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The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate Stirling numbers of both kinds associated with degenerate hyperharmonic numbers and also with degenerate Bernolli, degenerate Euler, degenerate Bell and degenerate Fubini polynomials., 16 pages

Broder introduced the r-Stirling numbers of the first kind and of the second kind which enumerate restricted permutations and respectively restricted partitions, the restriction being that the first r elements must be in distinct cycles and respectively in distinct subsets. Kim-Kim-Lee-Park constructed the degenerate r-Stirling numbers of both kinds as degenerate versions of them. The aim of this paper is to derive some identities and recurrence relations for the degenerate r-Stirling numbers of the first kind and of the second kind via boson operators. In particular, we obtain the normal ordering of a degenerate integral power of the number operator multiplied by an integral power of the creation boson operator in terms of boson operators where the degenerate r-Stirling numbers of the second kind appear as the coefficients., Comment: 12 pages

The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate hyperharmonic numbers, hyperharmonic numbers and degenerate harmonic numbers. In particular, we derive an explicit expression of the degenerate hyperharmonic numbers in terms of the degenerate harmonic numbers. This is a degenerate version of the corresponding identity representing the hyperharmonic numbers in terms of harmonic numbers due to Conway and Guy., Comment: 10 pages

Recently, the degenerate harmonic and the degenerate hyperharmonic numbers are introduced respectively as degenerate versions of the harmonic and the hyperharmonic numbers. The aim of this paper is to introduce the degenerate harmonic-Fubini polynomials and numbers related to the degenerate harmonic numbers and to study their properties, explicit expressions and some identities. In addition, as generalizations of those polynomials and numbers, we also introduce the degenerate hyperharmonic-Fubini polynomials and numbers related to the degenerate hyperharmonic numbers and derive similar results to the degenerate harmonic-Fubini polynomials and numbers. \end{abstract}, 12 pages

Recently, Freud-Rodriguez proposed a new counting process which is called the Bell-Touchard process and based on the Bell-Touchard probability distribution. This process was developed to solve the problem of rare events hypothesis which is one of the limitations of the Poisson process. In this paper, we consider the discrete degenerate Bell distributions and the degenerate Bell process which are 'degenerate versions' of the Bell-Touchard probability distributions and the Bell-Touchard process, respectively. We investigate several properties of the degenerate Bell distribution. We introduce the degenerate Bell process by giving two equivalent definitions and show one method of constructing a new infinite family of degenerate Bell process out of a given infinite family of degenerate Bell process., 9 pages

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., 7 pages. arXiv admin note: text overlap with arXiv:2204.01252

In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials and more generally of the higher-order degenerate Bernoulli polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples., Comment: 19 pages

The multi-Stirling numbers of the second kind, the unsigned multi-Stirling numbers of the first kind, the multi-Lah numbers and the multi-Bernoulli numbers are all defined with the help of the multiple logarithm, and generalize respectively the Stirling numbers of the second kind, the unsigned Stirling numbers of the first kind, the unsigned Lah numbers and the higher-order Bernoulli numbers . The aim of this paper is to introduce the multi-Stirling numbers of the second kind and to find several identities involving those four numbers defined by means of the multiple logarithm and some other special numbers., 9 pages

The Stirling numbers of the second kind are related to normal orderings in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal orderings in the shift algebra. Kim-Kim introduced a {\lambda}-analogue of the unsigned Stirling numbers of the first kind and that of the r-Stirling numbers of the first kind. In this paper, we introduce a {\lambda}-analogue of the shift algebra (called {\lambda}-shift algebra) and investigate normal orderings in the {\lambda}-shift algebra. From the normal orderings in the {\lambda}-shift algebra, we derive some identities about the {\lambda}-analogue of the unsigned Stirling numbers of the first kind ., Comment: 9

The degenerate exponentials play an important role in recent study 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 several special numbers and to find either their values or some other expressions of them as finite sums., 7 pages

Harmonic numbers have been studied since antiquity, while hyperharmonic numbers were intoduced by Conway and Guy in 1996. The degenerate harmonic numbers and degenerate hyperharmonic numbers are their respective degenerate versions. The aim of this paper is to further investigate some properties, recurrence relations and identities involving the degenerate harmonic and degenerate hyperharmonic numbers in connection with degenerate Stirling numbers of the first kind, degenerate Daehee numbers and degenerate derangements., 10pages

