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1: 20.11 Generalizations and Analogs
For applications to rapidly convergent expansions for π see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. …
2: 25.12 Polylogarithms
§25.12(i) Dilogarithms
For graphics see Figures 25.12.1 and 25.12.2, and for further properties see Maximon (2003), Kirillov (1995), Lewin (1981), Nielsen (1909), and Zagier (1989). …
§25.12(ii) Polylogarithms
The series also converges when | z | = 1 , provided that s > 1 . … Further properties include …
3: 8.17 Incomplete Beta Functions
§8.17(i) Definitions and Basic Properties
The 4 m and 4 m + 1 convergents are less than I x ( a , b ) , and the 4 m + 2 and 4 m + 3 convergents are greater than I x ( a , b ) . … The expansion (8.17.22) converges rapidly for x < ( a + 1 ) / ( a + b + 2 ) . For x > ( a + 1 ) / ( a + b + 2 ) or 1 x < ( b + 1 ) / ( a + b + 2 ) , more rapid convergence is obtained by computing I 1 x ( b , a ) and using (8.17.4). …
§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties
4: 5.11 Asymptotic Expansions
The scaled gamma function Γ ( z ) is defined in (5.11.3) and its main property is Γ ( z ) 1 as z in the sector | ph z | π δ . Wrench (1968) gives exact values of g k up to g 20 . … For similar results including a convergent factorial series see, Nemes (2013c). …
5: Bibliography S
  • D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • I. M. Sheffer (1939) Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.
  • I. Sh. Slavutskiĭ (1995) Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.
  • R. P. Stanley (1989) Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.
  • 6: Bibliography N
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • G. Nemes (2014b) The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.
  • G. Nemes (2014c) The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.
  • G. Nemes (2015c) The resurgence properties of the incomplete gamma function II. Stud. Appl. Math. 135 (1), pp. 86–116.
  • Y. Nievergelt (1995) Bisection hardly ever converges linearly. Numer. Math. 70 (1), pp. 111–118.