Abstract
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 characteriza tions 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.Â
Highlights
Introduction and PreliminariesThe 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
We depend on the Laplace distribution to calculate the moment representation of the generalized higher-order Genocchi polynomials
Summary
Introduction and PreliminariesThe classical Genocchi numbers and polynomials play important roles in combinatorics. Some properties and identities of generalized higher-order Genocchi polynomials and numbers are given on the foundation of the moment expression. 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(nα)(x; a, b, c) satisfy the following moment representation: G(nα)(x; a, b, c)
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