The Arithmetic Properties And Generalization Of Genocchi Polynomials

In this paper, we obtain four higher-order convolution identities for Genocchi polynomials and Apostol-Genocchi polynomials by using Beta functions, multino-mial coefficients and generating functions. These convolution identities are the generalization of some existing identities, we also derive some corollaries. In par-ticular,Firstly, let k,n be positive integers such that k≥2. Suppose that a1,a2,…,ak are positive real numbers, Dilcher and Vignat [14] obtained the following identity:(6)By using difference operators and tools from probability, we obtain k-th order combinatorial convolution identities for Genocchi polynomials, that isTheorem3.1. Let n, k be positive integers such that 2 |k. Suppose that a1, a2,…ak are positive real numbers, we have(7)Theorem3.2. Let n, k be positive integers such that 2 k. Suppose that a1, a2,…,ak are positive real numbers, we have(8)Secondly, let k, m, n be non-negative integers. He and Wang [18] obtained the following identity:(9)By using generating functions and combinatorial techniques, we obtained k-th exponent type convolution identities for Apostol-Genocchi polynomials, that isTheorem3.7. let k, m,n be non-negative integers. Suppose that λ and μ are arbitrary real numbers, then(10)...