Study of degenerate derangement polynomials by λ-umbral calculus

1 Intorduction

For a given set A with n elements, a derangement is a permutation of the elements in A such that no element appears in its original position. The number of derangements of A is called the n th derangement number, denoted by d n . By the definition of the derangement, we see that

Derangement numbers are computed recursively or by means of their exponential generating function in general, and one of the important and widely covered fields in various fields such as combinatorics, applied mathematics, and engineering (see [1–3]). In [4], the authors found closed forms for derangement numbers with Hessenberg determinants and derivatives of the exponential generating function of these numbers. In [5], the authors investigated several interesting combinatorial, analytic, and number theoretic properties of r -derangement numbers, which are a permutation on n + r elements, such that in its cycle decomposition, the first r elements appear in distinct cycles. Clarke and Sved found a relation between the derangement numbers and the Bell numbers by using the inclusion-exclusion principle in [6]. In [7], the authors defined p , q -analog of the derangement numbers, which is a generalization of derangement numbers, and derived a recurrence for them.

For a given λ ∈ R − { 0 } , the degenerate exponential function is defined to be

Note that lim λ → 0 e λ x = e x t and lim λ → 0 e λ ( t ) = e t .

The study of the degenerate version of functions was first initiated by Carlitz (see [8]), and since then, degenerations of various special functions have been defined and their properties have been actively studied by many researchers. In [9], the authors defined the partially degenerate Laguerre-Bernoulli polynomials of the first kind and derived some theorems on implicit summation formulae and symmetry identities for these polynomials. In [10], the authors derived some identities and properties on the degenerate Fubini polynomials and the higher-order degenerate Fubini polynomials by using generating functions and certain differential operators. Kwon et al. defined the modified type 2 degenerate poly-Bernoulli polynomials and derived some explicit expressions and their representations by using λ -umbral calculus (see [11]). In [12], the authors defined the degenerate poly-Genocchi polynomials and numbers by using the degenerate polylogarithm function and gave some identities of those polynomials and explicit expressions of degenerate unipoly polynomials related to special polynomials. Kim and Khan introduced a new type of degenerate poly-Frobenius-Euler polynomials and numbers, and derived some combinatorial identities related to these polynomials in [13]. Acikgoz and Duran introduced unified degenerate central Bell polynomials and studied many relations and formulae including the summation formula and derivative properties (see [14]).

For nonzero integers n and k , the Stirling numbers of the first kind S 1 ( n , k ) and the Stirling numbers of the second kind S 2 ( n , k ) , respectively, are given by

where ( x ) 0 = 1 , ( x ) n = x ( x − 1 ) ⋯ ( x − n + 1 ) , ( n ≥ 1 ) are the falling factorial sequences.

By using the degenerate exponential function, the degenerate derangement polynomials are defined by the generating function to be

When x = 0 , d n , λ ( 0 ) = d n , λ are called the degenerate derangement numbers.

By the definition of degenerate derangement polynomials,

and thus, we see that

As the degenerate version of the Stirling numbers of the first and the second kind, the degenerate Stirling numbers of the first kind S 1 , λ ( n , k ) and the degenerate Stirling numbers of the second kind S 2 , λ ( n , k ) are, respectively, introduced by Kim et al. (see [9,16]) as follows:

Let C be the field of complex numbers,

and let

Let P ∗ be the vector space of all linear functionals on P .

For given λ ∈ R − { 0 } , the linear functional ⟨ f ( t ) ∣ ⋅ ⟩ λ on P , called λ -linear functional given by f ( t ) , is defined by

where ( x ) 0 , λ = 1 , ( x ) n , λ = x ( x − λ ) ⋯ ( x − ( n − 1 ) λ ) , ( n ≥ 1 ) . From (5), we have

where δ n , k is the Kronecker’s symbol (see [17]).

For each real number λ and each positive integer k , Kim and Kim defined the differential operator on P in [17] by

and for any f ( t ) = ∑ k = 0 ∞ a k t k k ! ∈ ℱ ,

In addition, they showed that for f ( t ) , g ( t ) ∈ ℱ , and p ( x ) ∈ P ,

The order o ( f ( t ) ) of f ( t ) ∈ ℱ − { 0 } is the smallest integer k for which the coefficient of t k does not vanish. If o ( f ( t ) ) = 0 , then f ( t ) is said to be invertible, and such series has a multiplicative inverse 1 f ( t ) of f ( t ) . If o ( f ( t ) ) = 1 , then f ( t ) is called delta series, and it has a compositional inverse f ¯ ( t ) of f ( t ) with f ¯ ( f ( t ) ) = f ( f ¯ ( t ) ) = t (see [17–19]).

