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1: 16.4 Argument Unity
Rogers–Dougall Very Well-Poised Sum
Dougall’s Very Well-Poised Sum
Denote, formally, the bilateral hypergeometric function …This is Dougall’s bilateral sum; see Andrews et al. (1999, §2.8).
2: 15.4 Special Cases
Dougall’s Bilateral Sum
15.4.25 n = - Γ ( a + n ) Γ ( b + n ) Γ ( c + n ) Γ ( d + n ) = π 2 sin ( π a ) sin ( π b ) Γ ( c + d - a - b - 1 ) Γ ( c - a ) Γ ( d - a ) Γ ( c - b ) Γ ( d - b ) .
3: 17.8 Special Cases of ψ r r Functions
§17.8 Special Cases of ψ r r Functions
Ramanujan’s ψ 1 1 Summation
Bailey’s Bilateral Summations
Sum Related to (17.6.4)
For similar formulas see Verma and Jain (1983).
4: 17.1 Special Notation
§17.1 Special Notation
The main functions treated in this chapter are the basic hypergeometric (or q -hypergeometric) function ϕ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , the bilateral basic hypergeometric (or bilateral q -hypergeometric) function ψ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , and the q -analogs of the Appell functions Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) , Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) , Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) , and Φ ( 4 ) ( a , b ; c , c ; q ; x , y ) . …
f ( χ 1 ; χ 2 , , χ n ) + idem ( χ 1 ; χ 2 , , χ n ) = j = 1 n f ( χ j ; χ 1 , χ 2 , , χ j - 1 , χ j + 1 , , χ n ) .
5: 14.18 Sums
§14.18 Sums
§14.18(iii) Other Sums
Dougall’s Expansion
For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2000, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). …
6: 17.18 Methods of Computation
§17.18 Methods of Computation
7: 17.7 Special Cases of Higher ϕ s r Functions
Sum Related to (17.6.4)
q -Pfaff–Saalschütz Sum
F. H. Jackson’s q -Analog of Dougall’s F 6 7 ( 1 ) Sum
Gasper–Rahman q -Analogs of the Karlsson–Minton Sums
Gosper’s Bibasic Sum
8: 17.10 Transformations of ψ r r Functions
§17.10 Transformations of ψ r r Functions
17.10.1 ψ 2 2 ( a , b c , d ; q , z ) = ( a z , d / a , c / b , d q / ( a b z ) ; q ) ( z , d , q / b , c d / ( a b z ) ; q ) ψ 2 2 ( a , a b z / d a z , c ; q , d a ) ,
17.10.2 ψ 2 2 ( a , b c , d ; q , z ) = ( a z , b z , c q / ( a b z ) , d q / ( a b z ) ; q ) ( q / a , q / b , c , d ; q ) ψ 2 2 ( a b z / c , a b z / d a z , b z ; q , c d a b z ) .
17.10.3 ψ 8 8 ( q a 1 2 , - q a 1 2 , c , d , e , f , a q - n , q - n a 1 2 , - a 1 2 , a q / c , a q / d , a q / e , a q / f , q n + 1 , a q n + 1 ; q , a 2 q 2 n + 2 c d e f ) = ( a q , q / a , a q / ( c d ) , a q / ( e f ) ; q ) n ( q / c , q / d , a q / e , a q / f ; q ) n ψ 4 4 ( e , f , a q n + 1 / ( c d ) , q - n a q / c , a q / d , q n + 1 , e f / ( a q n ) ; q , q ) ,
17.10.4 ψ 2 2 ( e , f a q / c , a q / d ; q , a q e f ) = ( q / c , q / d , a q / e , a q / f ; q ) ( a q , q / a , a q / ( c d ) , a q / ( e f ) ; q ) n = - ( 1 - a q 2 n ) ( c , d , e , f ; q ) n ( 1 - a ) ( a q / c , a q / d , a q / e , a q / f ; q ) n ( q a 3 c d e f ) n q n 2 .
9: 17.4 Basic Hypergeometric Functions
§17.4 Basic Hypergeometric Functions
In these references the factor ( ( - 1 ) n q ( n 2 ) ) s - r is not included in the sum. …
§17.4(ii) ψ s r Functions
17.4.3 ψ s r ( a 1 , a 2 , , a r b 1 , b 2 , , b s ; q , z ) = ψ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) = n = - ( a 1 , a 2 , , a r ; q ) n ( - 1 ) ( s - r ) n q ( s - r ) ( n 2 ) z n ( b 1 , b 2 , , b s ; q ) n = n = 0 ( a 1 , a 2 , , a r ; q ) n ( - 1 ) ( s - r ) n q ( s - r ) ( n 2 ) z n ( b 1 , b 2 , , b s ; q ) n + n = 1 ( q / b 1 , q / b 2 , , q / b s ; q ) n ( q / a 1 , q / a 2 , , q / a r ; q ) n ( b 1 b 2 b s a 1 a 2 a r z ) n .
10: Bibliography D
  • K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.
  • A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
  • J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
  • T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.