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1: 16.22 Asymptotic Expansions
Asymptotic expansions of G p , q m , n ( z ; a ; b ) for large z are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). …
2: 13.8 Asymptotic Approximations for Large Parameters
§13.8(i) Large | b | , Fixed a and z
When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when | b | is large, and | b - a | and | z | are bounded.
§13.8(ii) Large b and z , Fixed a and b / z
For other asymptotic expansions for large b and z see López and Pagola (2010). …
3: 15.12 Asymptotic Approximations
For large b and c with c > b + 1 see López and Pagola (2011). … If | ph ( z - 1 ) | < π , then as λ with | ph λ | π - δ , … If | ph z | < π , then as λ with | ph λ | π - δ , … If | ph z | < π , then as λ with | ph λ | 1 2 π - δ , … For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).
4: 28.16 Asymptotic Expansions for Large q
§28.16 Asymptotic Expansions for Large q
5: Bibliography L
  • J. L. López and P. J. Pagola (2010) The confluent hypergeometric functions M ( a , b ; z ) and U ( a , b ; z ) for large b and z . J. Comput. Appl. Math. 233 (6), pp. 1570–1576.
  • J. L. López and N. M. Temme (2010b) Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.
  • 6: Bibliography Y
  • T. Yoshida (1995) Computation of Kummer functions U ( a , b , x ) for large argument x by using the τ -method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).
  • 7: Bibliography H
  • B. A. Hargrave and B. D. Sleeman (1977) Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.
  • C. J. Howls and A. B. Olde Daalhuis (1999) On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.
  • 8: Bibliography O
  • A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.
  • A. B. Olde Daalhuis (2003b) Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.
  • A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.
  • F. W. J. Olver (1959) Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.
  • 9: 8.18 Asymptotic Expansions of I x ( a , b )
    Large a , Fixed b
    For asymptotic expansions for large values of a and/or b of the x -solution of the equation …
    10: 13.9 Zeros
    For fixed a , b the large z -zeros of M ( a , b , z ) satisfy … For fixed b and z in the large a -zeros of M ( a , b , z ) are given by … For fixed b and z in the large a -zeros of U ( a , b , z ) are given by …