Next, we will consider the polynomials (212).
Now, one considers the polynomials of by (2.29).
Proof of Theorem 3.1 Let m ≥ 2, x = ( x 1, x 2, …, x m ) ∈ R m, and we consider the polynomials.
Let us consider the polynomials T n ( r, k ) ( x | λ ), called higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials, as follows: ( 1 − λ e t − λ ) r L i k ( 1 − e − t ) 1 − e − t e x t = ∑ n = 0 ∞ T n ( r, k ) ( x | λ ) t n n !, (2.1).
If for λ ∈ { 0, 1 } one considers the polynomials (the sum of binary variables is understood as the sum by mod ( 2 ) ). then denoting − x = ( − x 1, …, − x m ) it follows that for λ ∈ { 0, 1 }, P m ( x ) = 1 2 [ P m ( x ) + ( − 1 ) λ P m ( − x ) ]. (21).
In this paper, we consider the polynomials D ˆ n ( k ) ( x | a 1, …, a r ) called the Barnes-type Daehee of the second kind and poly-Cauchy of the second kind mixed-type polynomials, whose generating function is given by ∏ j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j ( 1 + t ) a j − 1 ) Lif k ( − ln ( 1 + t ) ) ( 1 + t ) x = ∑ n = 0 ∞ D ˆ n ( k ) ( x | a 1, …, a r ) t n n !, (1).
In this paper, we consider the polynomials D n ( k ) ( x | a 1, …, a r ) called the Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type polynomials, whose generating function is given by ∏ j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j − 1 ) Lif k ( ln ( 1 + t ) ) ( 1 + t ) x = ∑ n = 0 ∞ D n ( k ) ( x | a 1, …, a r ) t n n !, (1).
Let r ∈ Z > 0. We consider the polynomials N n ( x | a 1, …, a r ) and N ˆ n ( x | a 1, …, a r ), respectively, called the Barnes-type Narumi polynomials of the first kind and those of the second kind and respectively given by ∏ j = 1 r ( ( 1 + t ) a j − 1 log ( 1 + t ) ) ( 1 + t ) x = ∑ n = 0 ∞ N n ( x | a 1, …, a r ) t n n !
In this paper, we consider the polynomials D n ( x | a 1, …, a r ) and D ˆ n ( x | a 1, …, a r ) called the Barnes-type Daehee polynomials of the first kind and of the second kind, whose generating functions are given by ∏ j = 1 r ( ln ( 1 + t ) ( 1 + t ) a j − 1 ) ( 1 + t ) x = ∑ n = 0 ∞ D n ( x | a 1, …, a r ) t n n !, (1).
Here we will consider the polynomials S n ( x ) = S n ( x | λ 1, …, λ r ; μ 1, …, μ r ) and S ˆ n ( x ) = S ˆ n ( x | λ 1, …, λ r ; μ 1, …, μ r ), which are called Barnes-type Peters polynomials of the first kind and of the second kind, respectively, and are given by ∏ j = 1 r ( 1 + ( 1 + t ) λ j ) − μ j ( 1 + t ) x = ∑ n ≥ 0 S n ( x ) t n n !, (1.1).
