q-functions of several variables
We begin with the vector notation for the most important functions and q-Taylor formulas for power series and functions of inverse q-shifted factorials. We continue with a historical introduction to the rest of this long and interesting chapter and to the
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q-functions of several variables
Symmetries correspond to the q-functions of many variables. SO(3) q-Appell functions SU(2) q-Lauricella functions As molecules strive for the configuration with the greatest symmetry, our transformations are often symmetrical. The multiple q-hypergeometric functions are defined by the q-shifted factorial and the tilde operator. By our method we are able to find q-analogues of corresponding formulas for the multiple hypergeometric case, which elucidate the integration properties of q-calculus and helps in trying to make a first systematic attempt to find summation and reduction theorems for multiple basic series, including the Jackson q function. We give two definitions of the generalized q-Kampé de Fériet function, in the spirit of Karlsson and Srivastava [482], which are symmetric in the variables and allow us to treat q as a vector. Various connections between transformation formulas for multiple q-hypergeometric series and Lie algebras and finite groups known from the literature are cited.
10.1 The corresponding vector notation Definition 162 We define vector versions of powers, q-shifted factorials, q-Pochhammer symbols and JHC q-additions, as follows: α
x ≡
n
α
xj j ,
(10.1)
j =1
1 1 ≡ , ( x ; q)β (xj ; qj )βj n
(10.2)
j =1
T. Ernst, A Comprehensive Treatment of q-Calculus, DOI 10.1007/978-3-0348-0431-8_10, © Springer Basel 2012
359
360
10
q ! ≡ {l}
n
q-functions of several variables
{lj }qj !,
(10.3)
j =1 n
{ α }m, q ≡
{αj }mj ,qj ,
(10.4)
j =1 k
q (2) ≡
n
kj
qj ( 2 ) ,
(10.5)
j =1
(−1)k ≡ (−1)|k| ,
≡
n
j ,
(10.6)
j =1 n
Pk, x , y) ≡ q (
Pkj ,qj (xj , yj ),
(10.7)
j =1
α ; qk ≡
n
αj ; qj kj .
(10.8)
j =1
The q-factorial (10.8) is usually written (α); q k . The partial q-derivative of a function of n variables is defined as an operator in the spirit of Hörmander [285, p. 12]: n
l Dqjj ,xj F ( x , q),
x , q) ≡ Dlq,x F (
(10.9)
j =1
n l t l Dq,x t F ( Dqj ,xj jj j F ( x , q) ≡ x , q ).
(10.10)
j =1
Closed and open intervals are defined by ≡ [ a , b]
n
[aj , bj ],
≡ ( a , b)
j =1
n
(aj , bj ).
(10.11)
j =1
Jackson [299, p. 145], [300], [304, p. 146] has shown certain connections be ak x f (k) tween power series in x and series of the form ∞ k=0 (x;q)k , where f (k) is an integer-valued function. In 1921 Ryde [444] showed that certain linear homogeneous q-difference equations have series ∞ ak k=0
as solutions.
(x; q −1 )k
10.1
The corresponding vector notation
361
We need a q-analogue of holomorphic functions, which will be useful to characterize the functions to the far right in operator expressions. q,n denote the class of functions F ( Definition 163 Let H x ) of n variables which can be written in the form F ( x) ≡
∞ 0 k=
xαk
γk δk
( x q ; q ±1 )β
,
k
where α k ∈ N , βk ∈ Rn , γk ∈ Cn , δk ∈ Nn . n
(10.12)
q,n (O). If the function is defined in an open region O
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