Let f ( t ) be a delta series, and let g ( t ) be an invertible series. Then there exists a unique sequence S n , λ ( x ) ( deg S n , λ ( x ) = n ) of polynomials satisfying the orthogonality conditions:

(see [17]). Here S n , λ ( x ) is called the λ -Sheffer sequence for ( g ( t ) , f ( t ) ) , which is denoted by S n , λ ( x ) ∼ ( g ( t ) , f ( t ) ) λ . The sequence S n , λ ( x ) is the λ -Sheffer sequence for ( g ( t ) , f ( t ) ) if and only if

for all y ∈ C , where f ¯ ( t ) is the compositional inverse of f ( t ) such that f ( f ¯ ( t ) ) = f ¯ ( f ( t ) ) = t (see [17]).

Let S n , λ ( x ) ∼ ( g ( t ) , f ( t ) ) λ and let h ( x ) = ∑ l = 0 n a l S l , λ ( x ) ∈ P . Then by (9), we have

and thus, we know that

The following theorem is proved by Kim and Kim [17] and is very useful tools for researching degenerate versions of special polynomials and numbers.

Let ( x ) n = ∑ k = 0 n c n , k ( x ) k , λ . Since

by Theorem 1.1, we obtain

and thus, we know that

In the similar way, we also know that

Umbral calculus consisted of primarily symbolic techniques for sequence manipulation with little mathematical rigor. In the 1970s, Rota began to build a completely rigid foundation for theories based on relatively modern ideas of linear functions and linear operators (see [19]). Umbral calculus contributed to the generalization of Lagrange inversion formula and has been applied in many fields such as combinatorial counting with linear recurrences and lattice path counting, graph theory using chromatic polynomials, probability theory, link invariant theory, statistics, topology, physics, etc. (see [18,19]). In addition, it is being actively applied in various fields by researchers (see [17–25]).

In the past few years, various different umbral calculus has been studied (see [17–20]). In particular, Kim and Kim defined the degenerate Sheffer sequences, λ -Sheffer sequence, a family of λ -linear functionals, and λ -differential operators in [17].

In this article, we derive some interesting identities related to degenerate derangement polynomials, degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Daehee polynomials, Changhee polynomials, degenerate Bell polynomials, degenerate Lah-Bell polynomials, Mittag-Leffer polynomials, and degenerate Frobenius-Euler polynomials by using λ -umbral calculus.

2 Main results

By the definition of degenerate derangement polynomials, we note that

In addition,

From (2) and (13), we obtain

Let d n , λ ( x ) = ∑ l = 0 n a n , l ( x ) l , λ . Since ( x ) n , λ ∼ ( 1 , t ) λ , by Theorem 1.1, we have

and by (15),

By (17) and (18), we have

Conversely, we may assume that ( x ) n , λ = ∑ l = 0 n b n , l d l , λ ( x ) . Note that

By (19) and (20),

By (17), (19), and (21), we obtain the following theorem.

The degenerate Bernoulli polynomials are defined by the generating function to be

In the special case x = 0 , β n , λ ( 0 ) = β n , λ are called the degenerate Bernoulli numbers. By the definition of degenerate Bernoulli polynomials, the λ -Sheffer sequences of these polynomials are as follows:

Note that

Let d n , λ ( x ) = ∑ l = 0 n a n , l β l , λ ( x ) . By Theorem 1.1 and (22), we have

In addition, by (3), we obtain

Conversely, we assume that β n , λ ( x ) = ∑ l = 0 n b n , l d l , λ ( x ) . Then, by (20), we obtain

By (23), (24), and (25), we obtain the following theorem.

The degenerate Euler polynomials are defined by the generating function to be

When x = 0 , E n , λ = E n , λ ( 0 ) are called the degenerate Euler numbers. By the definition of degenerate Euler polynomials, we see that

Let d n , λ ( x ) = ∑ l = 0 n a n , l E l , λ ( x ) . Since

by Theorem 1.1, (22), and (27), we obtain

Conversely, assume that E n , λ ( x ) = ∑ l = 0 n a n , l d l , λ ( x ) . Then, by (20), we obtain

By (28) and (29), we obtain the following theorem.

The degenerate Daehee polynomials are defined by the generating function to be

In the special case x = 0 , D n , λ = D n , λ ( 0 ) are called the degenerate Daehee numbers. Note that, by (4),