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1: 13.14 Definitions and Basic Properties
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Whittaker’s Equation
โ–บStandard solutions are: …
2: 13.2 Definitions and Basic Properties
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13.2.1 z โข d 2 w d z 2 + ( b z ) โข d w d z a โข w = 0 .
3: 13.6 Relations to Other Functions
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§13.6(i) Elementary Functions
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§13.6(ii) Incomplete Gamma Functions
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§13.6(iv) Parabolic Cylinder Functions
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§13.6(v) Orthogonal Polynomials
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§13.6(vi) Generalized Hypergeometric Functions
4: 13.3 Recurrence Relations and Derivatives
§13.3 Recurrence Relations and Derivatives
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§13.3(i) Recurrence Relations
โ–บKummer’s differential equation (13.2.1) is equivalent toโ–บ
13.3.14 ( a + 1 ) โข z โข U โก ( a + 2 , b + 2 , z ) + ( z b ) โข U โก ( a + 1 , b + 1 , z ) U โก ( a , b , z ) = 0 .
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§13.3(ii) Differentiation Formulas
5: 10.16 Relations to Other Functions
§10.16 Relations to Other Functions
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Elementary Functions
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Parabolic Cylinder Functions
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Confluent Hypergeometric Functions
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Generalized Hypergeometric Functions
6: 10.39 Relations to Other Functions
§10.39 Relations to Other Functions
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Elementary Functions
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Parabolic Cylinder Functions
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Confluent Hypergeometric Functions
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Generalized Hypergeometric Functions and Hypergeometric Function
7: 18.34 Bessel Polynomials
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18.34.1 y n โก ( x ; a ) = F 0 2 โก ( n , n + a 1 ; x 2 ) = ( n + a 1 ) n โข ( x 2 ) n โข F 1 1 โก ( n 2 โข n a + 2 ; 2 x ) = n ! โข ( 1 2 โข x ) n โข L n ( 1 a 2 โข n ) โก ( 2 โข x 1 ) = ( 1 2 โข x ) 1 1 2 โข a โข e 1 / x โข W 1 1 2 โข a , 1 2 โข ( a 1 ) + n โก ( 2 โข x 1 ) .
8: 13.29 Methods of Computation
โ–บThe 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. … โ–บ
§13.29(iv) Recurrence Relations
โ–บThe recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. … โ–บnormalizing relationโ–บnormalizing relation
9: Errata
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  • 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.

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  • Equation (9.7.2)

    Following a suggestion from James McTavish on 2017-04-06, the recurrence relation u k = ( 6 โข k 5 ) โข ( 6 โข k 3 ) โข ( 6 โข k 1 ) ( 2 โข k 1 ) โข 216 โข k โข u k 1 was added to Equation (9.7.2).

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  • 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.

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  • Equation (13.2.7)
    13.2.7 U โก ( m , b , z ) = ( 1 ) m โข ( b ) m โข M โก ( m , b , z ) = ( 1 ) m โข s = 0 m ( m s ) โข ( b + s ) m s โข ( z ) s

    The equality U โก ( m , b , z ) = ( 1 ) m โข ( b ) m โข M โก ( 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.

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  • Equation (13.2.8)
    13.2.8 U โก ( a , a + n + 1 , z ) = ( 1 ) n โข ( 1 a n ) n z a + n โข M โก ( n , 1 a n , z ) = z a โข s = 0 n ( n s ) โข ( a ) s โข z s

    The equality U โก ( a , a + n + 1 , z ) = ( 1 ) n โข ( 1 a n ) n z a + n โข M โก ( 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.

  • 10: 18.5 Explicit Representations
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    Chebyshev
    โ–บRelated formula: … โ–บ
    §18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions
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    Laguerre
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    Hermite