Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special numbers

The main purpose of this paper is to provide a novel approach to deriving formulas for the p-adic q-integral including the Volkenborn integral and the p-adic fermionic integral. By applying integral equations and these integral formulas to the falling factorials, the rising factorials and binomial coefficients, we derive some new and old identities and relations related to various combinatorial sums, well-known special numbers such as the Bernoulli and Euler numbers, the harmonic numbers, the Stirling numbers, the Lah numbers, the Harmonic numbers, the Fubini numbers, the Daehee numbers and the Changhee numbers. Applying these identities and formulas, we give some new combinatorial sums. Finally, by using integral equations, we derive generating functions for new families of special numbers and polynomials. We also give further comments and remarks on these functions, numbers and integral formulas.


Introduction
This section deals with comprehensive study of analytic objects linked to theory of the Volkenborn integral, the p-adic fermionic integral and the generating functions for special numbers and polynomials.The p-adic integral and generating functions have been used in mathematics, in mathematical physics and in others sciences.Especially the p-adic integral and p-adic numbers are used in the theory of ultrametric calculus, the p-adic quantum mechanics and the p-adic mechanics.
In this paper, by using the Volkenborn integral and the fermionic integral and their integral equations, we give generating functions for special numbers and polynomials.By applying these integrals to the falling and rising factorials with their identities and relations, we derive both to standard and to new formulas, identities and relations closely related to the Volkenborn integral, the fermionic integral, combinatorial sums and special numbers.These formulas in particular will allow us to solve and compute these integrals, including the falling and rising factorials, numerically much more efficiently.We can simply give some definitions and results for the p-adic integrals, which are detailed study in the following references: [3], [36], [38], [40], [62], [75], [76]; and the references cited therein.
To state the p-adic q-Volkenborn integral, it is useful to introduce the following notations.Let Z p be a set of p-adic integers.Let K be a field with a complete valuation and C 1 (Z p → K) be a set of continuous derivative functions.That is C 1 (Z p → K) is contained in f : X → K : f (x) is differentiable and d dx f (x) is continuous .Kim [38] introduced and systematically studied the following family of the p-adic q-integral which provides a unification of the Volkenborn integral: where q ∈ C p , the completion of the algebraic closure of Q p , set of p-adic rational numbers, with 1−q , q = 1 x, q = 1 and µ q (x) = µ q x + p N Z p denotes the q-distribution on Z p , defined by µ q x + p N Z p = q x [p N ] q , (cf.[38]).Let X be a compact-open subset of Q p .A p-adic distribution µ on X is a Q p -linear vector space homomorphism from the Q p -vector space of locally constant functions on X to Q p .
Remark 2. If q → 1, then (1.1) reduces to the Volkenborn integral.That is where and µ 1 (x) denotes the Haar distribution.I 1 (f (x)) is also so-called the bosonic integral (cf. [3], [36], [62], [75], [76]); see also the references cited in each of these earlier works).This integral has been many applications not only in mathematics, but also in mathematical physics.By using this integral and its integral equations, various different generating functions related to the Bernoulli type numbers and polynomials have been constructed.
Remark 3. If q → −1, then (1.1) reduces to the p-adic fermionic integral which is defined by Kim [40].That is lim q→−1 where 3) and µ −1 x + p N Z p = (−1) x p N (cf.[40]).By using the p-adic fermionic integral and its integral equations, various different generating functions related to the Euler and Genocchi type numbers and polynomials have been constructed.
In order to give our paper results, we need some properties, identities and definitions for generating functions of the special numbers and polynomials.These functions have many valuable applications in almost all areas of mathematics, in mathematical physics and in computer and in engineering problems and in other areas of science.
It is well-known that the λ-Bernoulli numbers and polynomials and the λ-Euler numbers and polynomials have been studied in different sets.Thus, we give the following observations for the λ parameter.When we study generating functions on the set of complex numbers, we assume that λ ∈ C. When we study generating functions by the p-adic integral, we assume that λ ∈ Z p .We start with definition of generating function for the Apostol-Bernoulli polynomials B n (x; λ) as follows: Substituting x = 0 into (1.4),we have denotes the Apostol-Bernoulli numbers (cf.[40], [25], [50], [71], [72]; see also the references cited in each of these earlier works), and also denotes the Bernoulli numbers of the first kind (cf.[4]- [77]; see also the references cited in each of these earlier works).Kim et al. [47] defined the λ-Bernoulli polynomials (Apostol-type Bernoulli polynomials) B n (x; λ) by means of the following generating function: (|t| < 2π when λ = 1 and |t| < |log λ| when λ = 1) with denotes the λ-Bernoulli numbers (Apostol-type Bernoulli numbers) (cf.[47], [27], [72], [70]).The Fubini numbers w g (n) are defined by means of the following generating functions: (cf. [21]).The Frobenius-Euler numbers H n (u) are defined by means of the following generating function: Let u be a complex numbers with u = 1.
