Combinatorial Identities with Generalized Higher-order Genocchi Sequences

In this paper, we make use of the probabilistic method to calculate the moment representation of generalized higher-order Genocchi polynomials. We obtain the moment expression of the generalized higher-order Genocchi numbers with a and b parameters. Some characterizations and identities of generalized higher-order Genocchi polynomials are given by the proof of the moment expression. As far as properties given by predecessors are concerned, we prove them by the probabilistic method. Finally, new identities of relationships involving generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell numbers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of the second kind are established. 2010 Mathematics Subject Classifications: 11B68, 60E07, 11B83, 62E15.


Introduction and Preliminaries
The classical Genocchi numbers and polynomials play important roles in combinatorics.It is widely used in combinatorial mathematics, function theory, graph theory, approximate calculation and theoretical physics, such as the diffusion of matter.In the present paper, a further investigation for the generalized higher-order Genocchi polynomials with a, b and c parameters is performed.We depend on the Laplace distribution to calculate the moment representation of the generalized higher-order Genocchi polynomials.Hence the moment representation can also be used to give some relative identities and characterizations of the generalized higher-order Genocchi polynomials.Recently, symmetry identities of the generalized Bernoulli, Euler and Genocchi polynomials are investigated by Waseem A. Khan and other predecessors [6] [7][8] [9].Here the probabilistic method provides great convenience to investigate symmetry identities of the generalized Genocchi polynomials and relations between the generalized higher-order Genocchi numbers and combinatorial numbers.
Throughout this paper, let t ∈ C and | t |<π/|lnb − lna|, α ∈ N + , x ∈ R, a, b and c are positive integers, a = b, we now turn to the generalized higher-order Genocchi polynomials G (α) n (x; a, b, c) with a, b and c parameters for nonnegative integer n, which are usually defined by means of the following generating function [2] [4,5]: Taking x = 0 in Eq.( 1), we get the generating function of the generalized higher-order Genocchi numbers G (α) n (a, b) with a and b parameters [4]: Setting a = 1, b = e, c = e in Eq.( 1), the generating function of the classical higherorder Genocchi polynomials is as follows [3][5] [9] [12]: ) then setting x = 0 in Eq.( 3), we get the generating function of the classical higher-order Genocchi numbers [1][12]: In this paper, we make use of the special combinatorial sequences of the generalized Euler polynomials E n (x; a, b, c) with a, b and c, which are defined by the following generating function [10] [11]: For α ∈ N + , we get the generalized higher-order Euler polynomials E (α) n (x; a, b, c) with a, b and c [11]: We also need the following notations: r.v denotes a random variable, i.i.d shows that a sequence of random variables are independent and identically distributed.The notation E denotes an expectation operator and definition is as follows: When f (x) is a measurable function with continuous random variables X and p(x) is a density function of X, we have Especially, setting f (x) = x n ,we obtain the moment of n-th order EX n of random variables.
Remark 1. [see 15] If f and g are exponential generating functions, and then the coefficients of t n n! in f g are given by Next we shall introduce several moment representations of some combinatorial sequences.
satisfy the following moment representation, Lemma 3. [see 14] Suppose that r.v u ∼ U [0, 1], then Fibonacci numbers F n have the following moment representation, Lemma 4. [see 14] Suppose that r.v X ∼ P (1), then Bell numbers b n have the following moment representation, 2kπ , then Bernoulli numbers B n satisfy the following moment representation, π , then Euler numbers E n have the following moment representation, Lemma 7. [see 14] Suppose that r.v X ∼ Γ(u, 1), u ∼ U [0, 1], and X, u are independent, then Cauchy numbers of the second kind c n satisfy the following moment representation, for all i, when n, k 1, then Stirling numbers of the second kind have the following moment representation, It is demanded that S(n, 0) = S(0, k) = 0, S(0, 0) = 1.

