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1: 16.4 Argument Unity
Watson’s Sum
2: 17.7 Special Cases of Higher ϕ s r Functions
Gasper–Rahman q -Analog of Watson’s F 2 3 Sum
Andrews’ q -Analog of the Terminating Version of Watson’s F 2 3 Sum (16.4.6)
3: 10.23 Sums
§10.23 Sums
see Watson (1944, §16.13), and for further generalizations see Watson (1944, Chapter 16) and Erdélyi et al. (1953b, §7.10.1). … This result is proved in Watson (1944, Chapter 18) and further information is provided in this reference, including the behavior of the series near x = 0 and x = 1 . … For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). …
4: 22.6 Elementary Identities
§22.6(i) Sums of Squares
5: 22.8 Addition Theorems
§22.8(i) Sum of Two Arguments
§22.8(ii) Alternative Forms for Sum of Two Arguments
A geometric interpretation of (22.8.20) analogous to that of (23.10.5) is given in Whittaker and Watson (1927, p. 530). … If sums/differences of the z j ’s are rational multiples of K ( k ) , then further relations follow. …
6: 11.10 Anger–Weber Functions
11.10.10 S 1 ( ν , z ) = k = 0 ( 1 ) k ( 1 2 z ) 2 k Γ ( k + 1 2 ν + 1 ) Γ ( k 1 2 ν + 1 ) ,
11.10.30 𝐉 ν ( z ) = 2 sin ( 1 2 ν π ) k = 0 ( 1 ) k J k 1 2 ν + 1 2 ( 1 2 z ) J k + 1 2 ν + 1 2 ( 1 2 z ) + 2 cos ( 1 2 ν π ) k = 0 ( 1 ) k J k 1 2 ν ( 1 2 z ) J k + 1 2 ν ( 1 2 z ) ,
11.10.31 𝐄 ν ( z ) = 2 cos ( 1 2 ν π ) k = 0 ( 1 ) k J k 1 2 ν + 1 2 ( 1 2 z ) J k + 1 2 ν + 1 2 ( 1 2 z ) + 2 sin ( 1 2 ν π ) k = 0 ( 1 ) k J k 1 2 ν ( 1 2 z ) J k + 1 2 ν ( 1 2 z ) ,
§11.10(x) Integrals and Sums
For sums see Hansen (1975, pp. 456–457) and Prudnikov et al. (1990, §§6.4.2–6.4.3).
7: 6.12 Asymptotic Expansions
6.12.5 f ( z ) = 1 z m = 0 n 1 ( 1 ) m ( 2 m ) ! z 2 m + R n ( f ) ( z ) ,
6.12.6 g ( z ) = 1 z 2 m = 0 n 1 ( 1 ) m ( 2 m + 1 ) ! z 2 m + R n ( g ) ( z ) ,
8: 10.44 Sums
§10.44 Sums
§10.44(i) Multiplication Theorem
§10.44(ii) Addition Theorems
For results analogous to (10.23.7) and (10.23.8) see Watson (1944, §§11.3 and 11.41). …
§10.44(iv) Compendia
9: 20.8 Watson’s Expansions
§20.8 Watson’s Expansions
20.8.1 θ 2 ( 0 , q ) θ 3 ( z , q ) θ 4 ( z , q ) θ 2 ( z , q ) = 2 n = ( 1 ) n q n 2 e i 2 n z q n e i z + q n e i z .
See Watson (1935a). …
10: Errata
  • Subsection 17.7(iii)

    The title of the paragraph which was previously “Andrews’ Terminating q -Analog of (17.7.8)” has been changed to “Andrews’ q -Analog of the Terminating Version of Watson’s F 2 3 Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating q -Analog” has been changed to “Andrews’ q -Analog of the Terminating Version of Whipple’s F 2 3 Sum (16.4.7)”.

  • Equation (26.7.6)
    26.7.6 B ( n + 1 ) = k = 0 n ( n k ) B ( k )

    Originally this equation appeared with B ( n ) in the summation, instead of B ( k ) .

    Reported 2010-11-07 by Layne Watson.