The Euler numbers of the second kind E * n are given by (cf. [59], [67], [73]; see also the references cited in each of these earlier works).
The Stirling numbers of the first kind S 1 (n, k) the number of permutations of n letters which consist of k disjoint cycles, are defined by means of the following generating function: These numbers have the following properties: (cf. [12], [4], [10], [59], [64], [66]; and see also the references cited in each of these earlier works).
A relation between he falling factorial and Stirling numbers of the first kind is given by [12], [14], [15], [73]).The Bernoulli polynomials of the second kind b n (x) are defined by means of the following generating function: (cf. [59, pp. 113-117]; see also the references cited in each of these earlier works).
The Bernoulli numbers of the second kind b n (0) are defined by means of the following generating function: The Bernoulli polynomials of the second kind are defined by Substituting x = 0 into the above equation, one has The numbers b n (0) are also so-called the Cauchy numbers (cf.[59, p. 116], [30], [51], [56], [68]; see also the references cited in each of these earlier works).
The second kind Stirling numbers S 2 (n, k; λ) are defined by means of the following generating function: where k ∈ N 0 and λ ∈ C (cf. [50], [64], [71]; see also the references cited in each of these earlier works).Substituting λ = 1 into (1.15), then we get the Stirling numbers of the second kind, the number of partitions of a set of n elements into k nonempty subsets, The Stirling numbers of the second kind are also given by the following generating function including the falling factorial: (cf. [1]- [74]; see also the references cited in each of these earlier works).
The Schlomilch formula associated with a the Stirling numbers of the first and the second kind numbers is given by where k ∈ N 0 , [11, pp. 123-127]).From the above generating function, we give the following functional equation: From this equation, we have The associated Stirling numbers of the first kind are defined by means of the following generating function: where k ∈ N 0 , and S 12 (n, k) = 0, k > n 2 (cf.[11, pp. 123-127]).The Lah numbers was discovered by Ivo Lah in 1955.These numbers are coefficients expressing rising factorials in terms of falling factorials (cf.[11], [12], [15], [57], [60], [80]).The unsigned Lah numbers have an interesting meaning especially in combinatorics.These numbers count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets.These numbers are related to the some special numbers such as the Stirling numbers the first and the second kind.The Laguerre polynomials and the others.
The Lah numbers are defined by means of the following generating function:The Lah numbers are defined by means of the following generating function: (cf. [60, p. 44], [7], [80], and the references cited therein).By using this equation, we have The unsigned Lah numbers are defined by Two recurrence relations of these numbers are given by with the initial conditions L(n, 0) = δ n,0 and L(0, k) = δ 0,k , for all k, n ∈ N and (−1) j s 1 (n, j)S 2 (j, k) (cf.[60, p. 44]).
Let L(n, k) denote the set of all distributions of n balls, labelled 1, . . ., n, among k unlabeled, contents-ordered boxes, with no box left empty.Such distributions is known Laguerre configurations.If [20]).
The Changhee numbers of the first kind and the second kind are defined by means of the following generating functions: Ch n t n n! (cf.[31]).
In Section 2, we give some properties of the p-adic q-integrals and the p-adic fermionic integral with their integral equations.Using these equations, we construct generating functions for special numbers and polynomials, some identities and formulas including combinatorial sums.We also give interpolation of these numbers.
In Section 3, we give some application of the Volkenborn integral to the falling and rising factorials.We define sequences of the Bernoulli numbers related to these applications.We give some integral formulas including the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Lah numbers and the combinatorial sums.
In Section 4, we give some formulas for the sequence Y 1 (n : B), sequence of the Bernoulli numbers.By using the Volkenborn integral and its integral equations, we give some formula identities of the sequence Y 1 (n : B).We also gives integral formulas including the falling factorials.
In Section 5, we also give some formulas for the sequence Y 2 (n : B).Using the Volkenborn integral, we derive some formula identities of this sequence and some integral formulas related to the falling factorials.
In Section 6, we give various integral formulas for the Volkenborn integral associated with the falling factorials, the combinatorial sums, the special numbers including the Bernoulli numbers, the Stirling numbers and the Lah numbers.
