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Pfaff?Saalsch�tz balanced sum

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1: 16.4 Argument Unity
When k = 1 the function is said to be balanced or Saalschützian. …
Pfaff–Saalschütz Balanced Sum
Contiguous balanced series have parameters shifted by an integer but still balanced. … …
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 balanced or Saalschützian when it terminates, r = s , z = q , and … The series (17.4.1) is said to be k-balanced when r = s and …
3: 17.9 Further Transformations of ϕ r r + 1 Functions
Sears’ Balanced ϕ 3 4 Transformations
§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: 16.24 Physical Applications
These are balanced F 3 4 functions with unit argument. …
5: 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).
6: 4.11 Sums
§4.11 Sums
7: 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).
8: 7.15 Sums
§7.15 Sums
For sums involving the error function see Hansen (1975, p. 423) and Prudnikov et al. (1986b, vol. 2, pp. 650–651).
9: 18.38 Mathematical Applications
18.38.3 m = 0 n P m ( α , 0 ) ( x ) = ( α + 2 ) n n ! F 2 3 ( n , n + α + 2 , 1 2 ( α + 1 ) α + 1 , 1 2 ( α + 3 ) ; 1 2 ( 1 x ) ) 0 , x 1 , α 2 , n = 0 , 1 , ,
The 6 j symbol (34.4.3), with an alternative expression as a terminating balanced F 3 4 of unit argument, can be expressend in terms of Racah polynomials (18.26.3). …
f ( z ) = c 0 + k = 1 n c k ( z k + z k ) .
10: Bibliography M
  • 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. C. Milne (1997) Balanced Θ 2 3 summation theorems for U ( n ) basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.
  • 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.
  • L. J. Mordell (1917) On the representation of numbers as a sum of 2 r squares. Quarterly Journal of Math. 48, pp. 93–104.