© Taekyun Kim et al. 2010
Received: 1 December 2009
Accepted: 14 March 2010
Published: 30 March 2010
Let be a fixed prime number. Throughout this paper, the symbols , , , and denote the ring of rational integers, the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers and . The -adic absolute value in is normalized in such a way that (see [1–17]). For with , let be the least common multiple of and . We set
When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . In this paper, we assume that with (see [1–6, 18–23]). As the definition of -number, we use the following notations:
with the usual convention of replacing by (see ). Thus, we note that
Recently Govil and Gupta  have introduced a new type of q-integrated Meyer-König-Zeller-Durrmeyer (q-MKZD) operators, obtained moments for these operators, and estimated the convergence of these integrated q-MKZD operators. In this paper, we consider the q-extension which is in a direction different than that of Govil and Gupta .
Let be a field over . Then we call a function a -measure on if is finitely additive function defined on open-closed subsets in , whose values are in the field . Any open-closed subset in is a disjoint union of some finite intervals in , where is prime to , and therefore a -measure is determined by its values on all intervals in . Let denote the set of all rational numbers, whose denominator is a divisor of for some . In Section 2, we derive the modified -adic -measures related to -Nasybullin's type lemma.
Thus, we have
Therefore we obtain the following theorem.
From (1.9), we note that
From (2.12) and (2.7), we can easily derive the following Witt's formula:
For (2.14), (2.16) and (2.17), we note that
Therefore, we obtain the following theorem.
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