In Section 7, we give applications of the p-adic fermionic integral associated with the falling and rising factorials.We give some integral formulas including the Euler numbers and polynomials, the Stirling numbers, the Lah numbers and the combinatorial sums.We define two the sequences related to the Euler numbers and the Euler polynomials.
In Section 8, by using the fermionic integral and its integral equations, we derive some formula for of the sequence (y 1 (n : E)) and the p-adic fermionic integral formulas related to the falling factorials.
In Section 9, using the fermionic integral, we give some interesting formulas for the sequence (y 2 (n : E)) and the p-adic fermionic integral including the raising factorials.
In Section 10, We give some novel identities for combinatorial sums including special numbers associated with the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Eulerian numbers and the Lah numbers.
2. Integral equations for the Volkenborn integral, the p-adic q -integrals and the fermionic integral In this section, we give and survey some properties of the p-adic q-integrals.We also give integral equations for this integrals.By using these equations, we derive generating functions for special numbers and polynomials, some identities and formulas including combinatorial sums.
2.1.Properties of the p-adic Volkenborn integral.Here, we give some standard notations for the Volkenborn integral with its integral equations.The Volkenborn integral in terms of the Mahler coefficients is given by the following formula: let We have where which is known as the Witt's formula for the Bernoulli numbers (cf.[38], [40], [62]; see also the references cited in each of these earlier works).
The following integral equation for the Volkenborn integral is given by where [38], [40], [62], [79]; see also the references cited in each of these earlier works).
The following a novel integral equation for equation (1.1) was given by Kim [42]: where In [62, p. 70], exponential function is defined as follows: the above series convergences in region E, which subset of field K with char(K) = 0. Let k be residue class field of K.If char(k) = p, then where Substituting λ = 1, we get Equation (2.5) in [55].If t = 1 and q → 1 into (2.4),we have Zp Substituting λ = 1 into the above equation, we arrive at an Exercise 55A-1 in [62, p. 170] as follows: where a ∈ C + p and a = 1.Remark 4. Substituting a = e, λ = 1 and q → 1 into (2.4),we arrive at equation (1.5).Substituting a = e and q → 1 into (2.4),we get Exercise 55A-2 in [62, p. 170] , which gives us generating function for the Bernoulli numbers B n : where t ∈ E and t = 0.By using (2.4), we define a new family of numbers, S n (a; λ, q) by means of the following generating function: The numbers S n (a; λ, q) are related to Apostol-type Bernoulli numbers with special values of parameters a, λ and q.
We define a new family of polynomials, C n (x; a, b; λ, q) as follows: A Witt's type formula for the numbers S n (a; λ, q) as follows: Remark 5. Setting q → 1 and a = e in (2.6), we arrive at the Witt's formula for the Bernoulli numbers: We also easily see that (cf. [38], [40], [55], [62]; see also the references cited in each of these earlier works).Theorem 1] gave the following integral equation: where n is a positive integer and Theorem 2. Let n be a positive integer.
We modify (2.10) as follows: which is known as the Daehee numbers [32].Theorem 2 was proved by Schikhof [62].By using (2.10), we have (cf. [31], [66], [32]).By using (2.11), we obtain (2.12) 2.2.Properties of the p-adic fermionic integral.Here, we introduce some properties the fermionic p-adic integral with its integral equations.By using these equations, we give generating function for special numbers and polynomials.We also derive interpolation function for these numbers.Let f ∈ C 1 (Z p → K).Kim [41] gave the following integral equation for the p-adic fermionic integral on Z p as follows: where n is a positive integer.Substituting n = 1 into (2.13),we have the following very useful integral equation, which is used to construct generating functions associated with Euler type numbers and polynomials: [41]).By using (1.3) and (2.14), the Witt's formula for the Euler numbers and polynomials are given as follows, respectively and (cf. [40], [25]; see also the references cited in each of these earlier works).
(cf. [31], [66], [32]).By using (2.18), we get In [42], Kim also gave q-analogies of integral equation in (2.13) as follows: where d is an positive odd integer.Substituting (2.3) into (2.20),we get Setting d = 1 into the above equation, we have By using (2.21, we define a new family of numbers, k n (a; λ, q) by the following generating function: If d as odd integer, we the above generating function reduces to Here we study on set of complex numbers.From the above equation, we get the following theorem: Theorem 4. We assume that λ and q are complex numbers with |λq| < 1.Then we have Now, by using (2.22), we also assume that s is a complex numbers with positive real part.We define an interpolation function for the numbers k n (a; λ, q) as follows: Substituting a = e and λ = 1 into the above equation, we have If q goes to 1, the function ζ(s; e; 1, q) reduces to the following interpolation function for the Euler numbers: (cf. [40], [47], [37], [73]).