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., Comment: 13

Recently, introduced are the generalized Euler-Genocchi and generalized degenerate Euler-Genocchi polynomials. 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., Comment: 9 pages

The Stirling numbers of the second kind are related to normal orderings in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal orderings in the shift algebra. Kim-Kim introduced a λ-analogue of the unsigned Stirling numbers of the first kind and that of the r-Stirling numbers of the first kind. In this paper, we introduce a λ-analogue of the shift algebra (called λ-shift algebra) and investigate normal orderings in the λ-shift algebra. From the normal orderings in the λ-shift algebra, we derive some identities about the λ-analogue of the unsigned Stirling numbers of the first kind ., 9

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.

In recent years, some degenerate versions of quite a few special numbers and polynomials are introduced and investigated by means of various methods. The aim of this paper is to study some results on degenerate harmonic numbers, degenerate hyperharmonic numbers, degenerate Fubi polynomials and degenerate r-Fubini polynomials from a general identity which is valid for any two formal power series and involves the degenerate r-Stirling numbers of the second kind., 12 pages

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 {\alpha}, as a degenerate version of the generalized Euler-Genocchi polynomials of order {\alpha}. 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 {\alpha}, the degenerate Stirling numbers of the second kind, and the alternating degenerate power sum of integers., Comment: 13 pages

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 |z| , 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., 9 pages

The Mersenne primes are primes which can be written as some prime power of 2 minus 1. These primes were studied from antiquity in that their close connection with perfect numbers and even to present day in that their easiness for primality test. In this paper, we introduce the dimorphic Mersenne numbers as a degenerate version of the Mersenne numbers and investigate some of their properties in connection with the degenerate Bernoulli polynomials and the incomplete Bell polynomials., 7 pages

Recently, the $\lambda$-analogues of $r$-Stirling numbers of the first kind were studied by Kim-Kim. The aim of this paper is to introduce the $\lambda$-analogues of $r$-Stirling numbers of the second kind and to investigate some properties, recurrence relations and certain identities on those numbers. We also introduce the $\lambda$-analogues of Whitney-type $r$-Stirling numbers of the second and derive similar results to the case of the $\lambda$-analogues of r-Stirling numbers of the second kind. In addition, we consider the $\lambda$-analogues of Dowling polynomials and deduce a Dobinski-like formula., Comment: 11 pages

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 oderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and from 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., 10 pages

In this paper, we investigate new class of sequences related to fully degenerate Bernoulli numbers and polynomials. From those sequences, we derive some formulae for the degenerate Bernoulli and Euler polynomials., 10 pages

The aim of this paper is by using generating functions to further study some identities and properties on the degenerate Stirling numbers of the second kind, the degenerate $r$-Stirling numbers of the second kind, the degenerate Stirling numbers of the first kind and the unsigned degenerate Stirling numbers of the first kind., 15 pages

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., Comment: 12 pages

For any positive integer r, the r-associated Stirling number of the second kind enumerates the number of partitions of the set{1,2,3,...,n} into k non-empty disjoint subsets such that each subset contains at least r elements. We introduce the degenerate r-associated Stirling numbers of the second kind and of the first kind. They are degenerate versions of the r-associated 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. The aim of this paper is to derive recurrence relations for both of those numbers., Comment: 8 pages

Recently, authors studied the unsigned degenerate r-Stirling number of the first kind and the degenerate r-Stirling number of the second kind, respectively of which are the degenerate versions of the unsigned r-Stirling numbers of the first kind and those of the r-Stirling numbers of the second kind. The aim of this paper is to derive some identities involving such special numbers from the inverse relations for them., Comment: 8 pages

Let P be the set of the sequence of polynomials of degree n. The aim of this paper is to study the Stirling numbers of the second kind associated with P and of the first kind associated with P, in a unified and systematic way with the help of umbral calculus technique. Our results are illustrated with many examples which give rise to interesting inverse relations in each case., Comment: 39 pages

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., Comment: 10 pages

Many mathematicians have been studying various degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our main focus here is a new type of degenerate poly-Euler polynomials and numbers. This focus stems from their nascent importance for applications in combinatorics, number theory and in other aspects of applied mathematics. we construct a new type of degenerate poly-Euler polynomials by using the degenerate polylogarithm functions. We also show several combinatorial identities related to this polynomials and numbers.

