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11: 13.8 Asymptotic Approximations for Large Parameters
§13.8(i) Large | b | , Fixed a and z
§13.8(ii) Large b and z , Fixed a and b / z
§13.8(iii) Large a
§13.8(iv) Large a and b
12: 13.9 Zeros
§13.9(i) Zeros of M ( a , b , z )
§13.9(ii) Zeros of U ( a , b , z )
13: 13.11 Series
13.11.1 M ( a , b , z ) = Γ ( a 1 2 ) e 1 2 z ( 1 4 z ) 1 2 a s = 0 ( 2 a 1 ) s ( 2 a b ) s ( b ) s s ! ( a 1 2 + s ) I a 1 2 + s ( 1 2 z ) , a + 1 2 , b 0 , 1 , 2 , ,
13.11.2 M ( a , b , z ) = Γ ( b a 1 2 ) e 1 2 z ( 1 4 z ) a b + 1 2 s = 0 ( 1 ) s ( 2 b 2 a 1 ) s ( b 2 a ) s ( b a 1 2 + s ) ( b ) s s ! I b a 1 2 + s ( 1 2 z ) , b a + 1 2 , b 0 , 1 , 2 , .
13.11.3 𝐌 ( a , b , z ) = e 1 2 z s = 0 A s ( b 2 a ) 1 2 ( 1 b s ) ( 1 2 z ) 1 2 ( 1 b + s ) J b 1 + s ( 2 z ( b 2 a ) ) ,
( n + 1 ) A n + 1 = ( n + b 1 ) A n 1 + ( 2 a b ) A n 2 , n = 2 , 3 , 4 , .
14: 13.7 Asymptotic Expansions for Large Argument
§13.7 Asymptotic Expansions for Large Argument
§13.7(ii) Error Bounds
§13.7(iii) Exponentially-Improved Expansion
For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).
15: 13.28 Physical Applications
§13.28 Physical Applications
16: 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).
  • 17: 13.3 Recurrence Relations and Derivatives
    §13.3(i) Recurrence Relations
    13.3.7 U ( a 1 , b , z ) + ( b 2 a z ) U ( a , b , z ) + a ( a b + 1 ) U ( a + 1 , b , z ) = 0 ,
    13.3.14 ( a + 1 ) z U ( a + 2 , b + 2 , z ) + ( z b ) U ( a + 1 , b + 1 , z ) U ( a , b , z ) = 0 .
    §13.3(ii) Differentiation Formulas
    13.3.29 ( z d d z z ) n = z n d n d z n z n , n = 1 , 2 , 3 , .
    18: 13.5 Continued Fractions
    §13.5 Continued Fractions
    13.5.1 M ( a , b , z ) M ( a + 1 , b + 1 , z ) = 1 + u 1 z 1 + u 2 z 1 + ,
    13.5.3 U ( a , b , z ) U ( a , b 1 , z ) = 1 + v 1 / z 1 + v 2 / z 1 + ,
    19: 16.6 Transformations of Variable
    16.6.2 F 2 3 ( a , 2 b a 1 , 2 2 b + a b , a b + 3 2 ; z 4 ) = ( 1 z ) a F 2 3 ( 1 3 a , 1 3 a + 1 3 , 1 3 a + 2 3 b , a b + 3 2 ; 27 z 4 ( 1 z ) 3 ) .
    For Kummer-type transformations of F 2 2 functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).
    20: 6.20 Approximations
  • Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric U -function13.2(i)) from which Chebyshev expansions near infinity for E 1 ( z ) , f ( z ) , and g ( z ) follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the U functions. If | ph z | < π the scheme can be used in backward direction.