Application of the Volkenborn integral to the falling and rising factorials
In this section, we give applications of the Volkenborn integral on Z p to the falling and rising factorials, we derive some integral formulas including the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Lah numbers and the combinatorial sums.
By using the same spirit of the Bernoulli polynomial of the second kind which are also called Cauchy numbers of the first kind, by applying the Volkenborn integral to the rising factorial and the falling factorial, respectively, we derive various formulas, identities, relations and combinatorial sums including the Bernoulli numbers, the Stirling numbers, the Lah numbers, the Daehee numbers and the Changhee numbers.
In [70], similar to the Cauchy numbers defined aid of the Riemann integral, we studied the Bernoulli numbers sequences by using p-adic integral.Let x j ∈ Z and j ∈ {1, 2, . . ., n − 1} with n > 1.
We define a sequence (Y 1 (n : B)), including the Bernoulli numbers as follows: Kim et al. [32] defined the Daehee numbers (of the first kind) by the following integral representation as follows: Combining (3.1) with (3.2), a explicit formula for the sequence of (Y 1 (n : B)) is given by Few values of (3.1) is computed by (3.2) as follows: By using the Bernoulli numbers of the first kind, the above table reduces to the following numerical values: Due to work of Kim et al. [32], we see that x j coefficients computed by the Stirling numbers of the first kind.We define a sequence (Y 2 (n : B)), including the Bernoulli numbers as follows: Kim et al. [32] defined the Daehee numbers (of the second kind) by the following integral representation as follows: By using the Bernoulli numbers of the first kind, the above table reduces to the following numerical values: , . . . .

