Dirichlet product (or convolution)

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11: 19.16 Definitions

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19.16.7 $$\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{{\prod }_{j=1}^5\sqrt{t+x_j}}.$$

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Symbols:

$dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/19.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

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19.16.9 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a^{\prime }-1}\prod _{j=1}^n{(t+z_j)}^{-b_j}dt=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a-1}\prod _{j=1}^n{(1+t{z}_j)}^{-b_j}dt,$$ $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$, $z_j\in ℂ\setminus (-\mathrm{\infty },0]$,

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Defines:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function

Symbols:

$\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $ℝ$: real line, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.16(ii), §19.16(ii), §19.19, §19.20(iv), Erratum (V1.0.18) for Equation (19.16.9)

Permalink:

http://dlmf.nist.gov/19.16.E9

Encodings:

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Clarification (effective with 1.0.18):

The constraint $a,a^{\prime }>0$, was replaced with $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$. It therefore follows from (19.16.10) that $a^{\prime }>0$.

Suggested 2017-07-25 by Bastien Roucariès

See also:

Annotations for §19.16(ii), §19.16 and Ch.19

… ►For generalizations and further information, especially representation of the $R$-function as a Dirichlet average, see Carlson (1977b). …

12: Bibliography G

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.

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External Links:

ISSN 0033-5614, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.6(iii).

Encodings:

BibTeX

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On the exceptional zeros of cross-products of derivatives of spherical Bessel functions. Z. Angew. Math. Phys. 36 (3), pp. 491–494.

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External Links:

ISSN 0044-2275, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§10.58.

Encodings:

BibTeX

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Table of Integrals, Series, and Products. 6th edition, Academic Press Inc., San Diego, CA.

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Notes:

Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. For a review of the fourth edition (1980), with errata list, see Math. Comp. v. 36 (1981), pp. 310–314. For a review of the fifth edition (1994) see Math. Comp. v. 64 (1995), pp. 439–441.

External Links:

Errata in the 4th, 5th, and 6th editions, ISBN 0-12-294757-6, Link, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §1.8(v), §10.22(vi), §10.23(iv), §10.43(vi), §10.60(iv), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(vi), §13.23(v), §14.18(iv), §15.14, §18.17(ix), §18.18(ix), §19.25(v), §19.29(i), §19.29(ii), §20.10(iii), §22.14(ii), §22.14(iii), §23.14, §24.7(iii), §28.10(iii), §28.28(v), §4.10(iii), §4.11, §4.26(v), §4.27, §4.40(v), §4.41, §5.13, §6.14(iii), §7.14(iii), §8.14.

Encodings:

BibTeX

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