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Gauss multiplication formula

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1: 5.5 Functional Relations
Gauss’s Multiplication Formula
2: Errata
  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • Usability

    Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

  • Equations (17.2.22) and (17.2.23)
    17.2.22 ( q a 1 2 , - q a 1 2 ; q ) n ( a 1 2 , - a 1 2 ; q ) n = ( a q 2 ; q 2 ) n ( a ; q 2 ) n = 1 - a q 2 n 1 - a
    17.2.23 ( q a 1 k , q ω k a 1 k , , q ω k k - 1 a 1 k ; q ) n ( a 1 k , ω k a 1 k , , ω k k - 1 a 1 k ; q ) n = ( a q k ; q k ) n ( a ; q k ) n = 1 - a q k n 1 - a

    The numerators of the leftmost fractions were corrected to read ( q a 1 2 , - q a 1 2 ; q ) n and ( q a 1 k , q ω k a 1 k , , q ω k k - 1 a 1 k ; q ) n instead of ( q a 1 2 , - a q 1 2 ; q ) n and ( a q 1 k , q ω k a 1 k , , q ω k k - 1 a 1 k ; q ) n , respectively.

    Reported 2017-06-26 by Jason Zhao.

  • Subsections 15.4(i), 15.4(ii)

    Sentences were added specifying that some equations in these subsections require special care under certain circumstances. Also, (15.4.6) was expanded by adding the formula F ( a , b ; a ; z ) = ( 1 - z ) - b .

    Report by Louis Klauder on 2017-01-01.

  • References

    An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).

  • 3: Bibliography V
  • H. C. van de Hulst (1980) Multiple Light Scattering. Vol. 1, Academic Press, New York.
  • A. J. van der Poorten (1980) Some Wonderful Formulas an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
  • A. Verma and V. K. Jain (1983) Certain summation formulae for q -series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).
  • R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
  • 4: Bibliography M
  • T. Masuda, Y. Ohta, and K. Kajiwara (2002) A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.
  • J. P. McClure and R. Wong (1987) Asymptotic expansion of a multiple integral. SIAM J. Math. Anal. 18 (6), pp. 1630–1637.
  • S. C. Milne (1988) A q -analog of the Gauss summation theorem for hypergeometric series in U ( n ) . Adv. in Math. 72 (1), pp. 59–131.
  • D. S. Moak (1984) The q -analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.
  • L. J. Mordell (1958) On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.
  • 5: 17.6 ϕ 1 2 Function
    q -Gauss Sum
    Related formulas are (17.7.3), (17.8.8) and … For similar formulas see Verma and Jain (1983). …
    17.6.13 ϕ 1 2 ( a , b ; c ; q , q ) + ( q / c , a , b ; q ) ( c / q , a q / c , b q / c ; q ) ϕ 1 2 ( a q / c , b q / c ; q 2 / c ; q , q ) = ( q / c , a b q / c ; q ) ( a q / c , b q / c ; q ) ,
    where | z | < 1 , | ph ( - z ) | < π , and the contour of integration separates the poles of ( q 1 + ζ , c q ζ ; q ) / sin ( π ζ ) from those of 1 / ( a q ζ , b q ζ ; q ) , and the infimum of the distances of the poles from the contour is positive. …