Degenerate Catalan-Daehee numbers and polynomials of order r arising from degenerate umbral calculus

: Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person’s psychological burden in recent years, and they’ve discovered some interesting results. Furthermore, one of the most important approaches for ﬁnding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers. In this paper, we ﬁrst deﬁne the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order r as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order r , the degenerate Euler polynomials and numbers of order r , the degenerate Daehee polynomials of order r , the degenerate Bell polynomials, and so on. degenerate ﬁrst kind; degenerate


Introduction
In combinatorics, the Catalan numbers are the numbers of Dyck words of length 2n or the numbers of different ways n + 1 factors completely parenthesized or the numbers of non-isomorphic ordered trees with n + 1 vertices or the numbers of monotonic lattice paths along the edges of a grid with n × n square cells, which do not pass above the diagonal or the numbers of noncrossing partitions of the set {1, ..., n} and arise in many other counting problems with real-world applications [3,5,7,24,28]. The Catalan-Daehee numbers are defined by assigning √ 1 − 4t − 1 instead of t in the definition of Daehee numbers which play important role in connecting relationship between special numbers [10,16]. Moreover, the generating function of Catalan numbers can be represented by the fermionic p-adic integral on Z p of (1 − 4t) x 2 and the generating function of Catalan-Daehee numbers can be represented by the p-adic Volkenborn integral on Z p of the same function (1 − 4t) x 2 [16,17]. Various identities of Catalan-Daehee polynomials have been studied in [5,16,17,29].
Many scholars in the field of mathematics have worked on degenerate versions of special polynomials and numbers which include the degenerate Stirling numbers of the first and second kinds, the degenerate Bernstein polynomials, the degenerate Bell numbers and polynomials, the degenerate gamma function, the degenerate gamma random variables, and so on [1, 2, 10-15, 19, 20, 30, 31]. We can find the motivation to study degenerate polynomials and numbers in the following real-world examples. Suppose the probability of a baseball player getting a hit in a match is p. We wonder if the probability that the player will succeed in the 11th trial after failing 9 times in 10 trials is still p. We can see cases where the probability is less than p because of the psychological burden that the player must succeed in the 11th trial [31].
In the 1970s, Rota and his collaborators [22][23][24] began to construct a rigorous foundation for the classical umbral calculus, which consisted of a symbolic technique for the manipulation of numerical and polynomial sequences. The umbral calculus has received much attention from researchers because of its numerous applications in many fields of mathematics, physics, chemistry, and engineering [4,6,9,11,13,15,16,[20][21][22][23][24][25][26]28]. For instance, the connection between Sheffer polynomials and Riordan array and the isomorphism between the Sheffer groups and the Riordan Groups are proved [25,26]. Recently, Kim-Kim [11] introduced the λ -Sheffer sequences and the degenerate Sheffer sequences by substituting λ -linear functionals and λ -differential operators, respectively, instead of linear functionals and differential operators.
With these points in mind, in this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and degenerate Catalan-Daehee polynomials of order r(≥ 1) as one of the generalizations of the degenerate Catalan-Daehee polynomials. It is difficult to study identities related to degenerate Catalan-Daehee polynomials and special polynomials using the p-adic integral on Z p or other properties. Thus, we explore various interesting identities related to the degenerate Catalan-Daehee polynomials of order r and special polynomials and numbers by using degenerate Sheffer sequences. At the same time we derive the inversion formulas of these identities. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the falling factorials, the degenerate Bernoulli polynomials and numbers of order r, the degenerate Euler polynomials and numbers of order r, the degenerate Daehee polynomials of order r, the degenerate Bell polynomials, etc. Now, we give some definitions and properties needed in this paper.
The degenerate Bernoulli polynomials of the second kind of order r are defined by the generating function , (see [8,11]). (1.5) n,λ (0), which are called the degenerate Bernoulli numbers of the second kind of order r.
Kim-Kim introduced the degenerate Bell polynomials given by the generating function Bel l,λ (x) t l l! , (see [13]).  For n ≥ 0, it is well known that the Stirling numbers of the first and second kind, respectively are given by (see [1,14]), and Moreover, the degenerate Stirling numbers of the first and second kind, respectively are given by [12,14]), (1.8) and [12,14]). (1.9) For k ≥ 0, as an extension of the notion of the degenerate Stirling numbers of the second kind, Kim et al. introduced Jindalrae-Stirling numbers of the second kind by j,λ (n, k) t n n! , (see [18]). (1.10) From (1.9) and (1.10), we note that Let C be the complex number field and let F be the set of all power series in the variable t over C with Let P = C[x] and P * be the vector space all linear functional on P: Then P n is an (n + 1)-dimensional vector space over C.
Recently, Kim-Kim [11] considered λ -linear functional and λ -differential operator as follows: a k t k k! ∈ F and a fixed nonzero real number λ , each λ gives rise to the linear functional f (t) | · λ on P, called λ -linear functional given by f (t), which is defined by f (t) | (x) n,λ λ = a n , for all n ≥ 0, (see [11]). (1.12) In particular t k | (x) n,λ λ = n!δ n,k , for all n, k ≥ 0, where δ n,k is the Kronecker's symbol. For λ = 0, we observe that the linear functional f (t) | · 0 agrees with the one in For each λ ∈ R and each nonnegative integer k, they also defined the differential operator on P by [11]). (1.13) and for any power series Let f (t) and g(t) be a delta series and an invertible series, respectively. Then there exists a unique sequence s n,λ (x) such that the orthogonality condition holds k | s n,λ (x) λ = n!δ n,k , (n, k ≥ 0), (see [11]). (1.14) The sequence s n,λ (x) is called the λ -Sheffer sequence for (g(t), f (t)), which is denoted by [11]). (1.15) Assume that for each λ ∈ R * of the set of nonzero real numbers, s n,λ (x) is λ -Sheffer for (g λ (t), f λ (t)). Assume also that lim λ →0 f λ (t) = f (t) and lim λ →0 g λ (t) = g(t), for some delta series f (t) and an invertible series g(t).
In this case, Kim-Kim called that the family {s n,λ (x)} λ ∈R−{0} of λ -Sheffer sequences s n,λ is the degenerate (Sheffer) sequences for the Sheffer polynomial s n (x).

