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 mathbb{Z}_{p}.
Highlights
Studies on degenerate versions of some special polynomials and numbers began with the papers by Carlitz in [3, 4]
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
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 analogues for the Bernoulli interpretation of powers of the first m positive integers in terms of Bernoulli polynomials
Summary
Studies on degenerate versions of some special polynomials and numbers began with the papers by Carlitz in [3, 4]. The research has been carried out by several different methods like generating functions, combinatorial 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]). 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 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 analogues for the Bernoulli interpretation of powers of the first m positive integers in terms of Bernoulli polynomials.
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