The aim of this paper is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on Zp. We find some explicit expressions for the fully degenerate Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind, the degenerate r-Stirling numbers of the second kind and of the degenerate Stirling polynomials. We also consider the degenerate poly-Bernoulli polynomials and derive explicit representations for them in terms of the same degenerate Stirling numbers and polynomials., Comment: 11 pages

In this paper, we consider the problem of representing any polynomial in terms of the ordered Bell and degenerate ordered Bell polynomials, and more generally of the higher-order ordered Bell and higher-order degenerate ordered Bell polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples., 20 pages. arXiv admin note: substantial text overlap with arXiv:2108.05049, arXiv:2109.13700

The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate version of the Bell numbers and polynomials. we derive some new identities and properties of those numbers and polynomials that are associated with the degenerate Stirling numbers of the both kinds., 13 pages

Carlitz [2] initiated a study on degenerate versions of Bernoulli and Euler numbers which has been extended recently to the researches on various degenerate versions of quite a few special numbers and polynomials. They have been explored by using several different tools including generating functions, combinatorial methods, $p$-adic analysis, umbral calculus, special functions, differential equations and probability theory as well. \par The degenerate Poisson random variables are degenerate versions of the Poisson random variables. In [6], studied are the degenerate binomial and degenerate Poisson random variables in relation to the degenerate Lah-Bell polynomials. Among other things, it is shown that the rising factorial moments of the degenerate Poisson random variable are expressed by the degenerate Lah-Bell polynomials. Also, it is shown that the probability-generating function of the degenerate Poisson random variable is equal to the generating function of the degenerate Lah-Bell polynomials. The zero-truncated Poisson distributions (also called the conditional or the positive Poisson distributions) are certain discrete probability distributions whose supports are the set of positive integers. In [10], the zero-truncated degenerate Poisson random variables, whose probability mass functions are a natural extension of the zero-truncated Poisson random variables, are introduced and various properties of those random variables are investigated. Specifically, for those distributions, studied are its expectation, its variance, its n-th moment, its cumulative distribution function and certain expressions for the probability function of a finite sum of independent degenerate zero-truncated Poisson random variables with equal and unequal parameters., 10 pages

In this paper, we derive five basic identities for Sheffer polynomials by using generalized Pascal functional and Wronskian matrices. Then we apply twelve basic identities for Sheffer polynomials, seven from previous results, to degenerate Bernoulli polynomials and Korobov polynomials of the first kind and get some new identities. In addition, letting $\lambda~\rightarrow~0$ in such identities gives usthose for Bernoulli polynomials and Bernoulli polynomials of the second kind.

Catalan-Daehee numbers and polynomials, generating functions of which can be expressed as p-adic Volkenborn integrals on Zp, were studied previously. The aim of this paper is to introduce q-analogues of the catalan-Daehee numbers and polynomials with the help of p-adic q-integrals von Zp. We derive, among other thing, some explicit expressions for the q-analogues of the Catalan-Daehee numbers and polynomials., Comment: 7 pages

The aim of this paper is to study a dimorphic property associated with two different sums of identically independent Bernoulli random variables having two different families of probability mass functions. In addition, we give two expressions on sums of products of degenerate Stirling numbers of the second kind and Stirling numbers of the first kind connected with those two different sums of identically independent Bernoulli random variables., Comment: 8 pages

Recently, several types of degenerate Bell polynomials have been introduced as degenerate versions of the ordinary Bell polynomials. The aim of this paper is to study some identities for the degenerate Bell polynomials and their related identities by making use of series transformation formula obtained by Boyadzhiev. In particular, this answers to the natural question about the relationship between two different types of degenerate Bell polynomials., Comment: 11 pages

The aim of this paper is to represent any polynomial in terms of the degenerate Frobenius-Euler polynomials and more generally of the higher-order degenerate Frobenius-Euler polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples., Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:2108.05049

Recently, Bovadzhiev studied a power series whose coefficients are binomial expressions and extended some known formulas involving classical special functions and polynomials. The aim of this paper is to adopt his ideas to express several identities involving " degenerate formal power series" as those including degenerate stirling numbers of the second kind, degenerate Bell polynomials, degenerate Fubini polynomials and degenerate poly-Bernoulli polynomials., Comment: 17 pages

