Mellin–Barnes type

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11: Errata

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18.35.1

$P_{-1}^{(\lambda )}(x;a,b,c)=0,$

$P_0^{(\lambda )}(x;a,b,c)=1$

These equations which were previously given for Pollaczek polynomials of type 2 has been updated for Pollaczek polynomials of type 3.

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The statements “If $c$ and $d$ are real” and “If $c$ and $d$ are not both real” have been further clarified (suggested by Alan Barnes on 2021-03-26).

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The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$. The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at $s=a$ for $\gamma (a,z)$. In the case of (8.6.12), it separates the poles of the gamma function from the poles at $s=0,1,2,\mathrm{\dots }$.

Reported 2017-07-10 by Kurt Fischer.

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18.27.6 $$P_n^{(\alpha ,\beta )}(x;c,d;q)=\frac{c^n{q}^{-(\alpha +1)n}{(q^{\alpha +1},-q^{\alpha +1}{c}^{-1}d;q)}_n}{{(q,-q;q)}_n}{P}_n(q^{\alpha +1}{c}^{-1}x;q^{\alpha },q^{\beta },-q^{\alpha }{c}^{-1}d;q)$$

Originally the first argument to the big $q$-Jacobi polynomial on the right-hand side was written incorrectly as $q^{\alpha +1}{c}^{-1}dx$.

Reported 2017-09-27 by Tom Koornwinder.

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There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$ℱ\left(f\right)\left(x\right)$	$ℱf\left(x\right)$
Fourier Cosine	${ℱ}_c\left(f\right)\left(x\right)$	${ℱ}_{c}f\left(x\right)$
Fourier Sine	${ℱ}_s\left(f\right)\left(x\right)$	${ℱ}_{s}f\left(x\right)$
Laplace	$ℒ\left(f\right)\left(s\right)$	$ℒf\left(s\right)$	$ℒ(f(t);s)$
Mellin	$ℳ\left(f\right)\left(s\right)$	$ℳf\left(s\right)$	$ℳ(f;s)$
Hilbert	$\mathscr{H}\left(f\right)\left(s\right)$	$\mathscr{H}f\left(s\right)$	$\mathscr{H}(f;s)$
Stieltjes	$𝒮\left(f\right)\left(s\right)$	$𝒮f\left(s\right)$	$𝒮(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

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12: Bibliography P

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Asymptotics and Mellin-Barnes Integrals. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-79001-8, MathReview, ZentralBlatt

Referenced by:

§1.14(iv), §10.32(iii), §10.32(ii), §10.46, §10.9(iii), §10.9(ii), §16.11(i), §16.11(ii), §2.10(iii), §2.11(iii), §2.11(v), §2.5(ii), §2.5(ii), Ch.2, §24.7(ii), §25.9, §5.11(ii), §5.11(iii), §5.13, §5.19(ii), §5.6(ii), Ch.5, §8.6(ii), Profile Richard B. Paris.

Encodings:

BibTeX

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Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput. Appl. Math. 41 (1-2), pp. 117–133.

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External Links:

ISSN 0377-0427, Document, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:

ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

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External Links:

ZentralBlatt

Referenced by:

§24.10(iii).

Encodings:

BibTeX

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