squares and products

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11: 10.65 Power Series

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§10.65(iii) Cross-Products and Sums of Squares

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12: 10.67 Asymptotic Expansions for Large Argument

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§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu=0$

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13: DLMF Project News

error generating summary

14: 20.7 Identities

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§20.7(i) Sums of Squares

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§20.7(iv) Reduction Formulas for Products

… ►In the following equations

\tau^{\prime}=-1/\tau

, and all square roots assume their principal values. … ►

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

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15: 19.21 Connection Formulas

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19.21.7

(x-y)R_{D}\left(y,z,x\right)+(z-y)R_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z% \right)-3y^{1/2}x^{-1/2}z^{-1/2},

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.21(ii), §19.25(i), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.21.E7
Encodings:: pMML, png, TeX
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

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19.21.8

R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3x^{-1% /2}y^{-1/2}z^{-1/2},

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.21.E8
Encodings:: pMML, png, TeX
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

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19.21.10

2R_{G}\left(x,y,z\right)=zR_{F}\left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)R_{D}% \left(x,y,z\right)+x^{1/2}y^{1/2}z^{-1/2},

z\neq 0

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.20(ii), §19.21(i), §19.21(ii), §19.21(ii), §19.36(i), §19.36(ii), Erratum (V1.1.3) for Chapter 19, Erratum (V1.1.7) for Equation (19.21.10)
Permalink:: http://dlmf.nist.gov/19.21.E10
Encodings:: pMML, png, TeX
Correction (effective with 1.1.7):: An incorrect factor of 3 was removed in the last term on the right-hand side.
Suggested 2022-06-27 by Abdulhafeez Abdulsalam
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

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16: 19.29 Reduction of General Elliptic Integrals

… ►These theorems reduce integrals over a real interval

(y,x)

of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over

(0,\infty)

containing the square root of a cubic polynomial (compare §19.16(i)). …If ►

19.29.3

s(t)=\prod_{\alpha=1}^{4}\sqrt{a_{\alpha}+b_{\alpha}t}

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Symbols:: $s(t)$ : product and $\alpha$ : parameter
Referenced by:: §19.29(i)
Permalink:: http://dlmf.nist.gov/19.29.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(i), §19.29 and Ch.19

… ►where

\alpha,\beta,\gamma

is any permutation of the numbers

1,2,3

, and … ►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as

R_{G}\left(a_{1}b_{2},a_{2}b_{1},0\right)

multiplied either by

-2/(b_{1}b_{2})

or by

-2/(a_{1}a_{2})

in the cases of (19.29.20) or (19.29.21), respectively. …

17: 26.15 Permutations: Matrix Notation

… ►The inversion number of

\sigma

is a sum of products of pairs of entries in the matrix representation of

\sigma

: … ►Let

r_{j}(B)

be the number of ways of placing

j

nonattacking rooks on the squares of

B

. … ►

26.15.11

\sum_{k=0}^{n}r_{n-k}(B){\left(x-k+1\right)_{k}}=\prod_{j=1}^{n}(x+b_{j}-j+1).

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $x$ : real variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $n$ : nonnegative integer, $B$ : subset and $r_{j}(B)$ : number of rook positions
Permalink:: http://dlmf.nist.gov/26.15.E11
Encodings:: pMML, png, TeX
Errata (effective with 1.1.2):: The Pochhammer symbol now links to its definition.
See also:: Annotations for §26.15, §26.15 and Ch.26

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18: Bibliography M

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J. Martinek, H. P. Thielman, and E. C. Huebschman (1966) On the zeros of cross-product Bessel functions. J. Math. Mech. 16, pp. 447–452.

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External Links:: MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

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L. C. Maximon (1991) On the evaluation of the integral over the product of two spherical Bessel functions. J. Math. Phys. 32 (3), pp. 642–648.

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External Links:: ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt
Referenced by:: §1.17(ii), §10.59.
Encodings:: BibTeX

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S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:: Document, ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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External Links:: ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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L. J. Mordell (1917) On the representation of numbers as a sum of

2r

squares. Quarterly Journal of Math. 48, pp. 93–104.

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External Links:: ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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19: Bibliography C

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B. C. Carlson and J. FitzSimons (2000) Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.

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Notes:: Code written by the second author to compute symmetric elliptic integrals numerically in the Derive computer algebra system is available.
External Links:: ISSN 0377-0427, Document, Code, MathReview, ZentralBlatt
Referenced by:: §19.29(ii), §19.36(i), §19.36(i), §19.39(iv).
Encodings:: BibTeX

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B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric

R

-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:: ISSN 0025-5718, Document, MathReview, ZentralBlatt
Referenced by:: §19.25(v), §19.29(i), §22.14(ii).
Encodings:: BibTeX

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J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.

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External Links:: MathReview (T. Hayashida)
Referenced by:: §27.13(ii).
Encodings:: BibTeX

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J. A. Cochran (1964) Remarks on the zeros of cross-product Bessel functions. J. Soc. Indust. Appl. Math. 12 (3), pp. 580–587.

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External Links:: Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §10.21(x), §10.21(x).
Encodings:: BibTeX

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J. A. Cochran (1966a) The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.

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External Links:: MathReview (R. G. Langebartel), ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

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20: 22.14 Integrals

… ►For indefinite integrals of squares and products of even powers of Jacobian functions in terms of symmetric elliptic integrals, see Carlson (2006b). …