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Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet -Euler numbers and twisted Euler polynomials at non-positive integers, respectively.

We will construct two-variable twisted -Euler-zeta function and two-variable - -function in Complex -plane.

In this section, we will construct two-variable twisted -Euler-zeta function and two-variable - -function in Complex -plane.

Note that if in (3.2), then we see that the twisted -Euler-zeta function is defined by (cf. [28, 55]) (3.3).

where σ k − 1 ∗ ( n ) is the twisted divisor function given by σ k − 1 ∗ ( n ) = ∑ d | n, n d odd d k − 1, χ 1 ( n ) = { 1, if  n  is odd, 0, otherwise and χ 0 ( n ) = { 1, if  n  is even, 0, otherwise.

The twisted part?

The objective of this paper is to introduce and prove convolution identities for the twisted divisor functions σ k − 1 ∗ ( n ) as well as for the twisted Eisenstein series S 2 k + 2, χ 0 and S 2 k + 2, χ 1, S 2 k + 2 ∗, S 2 k + 2, χ 0 ∗, and S 2 k + 2, χ 1 ∗.

From (3.3), we derive multiple twisted -Euler Zeta function as follows: (34).

The following theorem means that multiple twisted -Euler Zeta functions interpolate multiple twisted -Euler polynomials at negative integers.

We also obtain the following theorem in which multiple twisted -Euler Zeta functions interpolate multiple twisted -Euler polynomials.

In Section 3, we consider multiple twisted -Euler Zeta functions which interpolate new multiple twisted -Euler polynomials at negative integers and investigate some characterizations of them.