Chu–Vandermonde sums (first and second)

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1: 17.6 ${}_2{}^{}\varphi _1^{}$ 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 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+n\right)\mathrm{\Gamma }\left(b+n\right)}{\mathrm{\Gamma }\left(c+n\right)\mathrm{\Gamma }\left(d+n\right)}=\frac{{\pi }^2}{\sin \left(\pi a\right)\sin \left(\pi b\right)}\frac{\mathrm{\Gamma }\left(c+d-a-b-1\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(d-a\right)\mathrm{\Gamma }\left(c-b\right)\mathrm{\Gamma }\left(d-b\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: 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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3: Bibliography S

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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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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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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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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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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: 1.3 Determinants

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1.3.4 $$det[a_{jk}]=\sum _{\mathrm{\ell }=1}^n{a}_{j\mathrm{\ell }}{A}_{j\mathrm{\ell }}.$$

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Symbols:

$det$: determinant, $j$: integer, $k$: integer, $n$: nonnegative integer and $A_{jk}$: cofactor

Permalink:

http://dlmf.nist.gov/1.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3 and Ch.1

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1.3.9 $$det{[a_{jk}]}^2\le \left(\sum _{k=1}^n{a}_{1k}^2\right)\left(\sum _{k=1}^n{a}_{2k}^2\right)\mathrm{\dots }\left(\sum _{k=1}^n{a}_{nk}^2\right).$$

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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 ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes. … ►

Vandermonde Determinant or Vandermondian

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1.3.19 $$\sum _{j,k=-\mathrm{\infty }}^{\mathrm{\infty }}|a_{j,k}-{\delta }_{j,k}|$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $j$: integer and $k$: integer

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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5: Bibliography T

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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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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

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Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, ${𝖯}_n,{𝖰}_n,P_n,Q_n,{𝑸}_n$ and the Laguerre polynomial, $L_n$, were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

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10.19.11 $$Q_3\left(a\right)=\frac{549}{28000}{a}^8-\frac{\mathrm{1\hspace{0.33em}10767}}{\mathrm{6\hspace{0.33em}93000}}{a}^5+\frac{79}{12375}{a}^2$$

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

Reported 2015-03-16 by Svante Janson.

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9.6.26 $${\mathrm{Bi}}^{\prime }\left(z\right)=\frac{3^{1/6}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(-\frac{1}{6};-\frac{1}{3};2\zeta )+\frac{3^{7/6}}{2^{7/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\zeta }^{4/3}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(\frac{7}{6};\frac{7}{3};2\zeta )$$

Originally the second occurrence of the function ${}_1{}^{}F_1^{}$ was given incorrectly as ${}_1{}^{}F_1^{}(\frac{7}{6};\frac{7}{3};\zeta )$.

Reported 2014-05-21 by Hanyou Chu.

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Two corrections have been made in this paragraph. First, the correct range of the initial displacement $a$ is . Previously it was $\sqrt{1/\beta }\le |a|\le \sqrt{2/\beta }$. Second, the correct period of the oscillations is $2K\left(k\right)/\sqrt{\eta }$. Previously it was given incorrectly as $4K\left(k\right)/\sqrt{\eta }$.

Reported 2014-05-02 by Svante Janson.

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Originally it was stated, incorrectly, that $Q_{\nu }^{\mu }\left(x\right)$ is real when $\nu ,\mu \in ℝ$ and $x\in (1,\mathrm{\infty })$. This statement is true only for $P_{\nu }^{\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$.

Reported 2012-07-18 by Hans Volkmer and Howard Cohl.

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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:

$P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: 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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