The nth r-extended Lah-Bell number is defined as the number of ways a set with $n+r$ elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah-Bell polynomials and complete $r$-extended Lah-Bell polynomials respectively as multivariate versions of $r$-Lah numbers and the r-extended Lah-Bell numbers and to investigate some properties and identities for these polynomials. From these investigations, we obtain some expressions for the r-Lah numbers and the r-extended Lah-Bell numbers as finite sums., Comment: 10 pages

In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals, Comment: 6 pages

Dowling constructed Dowling lattice Qn(G), for any finite set with n elements and any finite multiplicative group G of order m, which is a finite geometric lattice. He also defined the Whitney numbers of the first and second kinds for any finite geometric lattice. These numbers for the Dowling lattice Qn(G) are the Whitney numbers of the first kind Vm(n,k) and those of the second kind Wm(n,k), which are given by Stirling number-like relations. In this paper, by `degenerating' such relations we introduce the degenerate Whitney numbers of the first kind and those of the second kind and investigate, among other things, generating functions, recurrence relations and various explicit expressions for them. As further generalizations of the degenerate Whitney numbers of both kinds, we also consider the degenerate r-Whitney numbers of both kinds., Comment: 22 pages

The aim of this paper is to study some basic properties of the r$-extended Lah-Bell numbers and polynomials associated with r-Lah numbers and to show the connection between the r-extended Lah-Bell polynomials and the rising factorial moments of the Poisson random variable shifted by an even nonnegative integer 2r. Here the nth r-extended Lah-Bell number counts the number of ways a set of n+r elements can be partitioned into non-empty linearly ordered subsets such that r distinguished elements have to be in distinct linearly ordered subsets and the r-extended Lah-Bell polynomials are natural extensions of the r-extended Lah-Bell numbers., 10 pages

The aim of this paper is to study the Poisson random variables in relation to the Lah-Bell polynomials and the degenerate binomial and degenerate Poisson random variables in connection with the degenerate Lah-Bell polynomials. Among other things, we show that the rising factorial moments of the degenerate Poisson random variable with parameter {\alpha} are given by the degenerate Lah-Bell polynomials evaluated at {\alpha}. We also show that the probability-generating function of the degenerate Poisson random variable is equal to the generating function of the degenerate Lah-Bell polynomials. Also, we show similar results for the Poisson random variables. Here the nth Lah-Bell number counts the number of ways a set of n elements can be partitioned into non-empty linearly ordered subsets, the Lah-Bell polynomials are natural extensions of the Lah-Bell numbers and the degenerate Lah-Bell polynomials are degenerate versions of the Lah-Bell polynomials., Comment: 10 pages

Objective: Brain edema after severe traumatic brain injury (sTBI) is a common occurrence in infants resulting in elevated intracranial pressure (ICP), and causing a severe or fatal state. Because of infant brain plasticity, however, appropriate treatments to improve ICP can lead to positive outcomes.[for full text, please go to the a.m. URL], 71. Jahrestagung der Deutschen Gesellschaft für Neurochirurgie (DGNC), 9. Joint Meeting mit der Japanischen Gesellschaft für Neurochirurgie

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., 7 pages

In this paper, we introduce the degenerate multiple polyexponential functions which are multiple versions of the degenerate modified polyexponential functions. Then we consider the degenerate multi-poly-Genocchi polynomials which are defined by using those functions and investigate explicit expressions and some properties for those polynomials., 8 pages

The aim of this paper is twofold. The first one is to find several relations between the type 2 higher-order degenerate Euler polynomials and the type 2 higher-order Changhee polynomials in connection with the degenerate stirling numbers of both kinds and Jindalrae-stirling numbers of both kinds. The second one is to introduce the discrete harmonic numbers and some related polynomials and numbers, and to derive their explicit expressions and an identity., 12 pages

Studying degenerate versions of various special polynomials have become an active area of research and yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of polylogarithm function, called the degenerate polylogarithm function. Then we construct new type degenerate Bernoulli polynomials and numbers, called degenerate poly-Bernoulli polynomials and numbers,by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Bernoulli numbers, 9 pages

In this paper, we introduce the Lah-Bell numbers and their natural extensions, namely the Lah-Bell polynomials, and derive some basic properties of such numbers and polynomials by using elementary methods. In addition, we consider the degenerate Lah-Bell numbers and polynomials as degenerate versions of the Lah-Bell numbers and polynomials., Comment: 10 pages