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Abstract In this paper, we present some identities relating the hyperharmonic, the Daehee and the derangement numbers, and we derive some nonlinear differential equations from the generating function of a hyperharmonic number. In addition, we use this differential equation to obtain some identities in which the hyperharmonic numbers and the Daehee numbers are involved.

Abstract Since Cauchy numbers were introduced, various types of Cauchy numbers have been presented. In this paper, we define degenerate Cauchy numbers of the third kind and give some identities for the degenerate Cauchy numbers of the third kind. In addition, we give some relations between four kinds of the degenerate Cauchy numbers, the Daehee numbers and the degenerate Bernoulli numbers.

Abstract In this paper, we study the Fourier series related to higher-order Bernoulli functions and give new identities for higher-order Bernoulli functions which are derived from the Fourier series of them.

In this paper, we consider Changhee polynomials of type two, which are motivated from the recent work of D. Kim and T. Kim. We investigate some symmetry identities for the Changhee polynomials of type two which are derived from the properties of symmetry for the fermionic p-adic integral on Z p .

In this paper, we define new Daehee numbers, the degenerate Daehee numbers of the third kind, using the degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee, and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation.

We will give the Lebesgue-Radon-Nikodym theorem with respect to weighted 𝑝-adic 𝑞-measure on ℤ𝑝. In special case, 𝑞=1, we can derive the same result as Kim, 2012; Kim et al., 2011.

We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. Let Pn={p(x)∈ℚ[x]∣deg p(x)≤n} be the (n+1)-dimensional vector space over ℚ. Then we show that {H0(x),H1(x),…,Hn(x)} is a good basis for the space Pn for our purpose of arithmetical and combinatorial applications.

We give some new identities on the Bernoulli and Euler numbers by using the bosonic p-adic integral on Zp and reflection symmetric properties of Bernoulli and Euler polynomials.

In 2009, Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher-order, recently. In this paper, we extend our result to the higher-order twisted q-Euler numbers and polynomials. The purpose of this paper is to establish various identities concerning higher-order twisted q-Euler numbers and polynomials by the properties of p-adic invariant integral on ℤp. Especially, if q=1, we derive the result of Kim et al. (2009).

We derive the modified p-adic q-measures related to q-Nasybullin's type lemma.

Recently, many mathematicians have studied various kinds of the q-analogue of Genocchi numbers and polynomials. In the work (New approach to q-Euler, Genocchi numbers and their interpolation functions, “Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105–112, 2009.”, Kim defined new generating functions of q-Genocchi, q-Euler polynomials, and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type q-zeta function. This function interpolates q-Genocchi polynomials at negative integers. Finally, we also give some identities related to these polynomials.

We give a new construction of the q-extensions of Euler numbers and polynomials. We present new generating functions which are related to the q-Euler numbers and polynomials. We also consider the generalized q-Euler polynomials attached to Dirichlet's character χ and have the generating functions of them. We obtain distribution relations for the q-Euler polynomials and have some identities involving q-Euler numbers and polynomials. Finally, we derive the q-extensions of zeta functions from the Mellin transformation of these generating functions, which interpolate the q-Euler polynomials at negative integers.

By the properties of p-adic invariant integral on ℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties of p-adic invariant integral on ℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

The main purpose of this paper is to study the distribution of Genocchi polynomials. Finally, we construct the Genocchi zeta function which interpolates Genocchi polynomials at negative integers.