On Generating Functions for Boole Type Polynomials and Numbers of Higher Order and Their Applications

: The purpose of this manuscript is to study and investigate generating functions for Boole type polynomials and numbers of higher order. With the help of these generating functions, many properties of Boole type polynomials and numbers are presented. By applications of partial derivative and functional equations for these functions, derivative formulas, recurrence relations and ﬁnite combinatorial sums involving the Apostol-Euler polynomials, the Stirling numbers and the Daehee numbers are given.

where v ∈ N 0 (cf. ). The definition of the Apostol-Euler polynomials of order v, shown by E n (x, λ), is given below.
When λ = 1, the above equation reduces to the well-known the Euler numbers E n = E n (1) (cf. ). The definition of the Stirling numbers of the first kind, shown by S 1 (n, k), is given below.
If k > n, then S 1 (n, k) = 0 (cf. 32]). The definition of the Stirling numbers of the second kind, shown by S 2 (n, k), is given below.
If k > n, then S 2 (n, k) = 0 (cf. 32]). The Peters polynomials are one of the members of the Sheffer polynomials, which are a very broad family of polynomial sequences. The definition of the Peters polynomials, shown by s n (x; λ, µ), is given below.
The definition of the Daehee numbers, shown by D n , is given below.
Here, brief information about notations and index for the above special combinatorial numbers and polynomials is given as follows: The first author has recently defined many different Peters and Boole type combinatorial numbers and polynomials. He gave some notations for these numbers and polynomials. For instance, in order to distinguish them from each other, these polynomials are labeled by the following symbols: y j,n (x; λ, q), j = 1, 2, ..., 7, and also Y n (x; λ). Therefore, the number 7 is only used for index representation for these polynomials (cf. [16][17][18][19][20][21]).
Results of this paper are briefly summarized below. Some fundamental properties of Boole type numbers of higher order and Boole type polynomials of higher order. We derive some fundamental properties of these numbers, and polynomials are given in Section 2.
PDEs and functional equations related to generating functions for Boole type polynomials of higher order, the Daehee numbers and logarithm function are given. Using these equations, derivative formulae and recurrence relations are given in Section 3.

2.
Generating Function for the Polynomials y 7,n (x; λ, q, d) of Order v and the Numbers y 7,n (λ, q, d) of Order v In this section, we define the generalization of the numbers y 7,n (λ, q, d) as follows: We also define the generalization of the polynomials y 7,n (x; λ, q, d) as follows: We investigate some properties of the polynomials y 7,n (x; λ, q, d) and the numbers y 7,n (λ, q, d). We give identities and formulas involving these numbers and polynomials, the Apostol-Euler numbers, and the Stirling numbers.
By (8) and (9), we have In order to give a computation formula for the numbers y Combining the above equation with (1) and (2), we get Comparing the coefficients of t m m! on both sides of the above equation, a computation formula for the numbers y (v) 7,m (λ, q, d) is given by the following theorem: Using (8), we obtain From the previous equation, we have Making some straightforward calculations in the previous equation, a recurrence relation for 7,n (λ, q, d) is obtained. This relation is given by the following theorem: With the help of Equation (8), setting the following equation: Making some calculations in the previous equation, another recurrence relation for y (v) 7,n (λ, q, d) is also obtained. This relation is given by the following theorem: Theorem 3. Let q > 0, v 1 , v 2 , d, n ∈ N and λ ∈ C. Then we have Setting v 1 = v 2 = 1 in (10), we compute the following few values of the numbers y 7,n (λ, q, d): A relation between the numbers y  Proof.
Comparing the coefficients of t n n! on both sides of the above equation, we have the derived result.
Using definition of the numbers y (v) 7,n (x; λ, q, d), we have Making some straightforward calculations in the previous equation, and after that comparing the coefficients of t n n! on both sides of the above equation, we have the following theorem: Theorem 5. Let q > 0, v, d, n ∈ N and λ ∈ C. Then we have Substituting λt = e z − 1 into (9), we have Combining the previous equation with (1) and (3), we obtain Since S 2 (n, m) = 0 for m > n, we have Comparing the coefficients of z n n! on the both sides of the above equation, we derive the following theorem: Theorem 6. Let q > 0, v, d, n ∈ N and λ ∈ C. Then we have Using the previous equation, we derive the following theorem: Let q > 0, v, d, n ∈ N and λ ∈ C. Then we have Proof.
Comparing the coefficients of t n n! on both sides of the above equation, we have the derived result.
Combining the following the Chu-Vandermonde identity with (11) we have and 7,n (λ, q, d).
Combining (12) with (11), we arrive at the following corollary: Let q > 0, v, d, n ∈ N and λ ∈ C. Then we have Kucukoglu [27] defined the following generating functions: and Combining (14) with (9), we have From the above equation, we get From the equality in (14) with 2d and 2x instead of d respectively x, we arrive at the following one: n,2d (2x; λ, q) t n n! .
Using the Cauchy product and comparing the coefficients of t n n! on both sides of the above equation, we have the following theorem: Theorem 8. Let q > 0, v, d, n ∈ N and λ ∈ C. Then we have

