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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: 27.19 Methods of Computation: Factorization
    The snfs can be applied only to numbers that are very close to a power of a very small base. …
    6: 17.18 Methods of Computation
    Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. …
    7: Diego Dominici
    Diego was very active in the SIAM activity group on Orthogonal Polynomials and Special Functions since 2010. …
    8: 27.13 Functions
    The basic problem is that of expressing a given positive integer n as a sum of integers from some prescribed set S whose members are primes, squares, cubes, or other special integers. Each representation of n as a sum of elements of S is called a partition of n , and the number S ( n ) of such partitions is often of great interest. … This conjecture dates back to 1742 and was undecided in 2009, although it has been confirmed numerically up to very large numbers. Vinogradov (1937) proves that every sufficiently large odd integer is the sum of three odd primes, and Chen (1966) shows that every sufficiently large even integer is the sum of a prime and a number with no more than two prime factors. … This problem is named after Edward Waring who, in 1770, stated without proof and with limited numerical evidence, that every positive integer n is the sum of four squares, of nine cubes, of nineteen fourth powers, and so on. …
    9: 37.21 Physical Applications
    A very important application is in lithography (see references in de Winter et al. (2020)), where Zernike polynomials are still used in the recent EUV (extreme ultraviolet) lithography developed by ASML and Zeiss companies. …
    10: About MathML
    That format displays mathematics as static images and is neither very scalable nor accessible, but serves as a workable fallback. …