Kim et al.
[32] defined the first and second kind Daehee polynomials, respectively, as follows: and We also define the following sequences for the Bernoulli polynomials, (Y 1 (x, n : B(x))) and (Y 2 (x, n : B(x))) related to the polynomials D n (x) and D n (x) as follows: and

Formulas for the sequence Y 1 (n : B)
To use the Volkenborn integral and its integral equations, we give some formula identities of the sequence (Y 1 (n : B)).We also gives some p-adic integral formulas including the falling factorials.
Proof.We know that numbers of the sequence (Y 1 (n : B)) are related ot the numbers D n .By using same computaion of the numbers D n , this theorem is proved.That is, we now briefly give this proof.Since by using (2.10), we get Thus, we get the desired result.
Theorem 6. Zp we have (cf. [59, p. 58]).By applying the Volkenborn integral on Z p to the above equation and combining with (4.2), we get the desired result.
By applying the Volkenborn integral on Z p to the above equation and combining with (4.2), we arrive at the following result: By applying the Volkenborn integral on Z p to equation (1.19), we obtain By using (4.2), we get Substituting (1.18) into the above equation, we arrive at the following theorem: Theorem 7.

Formulas for the sequence Y 2 (n : B)
To use the Volkenborn integral and its integral equations, we also give some formula identities of the sequence (Y 2 (n : B)).We gives some p-adic integral formulas including the raising factorials.
We give another formula for the numbers Y 2 (n : B) by the following theorem: where C(n, k) = |s 1 (n, k)| and B k denotes the Bernoulli numbers.
Proof.By applying the Volkenborn integral on Z p to the following equation and using (2.7), we get the desired result.
Proof.By applying the Volkenborn integral on Z p to the equation (1.20), we get By substituting (4.2), we get the desired result.
By applying the Volkenborn integral on Z p to (1.20), we get the following formula for the numbers Y 2 (n : B) : Substituting (1.10) into (1.20),we have (5.5) By applying the Volkenborn integral on Z p to the above equation, we arrive at the following formula for the numbers Y 2 (n : B): By combining (4.4) and (5.4), we get the following results: (5.6) Proof.By applying the Volkenborn integral on Z p to the following equation: and using (5.3), after some elementary calculations, we get the desired result. (5.7) Proof.By applying the Volkenborn integral on Z p to the following equation By combining the above equation with (2.7), we get the desired result.
A recurrence relation of the numbers Y 2 (n : B) is given by the following theorem.
Theorem 11. (5.9) Proof.We set From the equation, we get By applying the Volkenborn integral on Z p to the above equation, and using (6.1), we get Therefore, the proof of the theorem is completed.

Integral formulas for the Volkenborn integral
In this section, we give some integral formulas for the Volkenborn integral including the falling factorials, the combinatorial sums, the special numbers including the Bernoulli numbers, the Stirling numbers and the Lah numbers.
2) By applying the Volkenborn integral on Z p to the both sides of the above equation, and using (2.10), we arrive at the desired result.
By combining (6.2) with (1.10), we have By combining the above equation with (1.9), since k < 0, S 1 (n, k) = 0, we get By applying the Volkenborn integral on Z p to the both sides of the above equation, and using (2.7), we also arrive at the following lemma: Remark 7. By using (1.20), we have By applying the Volkenborn integral on Z p to the both sides of the above equation, and using (6.1), we get another proof of (5.6).
Theorem 12. Zp Proof.In order to prove this theorem, we need the following identity: (cf. [59, p. 58]).By applying the Volkenborn integral on Z p to the both sides of the above equation, and using (4.3), we arrive at the desired result.

Since
and Combining the above equation with (6.1) and (4.2), we get By using the above equation, we arrive at the following result: A recurrence relation for the numbers Y 1 (n : B) is given by the following theorem: or . By applying the Volkenborn integral on Z p to the both sides of the above equation, and using (3.2), (6.1) and ( 6.4), we arrive at the desired result. .
Proof.The well-known Chu-Vandermonde identity is defined as follows n k=0 x k By applying the Volkenborn integral on Z p wrt x and y to the LHS of Equation (6.7), and using (2.10), we get By applying the Volkenborn integral on Z p wrt x and y to the RHS of Equation (6.7), and using (3.6) and ( 3.7), we obtain Zp Y 1 (y, n : B(y))dµ 1 (y) .(6.9) Combining (6.8) with (6.9), we arrived at the desired result.

