THE PARTIALLY DEGENERATE CHANGHEE-GENOCCHI POLYNOMIALS AND NUMBERS

In this paper, we introduce the partially degenerate ChangheeGenocchi polynomials and numbers and investigated some identities of these polynomials. Furthermore, we investigate some explicit identities and properties of the partially degenerate Changhee-Genocchi arising from the nonlinear differential equations.


Introduction
As is well known, the Genocchi polynomials G n (x) are defined by the generating function as follows: G n (x) t n n! (see [1,3,6,17]). (1.1) When x = 0, G n = G n (0) are called the Genocchi numbers.
By replacing t by e t − 1, we get Ch m (x)S 2 (n, m) t n n! , (1.3) where E m (x) are ordinary Euler polynomials.Thus, we have Ch m (x)S 2 (n, m). (1.4) Now, we define the degenerate exponential function as follow: e t λ = (1 + λt) When x = 0, CG n = CG n (0) are called the Changhee-Genocchi numbers.
The Genocchi-Changhee polynomials GCh n (x) are defined by the generating function to be GCh n (x) t n n! .
We recall the Stirling numbers of the first kind S 1 (n, m) and S 2 (n, m) are defined by [4,7,14]).
(1.12) and [10,12,15]). (1.13) Recently, B-M.Kim et al. studied Changhee-Genocchi polynomials and some identities of these polynomials.They also introduced Changhee-Genocchi polynomials and investigated some identities of these polynomials ([?]).Also, H. -I.Kwon et al. introduced degenerate Changhee-Genocchi polynomials and some identities of these polynomials and investigated some identities of these polynomials ([?]).In this paper, we introduce the partially degenerate Changhee-Genocchi polynomials and numbers and investigated some identities of these polynomials.Furthermore, we investigate some explicit identities and properties of the partially degenerate Changhee-Genocchi arising from the nonlinear differential equations.

The partially degenerate Changhee-Genocchi polynomials and numbers
In this section, we define the partially degenerate Changhee-Genocchi polynomials and numbers and investigate some identities of the partially degenerate Changhee-Genocchi polynomials.Now, we consider the degenerate Genocchi polynomials which are given by the generating function to be 2t e t λ + 1 t n n! . (2.1) Thus, t n+1 n! . (2.3) Comparing the coefficients on the both sides in (??), we have the following result.
Now, we define the partially degenerate Changhee-Genocchi polynomials which are given by 2 log( When x = 0, CG n,λ = CG n,λ (0) are called the partially degenerate Changhee-Genocchi numbers.Also, we define the higher-order partially degenerate Changhee-Genocchi numbers which are given by the generating function to be (2.9) Now, we observe that lim Comparing the coefficients on the both sides in (??), we have the following result.
Now, we observe that (2.12) Comparing the coefficients on the both sides in (??), we have the following result.

The partially degenerate Changhee-Genocchi numbers arising from differential equations
In this section, we investigate some identities of the partially degenerate Changhee-Genocchi numbers arising from differential equations. Let Then, by taking the derivative with respect to t of (??), we obtain From (??), we have λF 2 = −(1 + λt)F (1) .
(3.4) Thus, by multiple (1 + λt) on the both sides of (??), we obtain From (??) and (??), we get From the above equation, we have Multiply (1 + λt) on the both sides of (??), we get From (??) and (??), we obtain Continuing this process, we get Let us take the derivative on the both sides of (??) with respect to t.Then we obtain Multiply (1 + λt) on the both sides of (??), we have Then, by (??) and (??), we obtain By substituting N by N + 1 given in (??), we have another equation.
Theorem 3.1.Let N ∈ N. Then the following differential equation, , where From (??), we get From the above equation, we get Multiply 2 N +1 λ(log(1 + t)) N +1 on the right sides of (??), we get Where S 1 (n, k) is the Stirling number of the first kind.