Degenerate Catalan-Daehee polynomials arising from degenerate Sheffer sequences
In this section, we define the degenerate Catalan-Daehee polynomials of order r, and derive several identities between the degenerate Catalan-Daehee polynomials of order r and some other polynomials arising from degenerate Sheffer sequences.
As is known, the Catalan numbers C n are given by the generating function C n t n , (see [5,16,17]).
The Catalan numbers C (r) n of order r, as a generalization of Catalan numbers, are given by the generating function n t n , (see [16,17]).
Kim-Kim introduced the Catalan-Daehee polynomials which are given by the generating function (see [16,17]).
(2.1) When x = 0, C n := C n (0), which are called Catalan-Daehee numbers. From (1.6) and (2.1), we note that We introduce the degenerate Catalan-Daehee polynomials C n,λ (x) which are given by the generating function When x = 0, C n,λ := C n,λ (0), which are called degenerate Catalan Daehee numbers. When λ → 0, we note that C n,λ (x) = C n (x). As a generalization of the degenerate Catalan-Daehee polynomials, we also introduce degenerate Catalan-Daehee polynomials C (r) n,λ (x) of order r are given by the generating function n,λ (0), which are called degenerate Catalan Daehee numbers of order r. It easy to see that the compositional inverse of f (t) = 1 and n!C (2.6) Proof. From (1.2), (1.15) and (2.6), we consider the following two Sheffer sequence as follows: n−m,λ .
(2) when r 1 r 2 , C where B when r 1 > r 2 , and when r 1 < r 2 , n−k,λ . The following theorem gives the inversion formula of Theorem 5.
Theorem 6. For n ∈ N ∪ {0}, we have Proof. From (2.22), we consider the two degenerate Sheffer sequences as follows: , t λ and n!C when r 1 > r 2 , and when r 1 < r 2 , where E Observe that where E  It is difficult to single out where and why these formulas play an important role, but we do not doubt that they will be helpful to researchers in need of these identities. Further research would be related with the degenerate versions of some special combinatorial numbers and polynomials and then contribution in mathematics and physics applications.