q-hypergeometric series
This chapter starts with the general definition of q-hypergeometric series. This definition contains the tilde operator and the symbol ∞, dating back to the year 2000. The notation △(q;l;λ), a q-analogue of the Srivastava notation for a multiple index, pl
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q-hypergeometric series
This is a modest attempt to present a new notation for q-calculus and, in particular, for q-hypergeometric series, which is compatible with the old notation. Also, a new method that follows from this notation is presented. This notation leads to a new method for computations and classifications of q-special functions. With this notation many formulas of q-calculus become very natural and the q-analogues of many orthogonal polynomials and functions assume a very pleasant form reminding directly of their classical counterparts.
7.1 Definition of q-hypergeometric series The stress tensor (matrix) corresponds to the generalized q-hypergeometric series. Matrices
p φr ((a); (b)|q; x)
Definition 134 Generalizing Heine’s series (6.181), we define a q-hypergeometric series by (compare with [220, p. 4], [261, p. 345]): aˆ 1 , . . . , aˆ p i fi (k) φ |q; z|| p+p r+r bˆ1 , . . . , bˆr j gj (k) ∞ aˆ 1 ; qk · · · aˆ p ; qk k (k2) 1+r+r −p−p k i fi (k) ≡ (−1) q z , (7.1) 1, bˆ1 ; qk · · · bˆr ; qk j gj (k) k=0
where m
a ≡ a ∨ a∨l a ∨k a ∨ (q; l; λ).
(7.2)
In case of (q; l; λ) the index is adjusted accordingly. It is assumed that the denominator contains no zero factors, i.e. bk = 0, and also that the fi (k) and gj (k) contain a (k); qk or (s(k); q)k respectively. p and r factors of the form The resonance corresponds to the Heine infinity. T. Ernst, A Comprehensive Treatment of q-Calculus, DOI 10.1007/978-3-0348-0431-8_7, © Springer Basel 2012
241
242
7
Resonance
q-hypergeometric series
∞H
In a few cases the parameter a in (7.1) will be the real plus infinity (0 < |q| < 1). This corresponds to multiplication by 1. If we want to be formal, we could introduce a symbol ∞H , with the property ∞H ; qn = ∞H + α; qn = 1,
α ∈ C, 0 < |q| < 1.
(7.3)
The symbol ∞H corresponds to the parameter 0 in [220, p. 4]. We will denote ∞H by ∞ in the rest of the book. The following notation is also used: n∞ ≡ ∞, . . . , ∞, n times,
n ∈ N.
(7.4)
Furthermore, bj = 0, j = 1, . . . , r, bj = −m, j = 1, . . . , r, m ∈ N, bj =
2mπi , log q
j = 1, . . . , r, m ∈ N, [467].
ˆ We Remark 29 There is a rule analogous to (7.3), but for the Riemann sphere C. find a q-analogue of this in [185]. The use of the extra sign || in (7.1) is best illustrated by the following example, and μ where the vectors λ have length p: Example 31 The following notation makes sense: ), (q; 3; ν + σ − 1), 9∞ (q; 2; λ − k; qk 2 1 φ || |q; −x . (7.5) 16+4p 15+4p (q; 2; μ, ν, σ, ν + σ − 1), ν, σ, 1 − The (q; 3; ν + σ − 1) corresponds to 3k q-shifted factorials, this explains the 9∞. k
The motivation for the extra factor (−1)q ±(2) is that we need a q-analogue of lim
x→∞
p Fr (a1 , . . . , ap ; b1 , . . . , br−1 , x; xz)
= p Fr−1 (a1 , . . . , ap ; b1 , . . . , br−1 ; z).
(7.6)
This q-analogue is given by
lim p φr a1 , . . . , ap ; b1 , . . . , br−1 , x|q; zq x x→−∞
= p φr−1 (a1 , . . . , ap ; b1 , . . . , br−1 |q; z)
0 < |q| < 1 .
(7.7)
However, the q-
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