘ

Symbols:: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $\Delta$ : forward difference operator, $y$ : real variable, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer
Referenced by:: §16.4(iii)
Permalink:: http://dlmf.nist.gov/18.26.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iii), §18.26 and Ch.18

►

18.26.17

\frac{\Delta_{y}\left(R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)% \right)}{\Delta_{y}\left(y(y+\gamma+\delta+1)\right)}=-\frac{n}{(\gamma+1)N}\*% R_{n-1}\left(y(y+\gamma+\delta+2);\gamma+1,\delta,N-1\right).

ⓘ

Symbols:: $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $\Delta$ : forward difference operator, $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer
Referenced by:: §18.26(iii)
Permalink:: http://dlmf.nist.gov/18.26.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iii), §18.26 and Ch.18

…

20: 31.16 Mathematical Applications

… ►

31.16.1

\mathit{Hp}_{n,m}\left(x\right)\mathit{Hp}_{n,m}\left(y\right)=\sum_{j=0}^{n}A% _{j}{\sin}^{2j}\theta\*P^{(\gamma+\delta+2j-1,\epsilon-1)}_{n-j}\left(\cos% \left(2\theta\right)\right)P^{(\delta-1,\gamma-1)}_{j}\left(\cos\left(2\phi% \right)\right),

… ►

31.16.3

A_{0}=\frac{n!}{{\left(\gamma+\delta\right)_{n}}}\mathit{Hp}_{n,m}\left(1% \right),\quad Q_{0}A_{0}+R_{0}A_{1}=0,

ⓘ

Symbols:: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\beta$ : real or complex parameter, $A_{j}$ : coefficients, $Q_{j}$ and $R_{j}$
Referenced by:: §31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3)
Permalink:: http://dlmf.nist.gov/31.16.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.0.21):: The initial data $A_{0}$ , previously missing, has now been included.
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

… ►

31.16.5

P_{j}=\frac{(\epsilon-j+n)j(\beta+j-1)(\gamma+\delta+j-2)}{(\gamma+\delta+2j-3% )(\gamma+\delta+2j-2)},

ⓘ

Defines:: $P_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

►

31.16.6

Q_{j}=-aj(j+\gamma+\delta-1)-q+\frac{(j-n)(j+\beta)(j+\gamma)(j+\gamma+\delta-% 1)}{(2j+\gamma+\delta)(2j+\gamma+\delta-1)}+\frac{(j+n+\gamma+\delta-1)j(j+% \delta-1)(j-\beta+\gamma+\delta-1)}{(2j+\gamma+\delta-1)(2j+\gamma+\delta-2)},

ⓘ

Defines:: $Q_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.16.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

►

31.16.7

R_{j}=\frac{(n-j)(j+n+\gamma+\delta)(j+\gamma)(j+\delta)}{(\gamma+\delta+2j)(% \gamma+\delta+2j+1)}.

ⓘ

Defines:: $R_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $j$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/31.16.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

…