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Harmonic numbers appear, for example, in many expressions involving Riemann zeta functions. Here, among other things, we introduce and study discrete versions of those numbers, namely the discrete harmonic numbers. The aim of this paper is twofold. The first is to find several relations between the Type 2 higher-order degenerate Euler polynomials and the Type 2 high-order Changhee polynomials in connection with the degenerate Stirling numbers of both kinds and Jindalrae–Stirling numbers of both kinds. The second is to define the discrete harmonic numbers and some related polynomials and numbers, and to derive their explicit expressions and an identity. [ABSTRACT FROM AUTHOR]

Correction: Kim, T.; Khan, W.A.; Sharma, S.K.; Ghayasuddin, M. A Note on Parametric Kinds of the Degenerate Poly-Bernoulli and Poly-Genocchi Polynomials. (This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications II) The authors wish to provide the following information: on page 15, add funding information [[1]]. Reference 1 Kim T., Khan W.A., Sharma S.K., Ghayasuddin M. A Note on Parametric Kinds of the Degenerate Poly-Bernoulli and Poly-Genocchi Polynomials. [Extracted from the article]

Recently, the parametric kind of some well known polynomials have been presented by many authors. In a sequel of such type of works, in this paper, we introduce the two parametric kinds of degenerate poly-Bernoulli and poly-Genocchi polynomials. Some analytical properties of these parametric polynomials are also derived in a systematic manner. We will be able to find some identities of symmetry for those polynomials and numbers. [ABSTRACT FROM AUTHOR]

In recent years, many mathematicians studied various degenerate versions of some special polynomials for which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specifically, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of the second kind, and an expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind. [ABSTRACT FROM AUTHOR]

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Recently, the so-called new type Euler polynomials have been studied without considering Euler polynomials of a complex variable. Here we study degenerate versions of these new type Euler polynomials. This has been done by considering the degenerate Euler polynomials of a complex variable. We also investigate corresponding ones for Bernoulli polynomials in the same manner. We derive some properties and identities for those new polynomials. Here we note that our result gives an affirmative answer to the question raised by the reviewer of the paper. [ABSTRACT FROM AUTHOR]

Recently, the degenerate λ -Stirling polynomials of the second kind were introduced and investigated for their properties and relations. In this paper, we continue to study the degenerate λ -Stirling polynomials as well as the r-truncated degenerate λ -Stirling polynomials of the second kind which are derived from generating functions and Newton's formula. We derive recurrence relations and various expressions for them. Regarding applications, we show that both the degenerate λ -Stirling polynomials of the second and the r-truncated degenerate λ -Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study type 2 degenerate q-Bernoulli polynomials and numbers by virtue of the bosonic p-adic q-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, identities regarding Carlitz's q-Bernoulli numbers, identities concerning degenerate q-Bernoulli numbers, and the representations of the fully degenerate type 2 Bernoulli numbers in terms of moments of certain random variables, created from random variables with Laplace distributions. It is expected that, as was done in the case of type 2 degenerate Bernoulli polynomials and numbers, we will be able to find some identities of symmetry for those polynomials and numbers. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and also as multivariate versions of the r- Stirling numbers of the second kind and the extended r-Bell polynomials. In this paper, we study several properties, some identities and various explicit formulas about these polynomials and their connections as well. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the extended degenerate r-central factorial numbers of the second kind and the extended degenerate r-central Bell polynomials. They are extended versions of the degenerate central factorial numbers of the second kind and the degenerate central Bell polynomials, and also degenerate versions of the extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, all of which have been studied by Kim and Kim. We study various properties and identities concerning those numbers and polynomials and also their connections. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to represent sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations we express each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, some of which involve terminating hypergeometric functions 1 F 1 and 2 F 1 . [ABSTRACT FROM AUTHOR]

In this paper, we introduce central complete and incomplete Bell polynomials which can be viewed as generalizations of central Bell polynomials and central factorial numbers of the second kind, and also as 'central' analogues for complete and incomplete Bell polynomials. Further, some properties and identities for these polynomials are investigated. In particular, we provide explicit formulas for the central complete and incomplete Bell polynomials related to central factorial numbers of the second kind. [ABSTRACT FROM AUTHOR]

In this paper, firstly the definitions of the families of three-variable polynomials with the new generalized polynomials related to the generating functions of the famous polynomials and numbers in literature are given. Then, the explicit representation and partial differential equations for new polynomials are derived. The special cases of our polynomials are given in tables. In the last section, the interesting applications of these polynomials are found. [ABSTRACT FROM AUTHOR]

In this paper, we consider sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the previously-studied ones. We represent each of them as linear combinations of Chebyshev polynomials of all kinds whose coefficients involve some terminating hypergeometric functions 2 F 1 . The results may be viewed as a generalization of the linearization problem, which is concerned with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. These representations are obtained by explicit computations. [ABSTRACT FROM AUTHOR]

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 . [ABSTRACT FROM AUTHOR]

The q-Bernoulli numbers and polynomials can be given by Witt's type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found. [ABSTRACT FROM AUTHOR]

We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Z p and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on Z p of the two variable q-Bernstein polynomials, recently introduced by Kim, and demonstrate that they can be written in terms of the q-analogues of Euler numbers. Further, from such p-adic integrals we will derive some identities for the q-analogues of Euler numbers. [ABSTRACT FROM AUTHOR]

Here, we consider the sums of finite products of Chebyshev polynomials of the third and fourth kinds. Then, we represent each of those sums of finite products as linear combinations of the four kinds of Chebyshev polynomials, which involve the hypergeometric function 3 F 2 . [ABSTRACT FROM AUTHOR]

We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on Z p . Then we investigate a quotient of such p-adic integrals on Z p , representing generating functions of three w-torsion Fubini polynomials and derive some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials. [ABSTRACT FROM AUTHOR]