About the Project
NIST

q-analogs

AdvancedHelp

(0.002 seconds)

6 matching pages

1: 17.7 Special Cases of Higher ϕ s r Functions
F. H. Jackson’s Terminating q -Analog of Dixon’s Sum
q -Analog of Dixon’s F 2 3 ( 1 ) Sum
Andrews’ Terminating q -Analog of (17.7.8)
Andrews’ Terminating q -Analog
Gasper–Rahman q -Analogs of the Karlsson–Minton Sums
2: 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 ) . …
3: 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 (1988) A q -analog of the Gauss summation theorem for hypergeometric series in U ( n ) . Adv. in Math. 72 (1), pp. 59–131.
  • S. C. Milne (1994) A q -analog of a Whipple’s transformation for hypergeometric series in U ( n ) . Adv. Math. 108 (1), pp. 1–76.
  • 4: 17.9 Further Transformations of ϕ r r + 1 Functions
    Watson’s q -Analog of Whipple’s Theorem
    Gasper’s q -Analog of Clausen’s Formula
    5: 24.16 Generalizations
    In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); p -adic integer order Bernoulli numbers (Adelberg (1996)); p -adic q -Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).
    6: 17.6 ϕ 1 2 Function
    First q -Chu–Vandermonde Sum