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Assume that the moment-generating function of the random variable $ Y $ exists in a neighborhood of the origin. The aim of this paper is to investigate the probabilistic type 2 Bernoulli polynomials associated with $ Y $ and the probabilistic type 2 Euler polynomials associated with $ Y $, along with the probabilistic type 2 cosine-Bernoulli polynomials associated with $ Y $, the probabilistic type 2 sine-Bernoulli polynomials associated with $ Y $, the probabilistic type 2 cosine-Euler polynomials associated with $ Y $, and the probabilistic type 2 sine-Euler polynomials associated with $ Y $. We deal with their properties, related identities and explicit expressions.

ABSTRACTHarmonic numbers and generalized harmonic numbers have been studied in connection with combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. We introduce Lerch-harmonic numbers which generalize the harmonic numbers and the generalized harmonic numbers. The aim of this note is to derive some identities expressing certain finite sums as the infinite sums involving the Lerch-harmonic numbers. Then, by taking limits of such identities we obtain the corresponding infinite sums of the finite sums as the infinite sums involving the Lerch transcendents.

The aim of this paper is to study the $ \lambda $-Stirling numbers of both kinds, which are $ \lambda $-analogues of Stirling numbers of both kinds. These numbers have nice combinatorial interpretations when $ \lambda $ are positive integers. If $ \lambda = 1 $, then the $ \lambda $-Stirling numbers of both kinds reduce to the Stirling numbers of both kinds. We derive new types of generating functions of the $ \lambda $-Stirling numbers of both kinds which are related to the reciprocals of the generalized rising factorials. Furthermore, some related identities are also derived from those generating functions. In addition, all the corresponding results to the $ \lambda $-Stirling numbers of both kinds are obtained for the $ \lambda $-analogues of $ r $-Stirling numbers of both kinds, which are generalizations of those numbers.

For any positive integer $ r $, the $ r $-truncated (or $ r $-associated) Stirling number of the second kind $ S_{2}^{(r)}(n, k) $ enumerates the number of partitions of the set $ \{1, 2, 3, \dots, n\} $ into $ k $ non-empty disjoint subsets, such that each subset contains at least $ r $ elements. We introduce the degenerate $ r $-truncated Stirling numbers of the second kind and of the first kind. They are degenerate versions of the $ r $-truncated Stirling numbers of the second kind and of the first kind, and reduce to the degenerate Stirling numbers of the second kind and of the first kind for $ r = 1 $. Our aim is to derive recurrence relations for both of those numbers.

The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic $ p $-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.

Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler polynomials. This paper is focused on the study of some properties, recurrence relations and identities related to the degenerate r-Bell polynomials, the two variable degenerate Fubini polynomials and the degenerate r-Stirling numbers of the second kind. Especially, we express the power series $ \sum _{n=0}^{\infty }\sum _{k=0}^{n}(k+r)_{p,\lambda }\frac {x^n}{n!} $ in terms of the degenerate r-Bell polynomials, of the degenerate r-Stirling numbers of the second kind and of the degenerate Fubini polynomials.

The degenerate exponentials are degenerate versions of the ordinary exponential and the truncated degenerate exponentials are obtained from the Taylor expansions of them by truncating the first finitely many terms. The degenerate exponentials play an important role in recent studies on degenerate versions of many special numbers and polynomials, the degenerate gamma function, the degenerate umbral calculus and the degenerate q-umbral calculus. The aim of this note is to consider infinite series whose terms involve truncated degenerate exponentials together with binomial coefficients, the generalized falling factorials and the Stirling numbers of the second kind, and to find either their values or some other expressions of them as finite sums.

We introduce the generalized degenerate Euler–Genocchi polynomials as a degenerate version of the Euler–Genocchi polynomials. In addition, we introduce their higher-order version, namely the generalized degenerate Euler–Genocchi polynomials of order α, as a degenerate version of the generalized Euler–Genocchi polynomials of order α. The aim of this paper is to study certain properties and identities involving those polynomials, the generalized falling factorials, the degenerate Euler polynomials of order α, the degenerate Stirling numbers of the second kind, and the ‘alternating degenerate power sum of integers’.

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable.

It is remarkable that, in recent years, intensive studies have been done for degenerate versions of many special polynomials and numbers and have yielded many interesting results. The aim of this paper is to study the generalized degenerate $ (r,s) $ -Stirling numbers of the second and their natural extensions to polynomials, namely the generalized degenerate $ (r,s) $ -Bell polynomials, arising from certain ‘degenerate boson normal ordering.’ We derive some properties, explicit expressions and generating functions for those numbers and polynomials. The generalized degenerate $ (r,s) $ -Stirling numbers of the second and the degenerate boson normal ordering are respectively degenerate versions of the generalized $ (r,s) $ -Stirling numbers of the second and the boson normal ordering studied earlier by Blasiak-Person-Solomon.