Since
x By applying the Volkenborn integral on Z p wrt x and y to equation (6.7), and using (4.3) and (4.4), we also get the following lemma: By applying the Volkenborn integral on Z p wrt x and y to equation (6.7), and using (4.3) and (4.4), we also get the following lemma: Lemma 6.

Zp Zp
x + y n dµ 1 (y) dµ Proof.In [62] the following integral formula is given where By substituting After some elementary calculation, we get the desired result.

Zp
x Proof.By applying the Volkenborn integral on Z p to the following relation and using (6.1) and (4.1), we get the desired result.
In [52], Osgood and Wu gave the following identity where 1,1 = 1, C 1,1 = 0, C 1,2 = 0 = C (3) 2,1 .By applying the Volkenborn integral on Z p to equation (6.17) wrt x and y, we arrive at the following lemma: Proof.By using (1.16), we get By applying the Volkenborn integral on Z p to equation (6.20) wrt x and y, and using (2.7), we get the desired result.
Proof.Gould [22,Vol. 3,] defined the following identity: By applying the Volkenborn integral on Z p to the above integral, and using (2.10), we get the desired result.
where H n denotes the harmonic numbers: Proof.Gould [22,Vol. 3,] defined the following identity: By applying the Volkenborn integral on Z p to the above integral, and using (2.10), we get the desired result.
Proof.Gould Theorem 19.Let n be a positive integer with n > 1.Then we have Proof.In [23, Eq. (2.15)], Gould gave the following identity for n > 1: By applying the Volkenborn integral to the above equation, and using (2.10), we arrive at the desired result.Proof.In [23,Eq. (2.64) and Eq-(6.17)],Gould gave the following identities: By applying the Volkenborn integral to the above equation, and using (2.10) and (2.7), respectively, we arrive at the desired result.
By applying the Volkenborn integral on Z p to the above integral, and using (2.10), we get the desired result.
Proof.Multiple both sides of equation (1.10) by x m , we get By applying the Volkenborn integral on Z p to the above integral, and using (2.7), we get the desired result.
In order to give formula for the following integral we need the following well-known identity where the coefficients of the x (n+n−k) , called connection coefficients, have a combinatorial interpretation as the number of ways to identify k elements each from a set of size m and a set of size n (cf. [78]).
By applying the Volkenborn integral on Z p to (6.29) and using (3.2), (6.1) and (4.1), we get the following lemma: Remark 11.Since D n = Y 1 (n : B), we rewrite (6.30) as follows: By using (1.10), we have By applying the Volkenborn integral on Z p to the above equation, and using (2.7), we get the following lemma: By combining (6.29) with (1.10), we get By applying the Volkenborn integral on Z p to the above equation, we get the following lemma: (cf. [29, p. 164]).By applying the Volkenborn integral on Z p to Equation (6.34), and using (2.7), we get the following identities: and By using (2.11), we also have Thus A relationship between the numbers Y 1 (n : B) and Y 2 (n : B) is given by the following theorem: Proof.By applying the Volkenborn integral on Z p to ( 1.19), and using (4.1) and (5.3), we get the desired result.

Application of the p-adic fermionic integral to the falling and rising factorials
In this section, we give applications of the p-adic fermionic integral on Z p to the falling and rising factorials, we derive some integral formulas including the Euler numbers and polynomials, the Stirling numbers, the Lah numbers and the combinatorial sums.
By using the same spirit of the Cauchy numbers of the first kind, by applying the padic fermionic integral to the rising factorial and the falling factorial, respectively, we derive various formulas, identities and relations.
In [70], similar to the Cauchy numbers defined aid of the Riemann integral, we also studied the Euler numbers sequences by using p-adic fermionic integral.Let x j ∈ Z and j ∈ {1, 2, . . ., n − 1} with n > 1.
We define the sequences (y 1 (n : E)), including the Euler numbers as follows: and The sequences (y 1 (n : E)) and (y 2 (n : E)) can be computed by the first and second kind Changhee numbers, which are defined by Kim et al. [31] as follows: and or Combining (7.1) with ( 7.3) and (3.1) with (7.4), we easily give the following relations for the general terms of the related sequences: By using the above formulas, sew values of the sequences (y 1 (n : E)) and (y 2 (n : E)) are computed, respectively, as follows: By using definition of Euler numbers, we have
8. Formulas for the sequence y 1 (n : E) Using the fermionic integral and its integral equations, we give some formula identities of the sequence (y 1 (n : E)).We also gives some p-adic fermionic integral formulas including the falling factorials.

