Watson identities

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11: 2.5 Mellin Transform Methods

… ►When $x=1$, this identity is a Parseval-type formula; compare §1.14(iv). … ►Then from Table 1.14.5 and Watson (1944, p. 403) …

12: 16.4 Argument Unity

… ►See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … ►

Watson’s Sum

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§16.4(iii) Identities

… ►Methods of deriving such identities are given by Bailey (1964), Rainville (1960), Raynal (1979), and Wilson (1978). … ► …

13: 2.11 Remainder Terms; Stokes Phenomenon

… ►Application of Watson’s lemma (§2.4(i)) yields … ►From (2.11.5) and the identity …

14: Bibliography S

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On an identity of Ramanujan based on the hypergeometric series ${}_2{}^{}F_1^{}(\frac{1}{3},\frac{2}{3};\frac{1}{2};x)$ . J. Number Theory 69 (2), pp. 125–134.

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External Links:

ISSN 0022-314X, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§20.11(iii).

Encodings:

BibTeX

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Asymptotic expansions of Mellin transforms and analogues of Watson’s lemma. SIAM J. Math. Anal. 16 (4), pp. 896–906.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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Some identities involving the Riemann zeta function. II. Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.

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External Links:

ISSN 0019-5588, MathReview (Jean-Marie De Koninck), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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15: Errata

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The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog of (17.7.8)” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Watson’s ${}_3{}^{}F_2^{}$ Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Whipple’s ${}_3{}^{}F_2^{}$ Sum (16.4.7)”.

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25.15.6 $$G(\chi )\equiv \sum _{r=1}^{k-1}\chi (r){\mathrm{e}}^{2\pi ir/k}.$$

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Reported by Gergő Nemes on 2021-08-23

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The overloaded operator $\equiv$ is now more clearly separated (and linked) to two distinct cases: equivalence by definition (in §§1.4(ii), 1.4(v), 2.7(i), 2.10(iv), 3.1(i), 3.1(iv), 4.18, 9.18(ii), 9.18(vi), 9.18(vi), 18.2(iv), 20.2(iii), 20.7(vi), 23.20(ii), 25.10(i), 26.15, 31.17(i)); and modular equivalence (in §§24.10(i), 24.10(ii), 24.10(iii), 24.10(iv), 24.15(iii), 24.19(ii), 26.14(i), 26.21, 27.2(i), 27.8, 27.9, 27.11, 27.12, 27.14(v), 27.14(vi), 27.15, 27.16, 27.19).

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Three new identities for Pochhammer’s symbol (5.2.6)–(5.2.8) have been added at the end of this subsection.

Suggested by Tom Koornwinder.

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26.7.6 $$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(k\right)$$

Originally this equation appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$.

Reported 2010-11-07 by Layne Watson.

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