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The aim of this paper is to introduce truncated degenerate Bell polynomials and numbers and to investigate some of their properties. In more detail, we obtain explicit expressions, identities involving other special polynomials, integral representations, a Dobinski-like formula and expressions of the generating function in terms of differential operators and the linear incomplete gamma function. In addition, we introduce truncated degenerate modified Bell polynomials and numbers and obtain similar results for those polynomials. As an application of our results, we show that the truncated degenerate Bell numbers can be expressed as a finite sum involving moments of a beta random variable with certain parameters.

Recently, the degenerate gamma functions were introduced as a degenerate version of the usual gamma function. In this paper, we investigate several properties of these functions. Namely, we obtain an analytic continuation as a meromorphic function on the whole complex plane, the difference formula, the values at positive integers, some expressions following from the Weierstrass and Euler formulas for the ordinary gamma function, and an integral representation as an integral along a Hankel contour.

Studying degenerate versions of various special polynomials has became an active area of research and has yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of the polylogarithm function, the so-called degenerate polylogarithm function. Then we construct a new type of degenerate Bernoulli polynomial and number, the so-called degenerate poly-Bernoulli polynomial and number, by using the degenerate polylogarithm function, and derive several properties concerning the degenerate poly-Bernoulli numbers.

The central factorial numbers of the second kind appear in the expansion of powers of x in terms of the central factorial sequence. In this paper, we introduce the central Bell numbers and polynomials associated with those central factorial numbers of the second kind and investigate some identities and properties of them.

In this paper, we introduce polyexponential functions as an inverse to the polylogarithm functions, construct type 2 poly-Bernoulli polynomials by using this and derive various properties of type 2 poly-Bernoulli numbers. Then we introduce unipoly functions attached to each suitable arithmetic function as a universal concept which includes the polylogarithm and polyexponential functions as special cases. By making use of unipoly functions, we define unipoly-Bernoulli polynomials, type 2 unipoly-Bernoulli numbers, and unipoly-Bernoulli numbers of the second kind, and show some basic properties for them.

In this paper, we study some p-adic invariant and fermionic p-adic integrals on ℤp associated with trigonometric functions. By using these p-adic integrals we represent several trigonometric functions as a formal power series involving either Bernoulli or Euler numbers. In addition, we obtain some identities relating various special numbers like zigzag, some ‘trigonometric’, Bernoulli, Euler numbers, and Euler numbers of the second kind.

The purpose of this paper is to exploit umbral calculus in order to derive some properties, recurrence relations, and identities related to the degenerate r-Stirling numbers of the second kind and the degenerate r-Bell polynomials. Especially, we will express the degenerate r-Bell polynomials as linear combinations of many well-known families of special polynomials.

In this paper, we introduce the degenerate Laplace transform and degenerate gamma function and investigate some properties of the degenerate Laplace transform and degenerate gamma function. From our investigation, we derive some interesting formulas related to the degenerate Laplace transform and degenerate gamma function., 10 pages

In this paper, we introduce some new λ-Bell polynomials and Bell polynomials of the second kind and investigate properties of these polynomials. Using our investigation, we derive some new identities for the two kinds of λ-Bell polynomials arising from umbral calculus.

We introduce a special nonlinear differential operator and, using its properties, reduce higher-order Frobenius–Euler Apostol-type polynomials to a finite series of first-order Apostol-type Frobenius–Euler polynomials and Stirling numbers. Interesting identities are established.

In this paper, we study nonlinear Changhee differential equations and derive some new and explicit identities of Changhee and Euler numbers from those nonlinear differential equations.

In this paper, we consider poly-Bernoulli and higher-order poly-Bernoulli polynomials and derive some new and interesting identities of those polynomials by using umbral calculus.

In this paper, we study some properties of several polynomials arising from umbral calculus. In particular, we investigate the properties of orthogonality type of the Frobeniustype Eulerian polynomials which are derived from umbral calculus. By using our properties, we can derive many interesting identities of special polynomials associated with Frobeniustype Eulerian polynomials. An application to normal ordering is presented.

In this paper, we consider a q-analogue of Laplace transform and we investigate some properties of q-Laplace transform. From our investigation, we derive some interesting formulae related to q-Laplace transform.

In this paper, we investigate some identities of Laguerre polynomials involving Bernoulli and Euler polynomials which are derived from umbral calculus., 12 pages

Recently, Agrawal and Thamer suggested a new sequence of summation-integral type operators using the Baskakov basis function. We find a generating function of the q-Baskakov basis function and suggest a q-analog of the operators. We estimate moments, give a recurrence formula, and obtain some direct results. We also mention the possibility of improving the q-operators and, finally, suggest a Bezier-type variant of the operators discussed here.

The q-Euler numbers and polynomials were recently constructed [T. Kim, “The Modified q-Euler Numbers and Polynomials,” Adv. Stud. Contemp. Math., 16, 161–170 (2008)]. These q-Euler numbers and polynomials have interesting properties. In this paper, we prove a theorem of the von Staudt-Clausen type for q-Euler numbers; namely, we prove that the q-Euler numbers are p-adic integers. Finally, we prove Kummer-type congruences for the q-Euler numbers.

In the paper, we derive an analog of the Lebesgue-Radon-Nikodým theorem with respect to fermionic p-adic invariant measure on ℤp.

In this paper, we present new generating functions related to q-Euler numbers and polynomials of higher order. Using those generating functions, we present new identities involving q-Euler numbers and polynomials of higher order.

A systemic study of some families of q-Euler numbers and families of polynomials of Norlund type is presented by using the multivariate fermionic p-adic integral on ℤ p . The study of these higher-order q-Euler numbers and polynomials yields an interesting q-analog of identities for Stirling numbers.

The objective of the paper is to indicate a symmetry of the multivariate p-adic invariant integral on ℤ p , which leads to a relation between the power sum polynomials and higher-order Euler polynomials.

The purpose of this paper is to present a systemic study of some families of multiple q-Genocchi and euler numbers by using multivariate q-Volkenborn integral. From the studies of those q-Genocchi numbers and polynomials of higher order we derive some interesting identities related to q-Genocchi numbers and polynomials of higher order., 11 pages

In this paper, we present a new generating function which is related to Daehee numbers. By using the Mellin transformation formula and the Cauchy theorem for this generating function, we define multiple Daehee q-l-functions and q-zeta functions We also give the values of this function at negative integers.

In this paper, we introduce a Daehee constant, the so-called q-extension of the Napier constant, and consider the Daehee formula associated with the q-extensions of trigonometric functions. That is, we derive the q-extensions of sine and cosine functions from our Daehee formula. Finally, we present the q-calculus related to the q-extensions of sine and cosine functions.

In this paper, we give some p-adic approximation of E n,x for certain n. Finally we will treat p-adic l-function of Kubota-Leopoldt’s type Euler numbers and p-adic measure for Euler numbers.

Recently, B. A. Kupershmidt constructed reflection symmetries of q-Bernoulli polynomials (see [12]). In this paper, we study new q-extensions of Euler numbers and polynomials by using the method of Kupershmidt. We also investigate the properties of symmetries of these q-Euler polynomials by using q-derivatives and q-integrals.

We construct multivariate p-adic L-function in the p-adic number fild by using Washington method.