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Dixon 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
Rogers–Dougall 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: Peter A. Clarkson
     Dixon and J. …
    6: 17.7 Special Cases of Higher ϕ s r Functions
    Sum Related to (17.6.4)
    q -Pfaff–Saalschütz Sum
    F. H. Jackson’s Terminating q -Analog of Dixon’s Sum
    q -Analog of Dixon’s F 2 3 ( 1 ) Sum
    Gosper’s Bibasic Sum
    7: 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.
  • A. L. Dixon and W. L. Ferrar (1930) Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.
  • J. M. Dixon, J. A. Tuszyński, and P. A. Clarkson (1997) From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics. Oxford University Press, Oxford.
  • 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.
  • 8: 4.27 Sums
    §4.27 Sums
    For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).
    9: 4.11 Sums
    §4.11 Sums
    10: 4.41 Sums
    §4.41 Sums
    For sums of hyperbolic functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §43), Prudnikov et al. (1986a, §5.3), and Zucker (1979).