On -Operators and Summation of Some -Series

Using Jackson's -derivative and the -Stirling numbers, we establish some transformation theorems leading to the values of some convergent -series. 1. Introduction The operator has many assets and plays a central role in arithmetic fields and in computation of some finite or infinite sums. For example, when we try to compute the sum , we use the operators , which give These operators are intimately related to the Stirling numbers of second kind by the formula (see [1]) where is a suitable function. We note that the -analogue of formula (1.2) has been studied by many authors (see [2, 3] and references therein) and has found applications in many fields such as arithmetic partitions and asymptotic expansions. This paper deals with the analogues of the operators in Quantum Calculus and some -transformation theorems that will be used to establish the sums of some -series. This paper is organized as follows. In Section 2, we present some preliminary notions and notations useful in the sequel. Section 3 gives three applications of a result proved in [2], states a transformation theorem using the -Stirling numbers, and presents some related applications. Section 4 attempts to give a new -analogue of formula (1.2) by studying the transformation theorem related to a -derivative operator. 2. Notations and Preliminaries To make this paper self-containing and easily decipherable, we recall some useful preliminaries about the Quantum Calculus and we select Gasper-Rahman’s book [4], for the notations and for a deep study in this way. Throughout this paper, we fix . 2.1. -Shifted Factorials For , the -shifted factorials are defined by We also write We put For and , we adopt the following notation [5]: The -analogue of the Jordan factorial is given by and the -binomial coefficient is defined by 2.2. The Jackson's -Derivative The -derivative of a function is defined by (see [4]) and provided exists. Note that when is differentiable, at , then tends to as tends to . It is easy to see that for suitable functions and , we have 2.3. Elementary -Special Functions Two -analogues of the exponential function are given by (see [4]) They satisfy the relations In 1910, F. H. Jackson defined a -analogue of the Gamma function by (see [4, 6]) It satisfies the following functional equations: 2.4. -Stirling Numbers of Noncentral Type In [7], Charalambides introduced the so-called noncentral -Stirling numbers, which are -analogues of the Stirling numbers and classified into two kinds. The noncentral -Stirling numbers of the first kind are defined by the following generating relation: and