Moment Representations of the Generalized Higher-order Genocchi polynomials
In this section, we derive the moment representation of the generalized higher-order Genocchi polynomials by means of the probabilistic method.Some properties and identities of generalized higher-order Genocchi polynomials and numbers are given on the foundation of the moment expression.
and for all j, L (j) = k≥1 L (j) k (2k−1)π be random variables and r.v {L (j) } 1≤ j ≤α are independent and obey the same distribution as L. Then let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, for α ∈ N + , x ∈ R, n ≥ α, the generalized higher-order Genocchi polynomials G (α) n (x; a, b, c) satisfy the following moment representation: Proof.From the generating function of the generalized higher-order Genocchi polynomials we discover All moments of the random variables exist, hence the discovery depends on the following identity: (2k−1)π be a random variable and for all j r.v {L (j) } 1≤ j ≤α are independent and obey the same distribution as L. According to Eq.( 16) and Eq.( 18), we have By comparing the coefficients t n n! , we arrive at the moment expression of G (α) n (x; a, b, c).
Corollary 1. Taking x = 0 in Eq.( 15), we get the moment representation of the higherorder Genocchi numbers with a and b parameters G For a = 1, b = e in Eq.( 19), we get the moment representation of the higher-order Genocchi numbers Corollary 2. For α = 1 in Eq.( 15), we get the moment representation of the generalized Genocchi polynomials G n (x; a, b, c), Corollary 3. Setting a = 1, b = e, c = e in Eq.( 15), we get the moment representation of the higher-order Genocchi polynomials Corollary 4. We can easily obtain the moment representation of the generalized higherorder Euler polynomials and get the relation between generalized higher-order Euler polynomials and generalized higher-order Gennocchi polynomials in the proof of theorem 1, The generalized higher-order Euler polynomials and the generalized higher-order Genocchi polynomials have similar properties and forms, we only aim at the properties of generalized higher-order Genocchi polynomials here.
Proof.In light of the theorem 1 we have This concludes the proof.
Theorem 3. Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, for α ∈ N + , x ∈ R, n ≥ α, then we get Proof.According to theorem 1, we have On substituting x with -x in Eq.( 25), then Eq.( 26) arises.
Proof.With the help of theorem 1 we have This concludes the proof.
Corollary 5. Taking α = 1 in Eq.(25), Eq.(26), Eq.( 27), we can easily get the following identities, Theorem 5. Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, for α ∈ N + , x ∈ R, n ≥ α, then we get Proof.From theorem 1 we have Thus we arrive at the desired result.
Theorem 6.Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, for α ∈ N + , x ∈ R, n ≥ α, then we get Proof.From Eq.( 15) in theorem 1, we have Therefor we derive the Eq.(32).
Theorem 7. Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, Proof.By theorem 1 we have and Thus, combining Eq.( 34) and Eq.( 35) gives the theorem 7.
Theorem 8. Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, for α ∈ N + , x, y ∈ R, n ≥ α, then we get Proof.It follows from Eq.( 15) that This concludes the proof.
Theorem 9. Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, for α ∈ N + , x, y, z ∈ R, n ≥ α, then we get Proof.By Eq.( 15), we have As for Eq.( 38), we have b, c).This concludes the proof.
Theorem 10.Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, Proof.Firstly, we have according to Eq.( 35), Secondly, we have By equating Eq.(40) and Eq.(41), we arrive at the desired result.
From the proof of theorem 10, we get the following identity: k (lx; a, b, c).

Identities of Generalized Higher-order Genocchi Numbers and Special Combinatorial Sequences
In this section, we establish some new identities involving the generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell numbers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of second kind by means of the moment representation of the generalized higher-order Genocchi polynomials.
Theorem 11.Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, for α ∈ N + , x ∈ R, n ≥ α, then we get Proof.By the Eq.(34) we can arrive at the Eq.( 45) directly.It follows from Eq.( 15) that from which we see that Hence we complete the proof of the theorem 11.
Theorem 12. Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, We now turn to the above identities of the generalized higher-order Genocchi numbers and x.For the case x is a constant variable, it seems normal.It is worth noticing that we can see x as a random variable which obeys to a certain distribution and then take mathematical expectation on the two sides of the equation of random variables.It turns out that some interesting identities including the generalized higher-order Genocchi numbers and some combinatorial sequences are derived.The probabilistic method is much simpler and more effective here.

Theorem 2 .
Let a, b and c be positive integers with conditions a = b, ab = 1 and c = 1, for α ∈ N + , x ∈ R, n ≥ α, then we get