Partial Derivative Equations and Their Applications
In this section, we deal with some partial derivative equations and functional equations involving generating functions for the polynomials y (v) 7,n (x; λ, q, d), the Daehee numbers and logarithm function. By using these equations, we derive derivative formulas for the polynomials y (v) 7,n (x; λ, q, d), and some identities including these polynomials, recurrence relations of these polynomials, the Daehee numbers and finite combinatorial sums.

Partial Derivative Equations and Derivative Formulas
Differentiating both side of (9) with respect to x, we get the following partial differential equations: and By using the above derivative equations, here we derive two derivative formulas for the polynomials y (v) 7,n (x; λ, q, d). Using these formulas, we derive a combinatorial sums including these polynomials and the Daehee numbers.
Combining (9) with (16), we get After some elementary calculations from the above equation, we arrive at the following theorem: Theorem 9. Let n ∈ N. Then we have Combining (9) with (17), we get Comparing the coefficients of t n n! on both sides of the above equation, we arrive at the following theorem: Using (5), the following well-known explicit formula for the Daehee numbers is given by [2,5]). Combining (19) and (18) with this formula, we derive the following finite combinatorial sum:

Recurrence Relations
Here, we give partial differential equations for generating functions G v (t, x; λ, q, d). With the help of these equations, two recurrence relations for the polynomials y (v) 7,n+1 (x; λ, q, d) are given. Differentiating both sides of (9) with respect to t, we obtain the following partial derivative equations: and Combining (9) with (21), we get Comparing the coefficients of t n n! on both sides of the above equation, we arrive at the following theorem: Theorem 11. Let n ∈ N 0 . Then we have Assume that |λt| < 1. Combining (9) with (22), we get, with y 7,n (d − 1; λ, q, d) = y (1) Comparing the coefficients of t n n! on both sides of the above equation, we arrive at the following theorem:

Conclusion
In the recent extensive written works about the theory of special functions, especially special numbers and polynomials, there are widespread manuscripts and books including special numbers and polynomials such as combinatorial numbers and polynomials, Apostol type numbers and polynomials, Peters type polynomials and numbers, Boole polynomials and numbers, Stirling numbers, Changhee numbers and Daehee numbers. In this paper, we give some new families of combinatorial numbers, which are generalizations and unifications of the Peters and Boole polynomials and numbers with the help of generating functions. By using these functions and their PDEs and functional equations, we derived various interesting properties and identities of these polynomials and numbers. Appropriate relationships of our polynomials and numbers and the results of this paper are compared with earlier results. Consequently, the results of this paper may potentially be used, not only in analytic number theory and for special numbers and polynomials, but also in other areas.