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1: 13.14 Definitions and Basic Properties
Whittaker’s Equation
Standard solutions are: …
2: 13.2 Definitions and Basic Properties
13.2.1 z d 2 w d z 2 + ( b - z ) d w d z - a w = 0 .
3: 13.6 Relations to Other Functions
§13.6(i) Elementary Functions
§13.6(ii) Incomplete Gamma Functions
§13.6(iv) Parabolic Cylinder Functions
§13.6(v) Orthogonal Polynomials
§13.6(vi) Generalized Hypergeometric Functions
4: 13.3 Recurrence Relations and Derivatives
§13.3 Recurrence Relations and Derivatives
§13.3(i) Recurrence Relations
Kummer’s differential equation (13.2.1) is equivalent to
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
5: 10.16 Relations to Other Functions
§10.16 Relations to Other Functions
Elementary Functions
Parabolic Cylinder Functions
Confluent Hypergeometric Functions
Generalized Hypergeometric Functions
6: 10.39 Relations to Other Functions
§10.39 Relations to Other Functions
Elementary Functions
Parabolic Cylinder Functions
Confluent Hypergeometric Functions
Generalized Hypergeometric Functions and Hypergeometric Function
7: Errata
  • Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

    The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

  • Chapter 25 Zeta and Related Functions

    A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

  • Equation (13.2.7)
    13.2.7 U ( - m , b , z ) = ( - 1 ) m ( b ) m M ( - m , b , z ) = ( - 1 ) m s = 0 m ( m s ) ( b + s ) m - s ( - z ) s

    The equality U ( - m , b , z ) = ( - 1 ) m ( b ) m M ( - m , b , z ) has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation a = - n has been changed to a = - m .

    Reported 2015-02-10 by Adri Olde Daalhuis.

  • Equation (13.2.8)
    13.2.8 U ( a , a + n + 1 , z ) = ( - 1 ) n ( 1 - a - n ) n z a + n M ( - n , 1 - a - n , z ) = z - a s = 0 n ( n s ) ( a ) s z - s

    The equality U ( a , a + n + 1 , z ) = ( - 1 ) n ( 1 - a - n ) n z a + n M ( - n , 1 - a - n , z ) has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.

    Reported 2015-02-10 by Adri Olde Daalhuis.

  • Equation (13.2.10)
    13.2.10 U ( - m , n + 1 , z ) = ( - 1 ) m ( n + 1 ) m M ( - m , n + 1 , z ) = ( - 1 ) m s = 0 m ( m s ) ( n + s + 1 ) m - s ( - z ) s

    The equality U ( - m , n + 1 , z ) = ( - 1 ) m ( n + 1 ) m M ( - m , n + 1 , z ) has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation a = - m , m = 0 , 1 , 2 , has been introduced.

    Reported 2015-02-10 by Adri Olde Daalhuis.

  • 8: 13.29 Methods of Computation
    The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. …
    §13.29(iv) Recurrence Relations
    The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. … normalizing relationnormalizing relation
    9: 31.3 Basic Solutions
    §31.3(i) Fuchs–Frobenius Solutions at z = 0
    §31.3(ii) Fuchs–Frobenius Solutions at Other Singularities
    §31.3(iii) Equivalent Expressions
    Each is related to the solution (31.3.1) by one of the automorphisms of §31.2(v). … The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).
    10: 7.11 Relations to Other Functions
    §7.11 Relations to Other Functions
    Incomplete Gamma Functions and Generalized Exponential Integral
    Confluent Hypergeometric Functions
    7.11.4 erf z = 2 z π M ( 1 2 , 3 2 , - z 2 ) = 2 z π e - z 2 M ( 1 , 3 2 , z 2 ) ,
    Generalized Hypergeometric Functions