Kim et al. Advances in Difference Equations ( 2019) 2019:190 https://doi.org/10.1186/s13662-019-2129-x RESEARCH Open Access Some identities of special numbers and polynomials arising from p-adic integrals on Z p Dae San Kim 1 , Han Young Kim 2 , Sung-Soo Pyo 3* and Taekyun Kim 2 * Correspondence: ssoopyo@gmail.com 3 Department of Mathematics Education, Silla University, Busan, Republic of Korea Full list of author information is available at the end of the article Abstract In recent years, studying degenerate versions of various special polynomials and numbers has attracted many mathematicians. Here we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials, and degenerate type 2 Euler polynomials, and their corresponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. Regarding those polynomials and numbers, we derive some identities, distribution relations, Witt type formulas, and analogues for the Bernoulli interpretation of powers of the first m positive integers in terms of Bernoulli polynomials. The present study was done by using the bosonic and fermionic p-adic integrals on Z p . MSC: 11B83; 11S80; 05A19 Keywords: Bosonic p-adic integral; Fermionic p-adic integral; Degenerate Carlitz type 2 Bernoulli polynomial; Fully degenerate type 2 Bernoulli polynomial; Degenerate type 2 Euler polynomial 1 Introduction Studies on degenerate versions of some special polynomials and numbers began with the papers by Carlitz in [3, 4]. In recent years, studying degenerate versions of various special polynomials and numbers has regained interest of many mathematicians. The research has been carried out by several different methods like generating functions, combina- torial approaches, umbral calculus, p-adic analysis, and differential equations. This idea of studying degenerate versions of some special polynomials and numbers turned out to be very fruitful so as to introduce degenerate Laplace transforms and degenerate gamma functions (see [11]). In this paper, we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials, and degenerate type 2 Euler polynomials, and their corre- sponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. We investigate those polynomials and numbers by means of bosonic and fermionic p-adic integrals and derive some identities, distribution relations, Witt type formulas, and ana- logues for the Bernoulli interpretation of powers of the first m positive integers in terms of Bernoulli polynomials. In more detail, our main results are as follows. © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.