q-analogues of general reduction formulas by Buschman and Srivastava and an important q-operator reminding of MacRobert

Abstract We find four q-analogues of general reduction formulas from Buschman and Srivastava together with some special cases, e.g. q-analogues of reduction formulas for Appell- and Kampé de Fériet functions. A proper q-analogue of the notation △(l; λ) by MacRobert, Meijer and Srivastava is given, and the definition of q-hypergeometric series is generalized accordingly.


Introduction
The umbral method for q-calculus [2] - [8], consisting of logarithmic qshifted factorials, the tilde operator, a comfortable notation for q-powers, the symbol for real infinity, equivalent to the zero in Gasper-Rahman [10], the q-Kampé de Fériet function, compare with [3], are the main ingredients in this new method, which will increase our knowledge of q-calculus, advocated in the beginning of the last century by the late Cambridge student, reverend F. H. Jackson. All the topics above are not new; they have been presented in the book [9].
In this article, the important notation △(l; λ) of MacRobert [11], Meijer and Srivastava [15] for a certain array of l parameters is given its proper q-analogue with the aid of a generalized tilde operator; in this paper we only consider the cases l = 2, 3, but a general definition is given. A deep knowledge of the △(l; λ) operator is necessary to grasp the subtleties of multiple hypergeometric functions. This △-operator has a very long history in the field of special functions, in particular in India, which we will come back to in later papers.

T. Ernst
Buschman and Srivastava [1] have proved a great number of double series identities with general terms. We will find q-analogues of most of these formulas like in [3]; the method of proof will be similar except that we now use the q-Dixon-and q-Watson summation formulas. Some of the obtained formulas are symmetric in two variables, just as in the undeformed case. We pick out a form of these formulas, which converges nicely for small values of x. A list of different formulations of the Buschman-Srivastava formulas and their q-analogues in various journals and books is given, for better orientation.
This paper is organized as follows: In this section we give a general introduction. In section 2, four q-analogues of Buschman-Srivastava formulas are given. In section 3, we apply the Buschman-Srivastava formulas to find q-analogues of reduction formulas for Appell and Kampé de Fériet functions; the △ operator appears only in the Heine function. In other papers, the △ operator can appear also in the q-Kampé de Fériet function.
Definition 1. The power function is defined by q a ≡ e alog(q) . Let δ > 0 be an arbitrary small number. We will use the following branch of the logarithm: −π + δ < Im (log q) ≤ π + δ. This defines a simply connected space in the complex plane.
The variables a, b, c, . . . ∈ C denote certain parameters. The variables i, j, k, l, m, n, p, r will denote natural numbers except for certain cases where it will be clear from the context that i will denote the imaginary unit.
Since products of q-shifted factorials occur so often, to simplify them we shall frequently use the more compact notation Let the Γ q -function be defined in the unit disk 0 < |q| < 1 by The following notation will prove convenient, since many of our formulas contain exponents with upper and lower indices, which become less legible in the Gasper-Rahman notation.
General reduction formulas
Assume that (m, l) = 1, i.e. m and l relatively prime. The operator m l : We will also need another generalization of the tilde operator. This leads to the following q-analogue of [12, p.22, (2)].
Definition 2. A q-analogue of a notation due to Thomas MacRobert (1884-1962) [11, p. 135] and Srivastava [15]. This notation was also often used for the Meijer G-function and the Fox H-function (q = 1).
When λ is a vector, we mean the corresponding product of vector elements. When λ is replaced by a sequence of numbers separated by commas, we mean the corresponding product as in the case of q-shifted factorials. The last factor in (10) corresponds to l nl .

Definition of the q-Kampé de Fériet function
We will give a definition reminding of [10], which allows easy confluence to diminish the dimension in (12), and has the advantage of beeing symmertic in the variables. Furthermore, q is allowed to be a vector and the full machinery of tilde operators and q-additions will be used. In the following two definitions we put (11) a ≡ a ∨ã ∨ m n a ∨ k a ∨ △(q; l; λ). The following definition is a q-analogue of [16, (24), p. 38], in the spirit of Srivastava.
. . , n. Then the generalized q-Kampé de Fériet function is defined by We assume that no factors in the denominator are zero. We assume that Definition 4. Generalizing Heine's series we shall define a q hypergeometric series by (13) p+p .

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We assume that the f i (k) and g j (k) contain p ′ and r ′ factors of the form a(k); q k or (s(k); q) k respectively. In case of △(q; l; λ), the index is adapted accordingly. When we have a sequence of elements a i , we can denote them by (A).

Two lemmata
In the following three proofs we will use the finite q-Dixon theorem.
In another proof we will use a q-analogue of the Watson formula [1].
2. q-analogues of Buschman-Srivastava double sums The Buschman-Srivastava paper [1] was a landmark for the studies of multiple q-hypergeometric series. Some of these formulas had previously been published in other form by Shanker and Saran [13]. Srivastava and Jain [17] have found q-analogues of some of these formulas, some of which are included in the book [9]. The following table summarizes the connection between the various formulas and the methods of proof; the four references are in chronological order.  We are now going to find a number of general double sums. Since the convergence problem is rather delicate, we try to choose the most proper form with respect to an arbitrary q-power. Sometimes we add this q-power afterwards, to save space in the proof. In the following, a statement like a = k will mean a = k, k ∈ N. Everywhere the symbol {C n } ∞ n=0 denotes a bounded sequence of complex numbers. It is assumed that both sides converge. Note that the formulas (18), (20) and (23) Proof.
Proof. We prove an equivalent formula.
where we have used (14) for the q-Dixon theorem.
Finally, multiply C n by QE n 2 − ng .