summation

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21: 8.17 Incomplete Beta Functions

… ►

8.17.24

I_{x}\left(m,n\right)=(1-x)^{n}\sum_{j=m}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+j-% 1}{j}x^{j},

m, n

positive integers;

0\leq x<1

ⓘ

Symbols:: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $n$ : nonnegative integer and $x$ : variable
A&S Ref:: 26.5.26 (The upper limit of summation has been corrected.)
Referenced by:: §8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36
Permalink:: http://dlmf.nist.gov/8.17.E24
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\infty$ .
Suggested 2011-03-23 by Stephen Bourn
See also:: Annotations for §8.17(vii), §8.17 and Ch.8

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22: Bibliography B

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Á. Baricz and T. K. Pogány (2013) Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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External Links:: ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt
Referenced by:: §11.7(v), §11.7(v).
Encodings:: BibTeX

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B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

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External Links:: Document, MathReview (T. M. Apostol), ZentralBlatt
Referenced by:: §24.16(ii).
Encodings:: BibTeX

►

B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.16(iii), §24.8(ii).
Encodings:: BibTeX

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J. G. Byatt-Smith (2000) The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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External Links:: ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

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23: Bibliography M

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A. R. Miller (1997) A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §16.10.
Encodings:: BibTeX

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S. C. Milne (1985a) A

q

-analog of the

{}_{5}F_{4}(1)

summation theorem for hypergeometric series well-poised in

\mathit{SU}(n)

. Adv. in Math. 57 (1), pp. 14–33.

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External Links:: ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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S. C. Milne (1988) A

q

-analog of the Gauss summation theorem for hypergeometric series in

U(n)

. Adv. in Math. 72 (1), pp. 59–131.

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External Links:: ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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S. C. Milne (1997) Balanced

\sideset{{}_{3}}{{}_{2}}{\mathop{\Theta}}

summation theorems for

U(n)

basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:: ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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24: Bibliography V

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A. Verma and V. K. Jain (1983) Certain summation formulae for

q

-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).

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External Links:: ISSN 0019-5839, MathReview Entry, ZentralBlatt
Referenced by:: (17.6.4_5), §17.6(i), (17.8.8), §17.8.
Encodings:: BibTeX

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25: 20.6 Power Series

… ►In the double series the order of summation is important only when

j=1

. …

26: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►where the summation is over all nonnegative integers

n_{1},n_{2},\ldots,n_{k}

such that

n_{1}+n_{2}+\cdots+n_{k}=n

. …

27: 34.2 Definition: $\mathit{3j}$ Symbol

… ►and the summation is over all nonnegative integers

s

such that the arguments in the factorials are nonnegative. …

28: 34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►

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

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29: Bibliography G

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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.

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

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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

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30: 18.3 Definitions

… ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials

T_{n}\left(x\right)

n=0,1,\dots,N

, are orthogonal on the discrete point set comprising the zeros

x_{N+1,n},n=1,2,\dots,N+1

, of

T_{N+1}\left(x\right)

: …