Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y1 (n, k; λ). Finally, we make some remarks and observations regarding these identities and relations.


Introduction
In order to give the results presented in this paper, we use two techniques which are p-adic integral technique and the umbral calculus technique.In [1][2][3][4][5], we constructed generating functions for families of combinatorial numbers which are used in counting techniques and problems and also computing negative order of the first and the second kind Euler numbers and other combinatorial sums.In this paper, by applying umbral algebra and umbral analysis methods and their operators to generating functions of the combinatorial numbers and the Apostol type polynomials and numbers, we give many identities and relations including the Fibonacci numbers, the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of higher order and the Apostol-Euler polynomials and numbers of higher order.
Throughout this paper, we use the following notations, definitions and relations.
Here and in the following, let C, R, Z, and N be the sets of complex numbers, real numbers, integers, and positive integers, respectively, and let N 0 : = N∪ {0}.We assume 0 0 = 1.

p-Adic Integrals
In the last section, we will give some combinatorial sums with p-adic integrals technique.Hence, let us give definitions of these integrals and a few properties of them.
Let f (x) ∈ C 1 (Z p → K), a set of continuous derivative functions, and K is a field with a complete valuation.

Umbral Algebra and Calculus
Throughout this section, we use the notations and definitions of the Roman's book (cf.[13]).Let P = C [x] be the algebra of polynomials in the single variable x over the field of complex numbers.Let P * be the vector space of all linear functionals on P. Let L | p(x) be the action of a linear functional L on a polynomial p(x).Let F denote the algebra of formal power series (cf. [13]).Furthermore, for all n ∈ N 0 , one has and also where f (t), g(t) are in F (cf. [13]).
For p (x) ∈ P, as a linear functional, we have and as a linear operator, we have (cf. [13]).The Sheffer polynomials for pair (g(t), f (t)), where g(t) must be invertible and f (t) must be delta series.The Sheffer polynomials for pair (g(t), t) is the Appell polynomials or Appell sequences for g(t).The Appell polynomials are defined by means of the following generating function (cf. [13]).Some properties of the Appell polynomials are given as follows.
(p. 86, Theorem 2.5.5 [13]), derivative formula (cf. p. 86, Theorem 2.5.6 [13]); and see also [6,30,31]).We summarize the results presented in this paper as follows: In Section 2, by applying the umbral algebra and umbral calculus methods to generating functions of the special numbers and polynomials, we derive some identities and relations including the numbers y 1 (n, k; λ), combinatorial sums, the Fibonacci numbers, Apostol-Bernoulli type numbers and polynomials, and the Apostol-Euler type numbers and polynomials.Finally, we give some remarks and observations.In Section 3, by using the p-adic integrals, we give many combinatorial sums related to the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y 1 (n, k; λ).

Numbers and Polynomials
In this section, by using the umbral algebra and umbral calculus methods, we derive many identities and relations containing the numbers y 1 (n, k; λ), combinatorial sums, the Fibonacci numbers, Apostol-Bernoulli type numbers and polynomials, and the Apostol-Euler type numbers and polynomials.
By applying the action of a linear operator λe t + 1 k to the Apostol-Euler polynomial E (a) n (x, λ), we obtain the following result.
Proof.By applying the action of a linear operator λe t + 1 k to the Apostol-Euler polynomial n (x, λ), we obtain Applying linear operators in (15) to the above equation, we have Combining the following relation with ( 23) (cf. p. 101 [13]), we have After some elementary calculation in the above equation, we have Combining ( 24) and ( 25), we arrive at the desired result.

Corollary 1. 2E
(a−1) n We assume that, λ = 1 and a ∈ N, we have the following well-known relationships between the polynomials B (a) n+a (x, −λ) .
Substituting the above relation into (22), we get the following result.
Setting k = 1 in (26), we get the following corollary.
The following theorem was proved in (cf.[1]).Here, we give a proof different from that in (cf.[1]).
Theorem 4. Let n and k be nonnegative integers.Then we have Proof.Using (12), we obtain From the above equation, we have Therefore, we arrive at the desired result.
Combining the above equation with (28), we get On the other hand Therefore, combining ( 29) with (30), we arrive at the desired result.Theorem 6.
Proof.We set the following functional equation By combining the above equation with ( 4) and ( 2), we get Comparing the coefficients of t n n! on both sides of the above equation yields the desired result.
Theorem 7. Let m ∈ N. Then we have Proof.We also set the following functional equation By combining the above equation with ( 1) and (3), we get Comparing the coefficients of t m m! on both sides of the above equation yields the desired result.
Proof.We define the following functional equation: By combining the above equation with ( 4), (2), and (7), we get Comparing the coefficients of t n n! on both sides of the above equation yields the desired result.

Combinatorial Sums via p-Adic Integral
In this section, by using the p-adic integrals, we derive some combinatorial sums containing the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y 1 (n, k; λ).Theorem 9.
Proof.Combining ( 2) with ( 4), we set the following functional equation: By using the above equation, we get Comparing the coefficients of t n n! on both sides of the above equation yields the following relation: By applying the Volkenborn integral to (31), we get Combining the above equation with (10), we arrive at the desired result. Since where E * (k) n (λ) denote the Apostol-type Euler numbers of the second kind of order k (cf. [25,33]),the Equation (32)  Theorem 10.

By combining the above equation with the following identity
Proof.By applying the fermionic p-adic integral to (31), we have x n dµ −1 (x) .
Combining the above equation with (11), we arrive at the desired result.x n dx.
After some calculations, we get the desired result.
Integrate Equation(31) with respect to x from 0 to 1, we obtain n