representation by squares

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1: 27.13 Functions

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§27.13(iv) Representation by Squares

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2: 22.16 Related Functions

… ►See Figure 22.16.2. …

3: Bibliography G

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Representations of Integers as Sums of Squares. Springer-Verlag, New York.

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External Links:

ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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4: Bibliography E

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On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.

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External Links:

Document, MathReview (S. Chowla), ZentralBlatt

Referenced by:

§20.12(i).

Encodings:

BibTeX

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5: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

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§1.17(ii) Integral Representations

… ►Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation … ►

Coulomb Functions (§33.14(iv))

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§1.17(iii) Series Representations

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6: Bibliography M

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On the representation of numbers as a sum of $2r$ squares. Quarterly Journal of Math. 48, pp. 93–104.

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Referenced by:

§27.13(iv).

Encodings:

BibTeX

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7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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§1.18(ii) $L^2$ spaces on intervals in $ℝ$

►Let $X=[a,b]$ or $[a,b)$ or $(a,b]$ or $(a,b)$ be a (possibly infinite, or semi-infinite) interval in $ℝ$ . For a Lebesgue–Stieltjes measure $d\alpha$ on $X$ let $L^2(X,d\alpha )$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $d\alpha$ , …Functions $f,g\in L^2(X,d\alpha )$ for which $⟨f-g,f-g⟩=0$ are identified with each other. The space $L^2(X,d\alpha )$ becomes a separable Hilbert space with inner product … ►Assume that ${\left\{{\varphi }_n\right\}}_{n=0}^{\mathrm{\infty }}$ is an orthonormal basis of $L^2\left(X\right)$ . The formulas in §1.18(i) are then: … ►for $f(x)\in L^2$ and piece-wise continuous, with convergence as discussed in §1.18(ii). …

8: Bibliography

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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

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On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.

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External Links:

ISSN 0002-9939, Fulltext, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.17.

Encodings:

BibTeX

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Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (2), pp. 411–437.

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External Links:

Document, MathReview (D. L. Colton), ZentralBlatt

Referenced by:

§18.17(iv).

Encodings:

BibTeX

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Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.

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External Links:

ISSN 1095-7154, Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.18(iv).

Encodings:

BibTeX

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9: 26.15 Permutations: Matrix Notation

… ►The sign of the permutation $\sigma$ is the sign of the determinant of its matrix representation. 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 $B$. …

10: Errata

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A sentence indicating that explicit representations for the coefficients $c_k$ are given in Volkmer (2023) was inserted just below (16.11.5).

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Factors inside square roots on the right-hand sides of formulas (19.18.6), (19.20.10), (19.20.19), (19.21.7), (19.21.8), (19.21.10), (19.25.7), (19.25.10) and (19.25.11) were written as products to ensure the correct multivalued behavior.

Reported by Luc Maisonobe on 2021-06-07

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Equations: (3.3.3_1), (3.3.3_2), (5.15.9) (suggested by Calvin Khor on 2021-09-04), (8.15.2), Pochhammer symbol representation in (10.17.1) for $a_k(\nu )$ coefficient, Pochhammer symbol representation in (11.9.4) for $a_k(\mu ,\nu )$ coefficient, and (12.14.4_5).

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Pochhammer symbol representations for the functions $F_k(\nu )$ and $G_k(\nu )$ were inserted.

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Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${}_2{}^{}F_1^{}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

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