with respect to summation

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11—17 of 17 matching pages

11: 18.20 Hahn Class: Explicit Representations

… ►

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

… ►Here we use as convention for (16.2.1) with

b_{q}=-N

a_{1}=-n

, and

n=0,1,\ldots,N

that the summation on the right-hand side ends at

k=n

. …(For symmetry properties of

p_{n}\left(x;a,b,\overline{a},\overline{b}\right)

with respect to

a

b

\overline{a}

\overline{b}

see Andrews et al. (1999, Corollary 3.3.4).) …

12: Bibliography G

… ►

F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

ⓘ

External Links:: ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt
Referenced by:: §17.7(iii).
Encodings:: BibTeX

… ►

G. Gasper (1981) Orthogonality of certain functions with respect to complex valued weights. Canad. J. Math. 33 (5), pp. 1261–1270.

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External Links:: ISSN 0008-414X, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.33(v).
Encodings:: BibTeX

… ►

W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §3.8(vi).
Encodings:: BibTeX

… ►

J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:: ISBN 0-8176-3837-7, MathReview, ZentralBlatt
Referenced by:: §15.17(v).
Encodings:: BibTeX

… ►

R. A. Gustafson (1987) Multilateral summation theorems for ordinary and basic hypergeometric series in

{\rm U}(n)

. SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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External Links:: ISSN 0036-1410, Document, MathReview, ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

…

13: 18.27 $q$ -Hahn Class

… ►For (17.4.1) with

b_{s}=q^{-N}

a_{0}=q^{-m}

, and

m=0,1,\ldots,N

we will use the convention that the summation on the right-hand side ends at

n=m

. … ►These families depend on further parameters, in addition to

q

. … ►Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. … ►

From Big $q$ -Jacobi to Jacobi

… ►

From Big $q$ -Jacobi to Little $q$ -Jacobi

…

14: 34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent

\mathit{9j}

symbols. … ►

34.7.4

\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\begin{pmatrix}j% _{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\*\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}.

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Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $r$ : nonnegative integer
Referenced by:: Erratum (V1.0.10) for Equation (34.7.4)
Permalink:: http://dlmf.nist.gov/34.7.E4
Encodings:: pMML, png, TeX
Errata (effective with 1.0.10):: Originally the third $\mathit{3j}$ symbol in the summation was written incorrectly as $\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}.$
Reported 2015-01-19 by Yan-Rui Liu
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

…

15: Bibliography

… ►

R. W. Abernathy and R. P. Smith (1993) Algorithm 724: Program to calculate F-percentiles. ACM Trans. Math. Software 19 (4), pp. 481–483.

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Notes:: Double-precision Fortran, maximum accuracy 13D.
External Links:: ISSN 0098-3500, Document, GAMS (module 724; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

G. B. Airy (1849) Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.

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Referenced by:: §9.16.
Encodings:: BibTeX

… ►

G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:: MathReview (R. N. Jain), ZentralBlatt
Referenced by:: §17.11.
Encodings:: BibTeX

… ►

A. Apelblat (1989) Derivatives and integrals with respect to the order of the Struve functions

\mathbf{H}_{\nu}(x)

and

\mathbf{L}_{\nu}(x)

. J. Math. Anal. Appl. 137 (1), pp. 17–36.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (A. McD. Mercer), ZentralBlatt
Referenced by:: §11.4(vi), §11.7(iv).
Encodings:: BibTeX

►

A. Apelblat (1991) Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.

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External Links:: ISSN 0044-2275, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt
Referenced by:: §10.61(v), §10.64.
Encodings:: BibTeX

…

16: 18.17 Integrals

… ►For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively. … ►Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of

n

-th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively. … ►Formulas (18.17.14) and (18.17.15) are fractional generalizations of

n

-th derivative formulas which are, after substitution of (13.6.19), special cases of (13.3.18) and (13.3.20), respectively. … ►Formulas (18.17.21_2) and (18.17.21_3) are respectively the limit case

c\to\tfrac{1}{2}

and the special case

c=1

of (18.17.21_1). … ►Formulas (18.17.45) and (18.17.49) are integrated forms of the linearization formulas (18.18.22) and (18.18.23), respectively. …

17: 2.10 Sums and Sequences

… ►

(a)

On the strip $a\leq\Re z\leq n$ , $f(z)$ is analytic in its interior, $f^{(2m)}(z)$ is continuous on its closure, and $f(z)=o\left(e^{2\pi|\Im z|}\right)$ as $\Im z\to\pm\infty$ , uniformly with respect to $\Re z\in[a,n]$ .

… ►In the present example it leads to … ►

§2.10(ii) Summation by Parts

►The formula for summation by parts is … ►where

\mathscr{C}_{1},\mathscr{C}_{2}

denote respectively the upper and lower halves of

\mathscr{C}

. …