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Rogers–Dougall very well-poised sum

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1: 16.4 Argument Unity
It is very well-poised if it is well-poised and a 1 = b 1 + 1 . …
Dixon’s Well-Poised Sum
RogersDougall Very Well-Poised Sum
Dougall’s Very Well-Poised Sum
2: 17.4 Basic Hypergeometric Functions
In these references the factor ( ( 1 ) n q ( n 2 ) ) s r is not included in the sum. …
17.4.3 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 .
17.4.5 Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) = m , n 0 ( a ; q ) m + n ( b ; q ) m ( b ; q ) n x m y n ( q ; q ) m ( q ; q ) n ( c ; q ) m + n ,
The series (17.4.1) is said to be well-poised when r = s and … The series (17.4.1) is said to be very-well-poised when r = s , (17.4.11) is satisfied, and …
3: 17.9 Further Transformations of ϕ r r + 1 Functions
Bailey’s Transformation of Very-Well-Poised ϕ 7 8
§17.9(iv) Bibasic Series
17.9.19 n = 0 ( a ; q 2 ) n ( b ; q ) n ( q 2 ; q 2 ) n ( c ; q ) n z n = ( b ; q ) ( a z ; q 2 ) ( c ; q ) ( z ; q 2 ) n = 0 ( c / b ; q ) 2 n ( z ; q 2 ) n b 2 n ( q ; q ) 2 n ( a z ; q 2 ) n + ( b ; q ) ( a z q ; q 2 ) ( c ; q ) ( z q ; q 2 ) n = 0 ( c / b ; q ) 2 n + 1 ( z q ; q 2 ) n b 2 n + 1 ( q ; q ) 2 n + 1 ( a z q ; q 2 ) n .
17.9.20 n = 0 ( a ; q k ) n ( b ; q ) k n z n ( q k ; q k ) n ( c ; q ) k n = ( b ; q ) ( a z ; q k ) ( c ; q ) ( z ; q k ) n = 0 ( c / b ; q ) n ( z ; q k ) n b n ( q ; q ) n ( a z ; q k ) n , k = 1 , 2 , 3 , .
4: Bibliography M
  • S. C. Milne (1985a) A q -analog of the F 4 5 ( 1 ) summation theorem for hypergeometric series well-poised in 𝑆𝑈 ( n ) . Adv. in Math. 57 (1), pp. 14–33.
  • S. C. Milne (1985d) A q -analog of hypergeometric series well-poised in 𝑆𝑈 ( n ) and invariant G -functions. Adv. in Math. 58 (1), pp. 1–60.
  • S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.
  • S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.
  • S. Moch, P. Uwer, and S. Weinzierl (2002) Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43 (6), pp. 3363–3386.
  • 5: 17.18 Methods of Computation
    Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …
    6: 17.12 Bailey Pairs
    17.12.1 n = 0 α n γ n = n = 0 β n δ n ,
    β n = j = 0 n α j u n j v n + j ,
    γ n = j = n δ j u j n v j + n .
    The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …
    7: 17.14 Constant Term Identities
    Rogers–Ramanujan Constant Term Identities
    17.14.2 n = 0 q n ( n + 1 ) ( q 2 ; q 2 ) n ( q ; q 2 ) n + 1 =  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 q 2 ; q 2 ) ( q ; q 2 ) ( z 1 q ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 q ; q ) = H ( q ) ( q ; q 2 ) ,
    17.14.3 n = 0 q n ( n + 1 ) ( q 2 ; q 2 ) n ( q ; q 2 ) n + 1 =  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 ; q 2 ) ( q ; q 2 ) ( z 1 q ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 ; q ) = G ( q ) ( q ; q 2 ) ,
    17.14.4 n = 0 q n 2 ( q 2 ; q 2 ) n ( q ; q 2 ) n =  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 ; q 2 ) ( q ; q 2 ) ( z 1 ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 2 ; q 4 ) = G ( q 4 ) ( q ; q 2 ) ,
    17.14.5 n = 0 q n 2 + 2 n ( q 2 ; q 2 ) n ( q ; q 2 ) n + 1 =  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( q 2 z 1 ; q 2 ) ( q ; q 2 ) ( z 1 q 2 ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( q 4 z 2 ; q 4 ) = H ( q 4 ) ( q ; q 2 ) .
    8: David M. Bressoud
    His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …
    9: 26.10 Integer Partitions: Other Restrictions
    where the last right-hand side is the sum over m 0 of the generating functions for partitions into distinct parts with largest part equal to m . … where the inner sum is the sum of all positive odd divisors of t . … where the inner sum is the sum of all positive divisors of t that are in S .
    §26.10(iv) Identities
    Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …
    10: 18.33 Polynomials Orthogonal on the Unit Circle
    18.33.2 ϕ n ( z ) = κ n z n + = 1 n κ n , n z n ,
    When a = 0 the Askey case is also known as the Rogers–Szegő case. See for a more general class Costa et al. (2012). …
    18.33.21 p ( z ) = k = 0 n c k z k , c n 0 ,
    18.33.22 p ( z ) z n p ( z ¯ 1 ) ¯ = k = 0 n c n k ¯ z k .