terminant function

(0.003 seconds)

11—20 of 33 matching pages

11: 16.4 Argument Unity

… ►See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … ►when

\Re\left(a-2b-2c\right)>-2

, or when the series terminates with

a=-n

: … ►when

\Re\left(2c-a-b\right)>-1

, or when the series terminates with

a=-n

. … ►when the series on the right terminates and the series on the left converges. When the series on the right does not terminate, a second term appears. …

12: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

►

§35.8(i) Definition

… ►If

-a_{j}+\tfrac{1}{2}(k+1)\in\mathbb{N}

for some

j, k

satisfying

1\leq j\leq p

1\leq k\leq m

, then the series expansion (35.8.1) terminates. … ►If

p>q+1

, then (35.8.1) diverges unless it terminates. ►

§35.8(ii) Relations to Other Functions

…

13: 18.38 Mathematical Applications

… ►

Complex Function Theory

… ►The

3j

symbol (34.2.6), with an alternative expression as a terminating

{{}_{3}F_{2}}

of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. … ►The

6j

symbol (34.4.3), with an alternative expression as a terminating balanced

{{}_{4}F_{3}}

of unit argument, can be expressend in terms of Racah polynomials (18.26.3). … ► … ►

Non-Classical Weight Functions

…

14: 10.49 Explicit Formulas

… ►

§10.49(i) Unmodified Functions

… ►

§10.49(ii) Modified Functions

… ►

§10.49(iii) Rayleigh’s Formulas

… ►

§10.49(iv) Sums or Differences of Squares

… ►

10.49.18

{\mathsf{j}_{n}}^{2}\left(z\right)+{\mathsf{y}_{n}}^{2}\left(z\right)=\sum_{k=% 0}^{n}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}.

ⓘ

Symbols:: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $n$ : integer, $k$ : nonnegative integer, $z$ : complex variable and $s_{k}(n)$
A&S Ref:: 10.1.27
Referenced by:: §10.49(iv)
Permalink:: http://dlmf.nist.gov/10.49.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §10.49(iv), §10.49 and Ch.10

…

15: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

… ►and … ►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at

k=n-1

(

k\geq 0

) and

z

is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. … ►

§5.11(iii) Ratios

… ►

16: 10.61 Definitions and Basic Properties

… ►

§10.61(i) Definitions

… ►

§10.61(ii) Differential Equations

… ►

§10.61(iii) Reflection Formulas for Arguments

►In general, Kelvin functions have a branch point at

x=0

and functions with arguments

xe^{\pm\pi i}

are complex. … ►

§10.61(iv) Reflection Formulas for Orders

…

17: 34.2 Definition: $\mathit{3j}$ Symbol

… ►

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

►where

{{}_{3}F_{2}}

is defined as in §16.2. ►For alternative expressions for the

\mathit{3j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{3}F_{2}}

of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

18: 34.4 Definition: $\mathit{6j}$ Symbol

… ►

34.4.3

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}={(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_% {1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l% _{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l% _{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3% }+l_{1}+l_{2}+1)!}\*{{}_{4}F_{3}}\left({-j_{1}+j_{2}-j_{3},j_{2}-l_{1}-l_{3},-% j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1\atop-j_{1}+j_{2}-l_{1}-l_{2},j_{2}-j_% {3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-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 $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : 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/34.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.4 and Ch.34

►where

{{}_{4}F_{3}}

is defined as in §16.2. ►For alternative expressions for the

\mathit{6j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{4}F_{3}}

of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

19: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

… ►

17.9.3

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(abz/c;q\right)_{% \infty}}{\left(bz/c;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,c/b,0\atop c,cq% /(bz)};q,q\right)+\frac{\left(a,bz,c/b;q\right)_{\infty}}{\left(c,z,c/(bz);q% \right)_{\infty}}{{}_{3}\phi_{2}}\left({z,abz/c,0\atop bz,bzq/c};q,q\right),

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Referenced by:: Erratum (V1.0.23) for Equation (17.9.3)
Permalink:: http://dlmf.nist.gov/17.9.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.0.23):: The second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ${{}_{2}\phi_{1}}$ is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)).
Suggested 2019-04-26 by Roberto S. Costas-Santos
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

…

20: 6.12 Asymptotic Expansions

… ►If the expansion is terminated at the

n

th term, then the remainder term is bounded by

1+\chi(n+1)

times the next term. For the function

\chi

see §9.7(i). … ►

6.12.3

\mathrm{f}\left(z\right)\sim\frac{1}{z}\left(1-\frac{2!}{z^{2}}+\frac{4!}{z^{4% }}-\frac{6!}{z^{6}}+\cdots\right),

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $!$ : factorial (as in $n!$ ), $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
A&S Ref:: 5.2.34
Referenced by:: §6.12(ii), §6.12(ii)
Permalink:: http://dlmf.nist.gov/6.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §6.12(ii), §6.12 and Ch.6

… ► … ►