The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. As a ‘degenerate version’ of this, we consider the normal ordering of a degenerate integral power of the number operator in terms of boson operators, which is represented by means of the degenerate Stirling numbers of the second kind. As an application of this normal ordering, we derive two equations defining the degenerate Stirling numbers of the second kind and a Dobinski-like formula for the degenerate Bell polynomials.

The nth Fubini number enumerates the number of ordered partitions of a set with n elements and is the number of possible ways to write the Fubini formula for a summation of integration of order n. Further, Fubini polynomials are natural extensions of the Fubini numbers. There are many variants of Fubini numbers and polynomials. Recently, the degenerate Fubini polynomials were introduced by Kim-Kim-Jang as a degenerate version of the Fubini polynomials. The aim of this article is by using generating functions and certain differential operators to further study some identities and properties on the degenerate Fubini polynomials and the higher-order degenerate Fubini polynomials.

Degenerate Dowling and degenerate r-Dowling polynomials were introduced earlier as degenerate versions and further generalizations of Dowling and r-Dowling polynomials. The aim of this paper is to show their connections with Poisson degenerate central moments for a Poisson random variable with a certain parameter and with Charlier polynomials.

In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind, which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal orderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirling numbers of the second kind, that of a degenerate integral power of the number operator is represented by means of the degenerate Stirling numbers of the second kind.

The n-th Fubini number counts the number of ordered partitions of a set with n elements and is the number of possible ways to write the Fubini formula for a summation of integration of order n. Further, Fubini polynomials are natural extensions of the Fubini numbers. Recently, the generalized Fubini polynomials were introduced by Sebaoui-Laissaoui-Guettai-Rahmani, as one of the variants of the Fubini polynomials. The aim of this paper is to study the degenerate generalized Fubini polynomials, which are a degenerate version of those polynomials, and to find an application of them to probability theory in connection with geometric random variable.

Bell polynomials are widely applied in many problems arising from physics and engineering. The aim of this paper is to introduce new types of special polynomials and numbers, namely Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.

Recently, the generalized Euler–Genocchi and generalized degenerate Euler–Genocchi polynomials are introduced. The aim of this note is to study the multi-Euler–Genocchi and degenerate multi-Euler–Genocchi polynomials which are defined by means of the multiple logarithm and generalize, respectively, the generalized Euler–Genocchi and generalized degenerate Euler–Genocchi polynomials. Especially, we express the former by the generalized Euler–Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind, and the latter by the generalized degenerate Euler–Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind.

Recently, Kim-Kim introduced the degenerate r-Bell polynomials and investigated some results which are derived from umbral calculus. The aim of this paper is to study some properties of the degenerate r-Bell polynomials and numbers via boson operators. In particular, we obtain two expressions for the generating function of the degenerate r-Bell polynomials in z2, and a recurrence relation and Dobinski-like formula for the degenerate r-Bell numbers. These are derived from the degenerate normal ordering of a degenerate integral power of the number operator in terms of boson operators where the degenerate r-Stirling numbers of the second kind appear as the coefficients.

The aim of this study is to represent any polynomial in terms of the degenerate Genocchi polynomials and more generally of the higher-order degenerate Genocchi polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

n recent years, some applications of Lah numbers were discovered in the real world problem of telecommunications and optics. The aim of this paper is to study the λ-analogues of Lah numbers and Lah-Bell polynomials which are λ-analogues of the Lah numbers and and LahBell polynomials. Here we note that λ-analogues appear when we replace the falling factorials by the generalized falling factorials in the defining equations. By using generating function method, we study some properties, explicit expressions, generating functions and Dobinski-like formulas for those numbers and polynomials. We also treat the more general λ-analogues of r-Lah numbers and r-extended λ-Lah-Bell polynomials. In addition, we show that the expectations of two random variables, both associated with the Poisson random variable with parameter α/λ, are equal to the λ-analogue of the Lah-Bell polynomial evaluated at α for one and the r-extended λ-Lah-Bell polynomial evaluated at α for the other. [ABSTRACT FROM AUTHOR]

In this paper, we consider various Stirling numbers of both kinds, including the unsigned degenerate Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, the unsigned degenerate r-Stirling numbers of the first kind and the degenerate r-Stirling numbers of the second kind. The aim of this paper is by using generating functions to further study explicit expressions, some identities and equivalent relations for those Stirling numbers. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic p-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind. [ABSTRACT FROM AUTHOR]

The aim of this paper is to further study some properties and identities on the degenerate Fubini and the degenerate Bell polynomials which are degenerate versions of the Fubini and the Bell polynomials, respectively. Especially, we find several expressions for the generating function of the sum of the values of the generalized falling factorials at positive consecutive integers. [ABSTRACT FROM AUTHOR]

In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials which are defined by means of the multiple polylogarithms and degenerate versions of the multi-poly-Bernoulli numbers and polynomials. We investigate some properties for those numbers and polynomials. In addition, we give some identities and relations for the degenerate multi-poly-Bernoulli numbers and polynomials. [ABSTRACT FROM AUTHOR]