Note on -Nasybullin's Lemma Associated with the Modified -Adic -Euler Measure

Abstract

We derive the modified -adic -measures related to -Nasybullin's type lemma.

1. Introduction

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

(1.1)

where lies in .

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:

(1.2)

(see [1–23]).

Let be the space of uniformly differentiable function on . For , the -adic -invariant integral on is defined as

(1.3)

(see [2, 3]).

The -Euler numbers, , can be determined inductively by

(1.4)

with the usual convention of replacing by (see [11]). The modified -Euler numbers of are defined in [2] as follows:

(1.5)

with the usual convention of replacing by . For any positive integer ,

(1.6)

is known as a measure on (see [9]). In [2], the Witt's type formulas for are given by

(1.7)

The modified -Euler polynomials are also defined by

(1.8)

with the usual convention of replacing by (see [2]). Thus, we note that

(1.9)

Recently Govil and Gupta [22] 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 [22].

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.

2. The Modified -Adic -Measure

Let be a -valued function defined on with the following property.

There exist two constants such that

(2.1)

for any number . Suppose that is a root of the equation . Then we define

(2.2)

for any interval . From (2.2), we note that

(2.3)

Thus, we have

(2.4)

Therefore we obtain the following theorem.

Theorem 2.1.

For with and , let be a -valued function defined on with the following properties.

There exist two constants such that

(2.5)

for any . Suppose that is a root of the equation . Then there exists a -measure on such that

(2.6)

for any interval .

From (1.9), we note that

(2.7)

Let be the th -Euler polynomials and let be the th -Euler functions, that is, for ,

(2.8)

Note that is the Euler function. By (2.7), we see that

(2.9)

Thus, the -Euler function satisfies the properties of Theorem 2.1 with constants

(2.10)

Then is equal to , as reduces simply to . Therefore, we obtain the following theorem.

Theorem 2.2.

For , let the function be defined on as follows:

(2.11)

Then is a -measure on .

For with and , let be a primitive Dirichlet character modulo . Then the generalized -Euler numbers are defined as follows:

(2.12)

From (2.12) and (2.7), we can easily derive the following Witt's formula:

(2.13)

We can compute a -analogue of the -adic --function by the following -adic -Mellin Mazur transform with respect to .

Let

(2.14)

Since the character is constant on the interval ,

(2.15)

where are the th generalized -Euler numbers attached to . For , we have

(2.16)

Assume that with . Let be the Teichmüller character mod . For , we set . Note that and are defined by for . For , we define

(2.17)

For (2.14), (2.16) and (2.17), we note that

(2.18)

Since for , we have . Let . Then we have

(2.19)

Therefore, we obtain the following theorem.

Theorem 2.3.

For , we have

(2.20)