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Fine transformations (first, second, third)

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1: 17.6 ϕ 1 2 Function
Fine’s First Transformation
17.6.11 1 - z 1 - b ϕ 1 2 ( q , a q b q ; q , z ) = n = 0 ( a q ; q ) n ( a z q / b ; q ) 2 n b n ( z q , a q / b ; q ) n - a q n = 0 ( a q ; q ) n ( a z q / b ; q ) 2 n + 1 ( b q ) n ( z q ; q ) n ( a q / b ; q ) n + 1 , | z | < 1 , | b | < 1 .
2: Bibliography F
  • H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
  • H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.
  • N. J. Fine (1988) Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.
  • C. H. Franke (1965) Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.
  • T. Fukushima (2010) Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.
  • 3: Bibliography H
  • P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
  • W. Hahn (1949) Über Orthogonalpolynome, die q -Differenzengleichungen genügen. Math. Nachr. 2, pp. 4–34 (German).
  • R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.
  • E. W. Hansen (1985) Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.
  • Harvard University (1945) Tables of the Modified Hankel Functions of Order One-Third and of their Derivatives. Harvard University Press, Cambridge, MA.