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11: 16.4 Argument Unity
See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … when ( 2 c - a - b ) > - 1 , or when the series terminates with a = - n . … when ( b + c + d - a ) < 1 , or when the series terminates with d = - n . … when the series on the right terminates and the series on the left converges. When the series on the right does not terminate, a second term appears. …
12: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8(i) Definition
If - a j + 1 2 ( k + 1 ) for some j , k satisfying 1 j p , 1 k m , then the series expansion (35.8.1) terminates. … If p > q + 1 , then (35.8.1) diverges unless it terminates.
§35.8(ii) Relations to Other Functions
13: 10.49 Explicit Formulas
§10.49(i) Unmodified Functions
§10.49(ii) Modified Functions
§10.49(iii) Rayleigh’s Formulas
§10.49(iv) Sums or Differences of Squares
10.49.18 j n 2 ( z ) + y n 2 ( z ) = k = 0 n s k ( n + 1 2 ) z 2 k + 2 .
14: 5.11 Asymptotic Expansions
§5.11 Asymptotic Expansions
and … If the sums in the expansions (5.11.1) and (5.11.2) are terminated at k = n - 1 ( k 0 ) and z is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. …
§5.11(iii) Ratios
15: 34.2 Definition: 3 j Symbol
34.2.6 ( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( - 1 ) j 2 - m 1 + m 3 ( j 1 + j 2 + m 3 ) ! ( j 2 + j 3 - m 1 ) ! Δ ( j 1 j 2 j 3 ) ( j 1 + j 2 + j 3 + 1 ) ! ( ( j 1 + m 1 ) ! ( j 3 - m 3 ) ! ( j 1 - m 1 ) ! ( j 2 + m 2 ) ! ( j 2 - m 2 ) ! ( j 3 + m 3 ) ! ) 1 2 F 2 3 ( - j 1 - j 2 - j 3 - 1 , - j 1 + m 1 , - j 3 - m 3 ; - j 1 - j 2 - m 3 , - j 2 - j 3 + m 1 ; 1 ) ,
where F 2 3 is defined as in §16.2. For alternative expressions for the 3 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 2 3 of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
16: 34.4 Definition: 6 j Symbol
34.4.3 { j 1 j 2 j 3 l 1 l 2 l 3 } = ( - 1 ) j 1 + j 3 + l 1 + l 3 Δ ( j 1 j 2 j 3 ) Δ ( j 2 l 1 l 3 ) ( j 1 - j 2 + l 1 + l 2 ) ! ( - j 2 + j 3 + l 2 + l 3 ) ! ( j 1 + j 3 + l 1 + l 3 + 1 ) ! Δ ( j 1 l 2 l 3 ) Δ ( j 3 l 1 l 2 ) ( j 1 - j 2 + j 3 ) ! ( - j 2 + l 1 + l 3 ) ! ( j 1 + l 2 + l 3 + 1 ) ! ( j 3 + l 1 + l 2 + 1 ) ! F 3 4 ( - j 1 + j 2 - j 3 , j 2 - l 1 - l 3 , - j 1 - l 2 - l 3 - 1 , - j 3 - l 1 - l 2 - 1 - j 1 + j 2 - l 1 - l 2 , j 2 - j 3 - l 2 - l 3 , - j 1 - j 3 - l 1 - l 3 - 1 ; 1 ) ,
where F 3 4 is defined as in §16.2. For alternative expressions for the 6 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 3 4 of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
17: 10.61 Definitions and Basic Properties
§10.61(i) Definitions
§10.61(ii) Differential Equations
§10.61(iii) Reflection Formulas for Arguments
In general, Kelvin functions have a branch point at x = 0 and functions with arguments x e ± π i are complex. …
§10.61(iv) Reflection Formulas for Orders
18: 17.9 Further Transformations of ϕ r r + 1 Functions
17.9.3 ϕ 1 2 ( a , b c ; q , z ) = ( a b z / c ; q ) ( b z / c ; q ) ϕ 2 3 ( a , c / b , 0 c , c q / ( b z ) ; q , q ) + ( a , b z , c / b ; q ) ( c , z , c / ( b z ) ; q ) ϕ 2 3 ( z , a b z / c , 0 b z , b z q / c ; q , q ) ,
19: 6.12 Asymptotic Expansions
If the expansion is terminated at the n th term, then the remainder term is bounded by 1 + χ ( n + 1 ) times the next term. For the function χ see §9.7(i). …
6.12.3 f ( z ) 1 z ( 1 - 2 ! z 2 + 4 ! z 4 - 6 ! z 6 + ) ,
20: 17.7 Special Cases of Higher ϕ s r Functions
F. H. Jackson’s Terminating q -Analog of Dixon’s Sum