Chu–Vandermonde sums (first and second)

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1: 17.6 ${{}_{2}\phi_{1}}$ Function

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First $q$ -Chu–Vandermonde Sum

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Second $q$ -Chu–Vandermonde Sum

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Heine’s First Transformation

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Fine’s First Transformation

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Nonterminating Form of the $q$ -Vandermonde Sum

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2: 15.4 Special Cases

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Chu–Vandermonde Identity

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Dougall’s Bilateral Sum

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15.4.25

\sum_{n=-\infty}^{\infty}\frac{\Gamma\left(a+n\right)\Gamma\left(b+n\right)}{% \Gamma\left(c+n\right)\Gamma\left(d+n\right)}=\frac{\pi^{2}}{\sin\left(\pi a% \right)\sin\left(\pi b\right)}\*\frac{\Gamma\left(c+d-a-b-1\right)}{\Gamma% \left(c-a\right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b% \right)}.

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $n$ : integer, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.4(ii)
Permalink:: http://dlmf.nist.gov/15.4.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §15.4(ii), §15.4(ii), §15.4 and Ch.15

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J. A. Stratton, P. M. Morse, L. J. Chu, and R. A. Hutner (1941) Elliptic Cylinder and Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients. John Wiley and Sons, Inc., New York.

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External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §28.1, 7th item, Notations S, Notations S.
Encodings:: BibTeX

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J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbató (1956) Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients. Technology Press of M. I. T. and John Wiley & Sons, Inc., New York.

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External Links:: ISBN 0262190028, 0262692910, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: 1st item, Ch.30.
Encodings:: BibTeX

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R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

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External Links:: ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

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R. Szmytkowski (2011) On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

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External Links:: ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

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R. Szmytkowski (2012) On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.

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External Links:: ISSN 0022-247X, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

4: Bibliography T

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L. N. Trefethen and D. Bau (1997) Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-361-7, MathReview (Moody T. Chu), ZentralBlatt
Referenced by:: §3.2(iii), §3.2(vii).
Encodings:: BibTeX

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5: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.4

\det[a_{jk}]=\sum^{n}_{\ell=1}a_{j\ell}A_{j\ell}.

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Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer, $n$ : nonnegative integer and $A_{\NVar{j}\NVar{k}}$ : cofactor
Permalink:: http://dlmf.nist.gov/1.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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1.3.9

\det[a_{jk}]^{2}\leq\left(\sum^{n}_{k=1}a^{2}_{1k}\right)\left(\sum^{n}_{k=1}a% ^{2}_{2k}\right)\dots\left(\sum^{n}_{k=1}a^{2}_{nk}\right).

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Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►for every distinct pair of

j, k

, or when one of the factors

\sum^{n}_{k=1}a^{2}_{jk}

vanishes. … ►

Vandermonde Determinant or Vandermondian

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1.3.19

\sum^{\infty}_{j,k=-\infty}\left|a_{j,k}-\delta_{j,k}\right|

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $j$ : integer, $k$ : integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.3.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(iii), §1.3 and Ch.1

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6: 26.8 Set Partitions: Stirling Numbers

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§26.8(i) Definitions

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§26.8(ii) Generating Functions

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§26.8(iv) Recurrence Relations

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§26.8(v) Identities

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§26.8(vii) Asymptotic Approximations

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7: Preface

… ►Chu, A. …

8: Errata

… ►For some classical polynomials we give some positive sums and discriminants. … ►

Subsection 19.2(ii) and Equation (19.2.9)

The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. In particular, (19.2.9) has been corrected to read

19.2.9

{K^{\prime}}\left(k\right)=\begin{cases}K\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ K\left(k^{\prime}\right)\mp 2\mathrm{i}K\left(-k\right),&\tfrac{1}{2}\pi<\pm% \operatorname{ph}k<\pi,\end{cases}

{E^{\prime}}\left(k\right)=\begin{cases}E\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ E\left(k^{\prime}\right)\mp 2\mathrm{i}(K\left(-k\right)-E\left(-k\right)),&% \tfrac{1}{2}\pi<\pm\operatorname{ph}k<\pi\end{cases}

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Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, $\mathsf{P}_{n},\mathsf{Q}_{n},P_{n},Q_{n},\boldsymbol{Q}_{n}$ and the Laguerre polynomial, $L_{n}$ , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

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Equation (10.19.11)

10.19.11

Q_{3}\left(a\right)=\tfrac{549}{28000}a^{8}-\tfrac{1\;10767}{6\;93000}a^{5}+% \tfrac{79}{12375}a^{2}

Originally the first term on the right-hand side of this equation was written incorrectly as $-\tfrac{549}{28000}a^{8}$ .

Reported 2015-03-16 by Svante Janson.

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Equation (9.6.26)

9.6.26

\operatorname{Bi}'\left(z\right)=\frac{3^{1/6}}{\Gamma\left(\tfrac{1}{3}\right% )}{\mathrm{e}}^{-\zeta}{{}_{1}F_{1}}\left(-\tfrac{1}{6};-\tfrac{1}{3};2\zeta% \right)+\frac{3^{7/6}}{2^{7/3}\Gamma\left(\tfrac{2}{3}\right)}\zeta^{4/3}{% \mathrm{e}}^{-\zeta}{{}_{1}F_{1}}\left(\tfrac{7}{6};\tfrac{7}{3};2\zeta\right)

Originally the second occurrence of the function ${{}_{1}F_{1}}$ was given incorrectly as ${{}_{1}F_{1}}\left(\tfrac{7}{6};\tfrac{7}{3};\zeta\right)$ .

Reported 2014-05-21 by Hanyou Chu.

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9: 14.18 Sums

§14.18 Sums

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§14.18(iii) Other Sums

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14.18.7

(x-y)\sum_{k=0}^{n}(2k+1)P_{k}\left(x\right)Q_{k}\left(y\right)=(n+1)\left(P_{% n+1}\left(x\right)Q_{n}\left(y\right)-P_{n}\left(x\right)Q_{n+1}\left(y\right)% \right)-1.

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Symbols:: $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind and $n$ : nonnegative integer
A&S Ref:: 8.9.2
Referenced by:: §14.18(iii), Erratum (V1.0.7) for Subsection 14.18(iii)
Permalink:: http://dlmf.nist.gov/14.18.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §14.18(iii), §14.18(iii), §14.18 and Ch.14

… ►For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2000, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). …

10: 10.44 Sums

§10.44 Sums

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§10.44(i) Multiplication Theorem

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§10.44(ii) Addition Theorems

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§10.44(iii) Neumann-Type Expansions

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§10.44(iv) Compendia

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