Theorem 24.
y 1 (n : E) = (−1) n 2 −n n!.Proof.We know that numbers of the sequence (y 1 (n : E)) are related ot the numbers Ch n .By using same computaion of the numbers Ch n , this theorem is also proved.That is, we now briefly give this proof.Since Since by using (2.17), we get Thus, we get the desired result.
By applying the fermionic integral on Z p to the above equation and combining with (8.1), we get the desired result.
Theorem 26. Zp Proof.By applying the Volkenborn integral on Z p to Equation (6.5), and using (8.1), we arrive at the desired result.
By using the above equation, we also get another proof of (8.2).
x.x (n) = x (n+1) + nx (n) .By applying the fermionic p-adic Volkenborn integral on Z p to the both sides of the above equation, and using (2.17), (7.3 ), respectively, we get Combining the above equation, we get the desired result.
By applying the p-adic fermionic integral on Z p to equation (6.17) wrt x and y, we arrive at the following lemma: Lemma 14.
Zp Zp Proof.By applying the p-adic fermionic integral on Z p to equation (6.20) wrt x and y, and using (2.15), we get the desired result.
Theorem 28.Let n be a positive integer with n > 1.Then we have Proof.By applying the fermionic Volkenborn integral to equation (6.24) with ( 2.17), we get desired result.Proof.By applying the fermionic Volkenborn integral to Equations (6.27) and( 6.28) with (2.17) and (2.15), we get desired result.
Proof.By applying the fermionic Volkenborn integral to Equation (6.21) with ( 2.17), we get desired result.

Zp
x n Proof.By applying the fermionic Volkenborn integral to Equation (6.23) with ( 2.17), we get desired result.

Zp
Proof.Gould [22,Vol. 3,] defined the following identity: By applying the fermionic p-adic integral on Z p to the above integral, and using (2.17), we get the desired result.
Proof.Gould [22,Vol. 3,] defined the following identity: By applying the fermionic p-adic integral on Z p to the above integral, and using (2.17), we get the desired result.
By applying the fermionic p-adic integral on Z p to the above integral, and using (2.17), we get the desired result.9. Formulas for the sequence y 2 (n : E) By using the fermionic integral and its integral equations, we derive some formula identities of the sequence (y 2 (n : E)).We also gives some p-adic fermionic integral formulas including the raising factorials.Proof.By applying the p-adic fermionic integral on Z p to the equation (1.20), we get By substituting (2.17), we get the desired result.
By applying the p-adic fermionic integral on Z p to equation (5.2) and using (2.15), we get the following theorem, which modify equation (7.5): By applying the p-adic fermionic integral on Z p to (1.20), and using (2.17), we get After some elementary calculations, we arrive at the following formula for the numbers y 2 (n : E): Theorem 37.
By applying the p-adic fermionic integral on Z p to equation (5.5), and using (2.15), we arrive at the following formula for the numbers y 2 (n : E): In [66], we gave

Zp
using the p-adic fermionic integral with (9.1), we have From the above equation, we arrive at the following theorem: Theorem 39.
By applying the fermionic p-adic integral on Z p to equation (6.34), and using (2.15), we get the following identities:

Identities for combinatorial sums including special numbers
In this section, by using integral formulas, we derive many novel the combinatorial sums including the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Eulerian numbers and the Lah numbers.
Combining (6.31) and (6.32), we arrive at the following theorem: Combining (6.32) and (6.33), we arrive at the following theorem: By combining (10.1) and (10.2), we get the following combinatorial sum by the following the following corollary: By combining (6.11) and (6.12), we arrive at the following theorem: .
Combining (6.1) with (6.4), we arrive at the following theorem: Substituting (5.1) into (5.9),we get we get After some elementary calculation in the above equation, we arrive at the following theorem: Combining (6.35) and (6.36), we get the following theorem: Theorem 46.Let n be a positive integer.Then we have Combining (5.6) and (5.7) we arrive at the following theorem: .
By combining (6.25) with (6.26), we get By substituting (4.1), into the above equation, we arrive at the following theorem: By combining (6.18) with (6.19), we arrive at the following theorem: By combining (8.3) with (8.4), we arrive at the following theorem: In [1, p. 30], Aigner gave the following valuable comments on the Eulerian number A n,k : Let σ = a 1 a 2 ...a n ∈ S(n), the set of all permutations of {1, 2, ..., n}, be given in word form.A run in σ is a largest increasing subsequence of consecutive entries.The Eulerian number A n,k is the number of σ ∈ S(n) with precisely k runs or equivalently with k − 1 descents a i > a i+1 .Consequently, A n,1 = A n,n = 1 with 12...n respectively nn − 1...1 as the only permutations.Aigner gave the following a recurrence relation: By using the following identity ]), we have By applying the Volkenborn integral and the p-adic fermionic integral to the above equation, respectively, we get relations between the Bernoulli, Euler, Eulerian and Stirling numbers by the following theorem: Theorem 51.
By combining (9.2) with (9.1), we obtain the following theorem: By combining left-hand side of (6.25) and (6.26), we get the following theorem: Theorem 53.Combining the above equation with (2.12) and (2.7), we get Proof of Theorem is completed.
By combining left-hand side of (8.5) and (8.6), we get the following theorem:  Proof.Substituting t = e t − 1 into (1.17), and combining with (1.15), we get the following functional equation: F L (e t − 1, k) = F S (t, k; 1)F F u (t, k) (10.4)where where the numbers w t n n! .
Comparing the coefficients of t n n! on both sides of the above equation, we get the desired result.By combining (6.13) and (6.1) with the above equation, we get the desired result.Proof.In order to prove the assertions of the theorem, we apply the Volkenborn integral and p-adic fermionic integral to the following identity (cf.[11, p. 123]): k=0 n j S 12 (n − j, k)t j+k .(10.6) After some elementary evaluation, we get the desired result.
Integrating both sides of (10.6) from 0 to 1 and using definition of the Cauchy numbers of the first kind, we arrive at the following corollary:  Applying the Volkenborn integral and the p-adic fermionic integral to the above identity with the Riemann integral from 0 to 1, respectively, we get the assertions of the theorem.
Applying the Riemann integral to equation (10.7) from 0 to 1, and using (1.13), we arrive at the following theorem: By applying the Volkenborn integral on Z p to above equation, and using (2.7), we get By substituting (4.2) and (6.1) into the above equation, after some elementary calculations, we arrive at the desired result.

[ 23 ,Remark 10 .
Eq. (2.65)] gave the following identity Volkenborn integral to the above equation, and using (2.10), we arrive at the desired result.Volkenborn integral to the above equation, and using (2.10), we arrive at the desired result.Substituting r = 1 into (6.22),since k−j n = 0 if k − j < n, we arrive at equation (2.10).

. 2 )(− 1 )
By combining the above equation with (9.1), we get the following formulas for the sequence y 2 (n : E): m+n S 1 (n, m)E m .

Theorem 52 .(− 1 )
Let n be a positive integer.Then we have m+n S 1 (n, m)E m .

1 .
Proof.By applying the Volkenborn integral to the following identity which was given by Gould[23,

g 2
(n) denotes the Fubini numbers of order k.By using (10.4), we get (n − m, k)w (k) g (m)

Remark 15 .S 2
Equation (10.8) has been proved by means of the different methods.For example, see[59].By using the falling factorial and the Volkenborn integral, we can give another proof.Theorem 63.Let n be a positive integer.Then we have (n, k) xx (k−1) − (k − 1)x (k−1) .