Basic hypergeometric series

On transformation formulae of ordinary hypergeometric series

“…The hypergeometric function and its generalizations, summation theorems and transformation formulae have been presented in many textbooks [1], [5], [6], and [11] where references to the extensive literature on the subject may be found. Mathematicians working in the area of ordinary and basic hypergeometric series were interested for transformation formulae among various generalised hypergeometric functions and they succeeded in their goal.…”

A Bilateral Series Involving Basic Hypergeometric Functions

“…We denote by r+1 φ r not only the convergent basic hypergeometric series, but also its analytic continuation to C \ R ≥1 ; cf. [GR,§4.5]. …”

“…Note also that interchanging the roles of k+1 φ k and l+1 φ l gives an alternative expression for the series, which is valid when (4) is replaced by the condition |qxd 1 · · · d l | < |yc 1 · · · c l+1 |. These two expressions are related by [GR,(4.10.10)]. …”

“…This is an integral of the form [GR,(4.9.4)], which is computed there using residue calculus. If |xq n | < 1, its value is given by [GR,(4.10.9)] as the k+1 φ k series in (7). Since (8) is analytic in x, this also holds for |xq n | ≥ 1 in the sense of analytic continuation.…”

Some Systems of Multivariable Orthogonal Askey-Wilson Polynomials

“…The Askey-Wilson polynomials defined as in [1] and [3] by p n (x|q) = p n (x; a, b, c, d|q) (1.9) = a −n (ab, ac, ad; q) n 4 φ 3 q −n , abcdq n−1 , ae iθ , ae −iθ ab, ac, ad ; q, q , where x = cos θ, are a q-analogue of the Wilson polynomials (for the definition of the q-shifted factorials and the basic hypergeometric series 4 φ 3 see [3]). These polynomials satisfy the orthogonality relation…”