Abstract
In this paper, authors found a new and interesting identity between Changhee polynomials and some degenerate polynomials such as degenerate Bernoulli polynomials of the first and second kind, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate Bell polynomials, degenerate Lah–Bell polynomials, and degenerate Frobenius–Euler polynomials and Mittag–Leffer polynomials by using -Sheffer sequences and -differential operators to find the coefficient polynomial when expressing the -th Changhee polynomials as a linear combination of those degenerate polynomials. In addition, authors derive the inversion formulas of these identities.
1. Introduction
Umbral calculus from 1850 to 1970 consisted primarily of symbolic techniques for sequence manipulation, and its mathematical rigor left little room for demands. In the 1970s, Gian-Carlo Rota began building a completely rigid foundation for theories based on relatively modern ideas of linear functions, linear operators, and adjacency functions (see [1–4]). 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, and physics (see [3]). It is being actively applied in various fields by researchers (see [1–16]).
In the past few years, many distinct umbral calculus types have begun to be studied (see [2, 4, 6, 10]). In particular, Kim–Kim defined the degenerate Sheffer sequences, -Sheffer sequence, a family of -linear functionals, and -differential operators as follows (see [2]).
Let be the field of complex numbers:and let
Let be the vector space of all linear functionals on .
Then, each real number gives rise to the linear functional on , called -linear functional given by , which is defined by (see [2])and by linear extension where , , . From (3), we havewhere is Kronecker’s symbol (see [2]).
For each real number and each positive integer , Kim and Kim defined the differential operator on in [2] as follows:and for any ,
In addition, they showed that for and ,
The order of is the smallest integer for which the coefficient of does not vanish. If , then is called invertible and such series has a multiplicative inverse of . If , then is called delta series and it has a compositional inverse of with (see [1, 2, 12, 16]).
Let be a delta series and let be an invertible series. Then, there exists a unique sequence of polynomials satisfying the orthogonality conditions (see [2])
Here, is called the -Sheffer sequence for , which is denoted by . The sequence is the -Sheffer sequence for if and only iffor all , where is the compositional inverse of such that (see [1, 2, 12, 16]).
Let and let . Then, by (8), we haveand thus we know that
The following theorem is proved by Kim and Kim [2] and is a very useful tool for researching degenerate versions of special polynomials and numbers.
Theorem 1. Let , . Then, we havewhere
For , the Stirling numbers of the first kind and Stirling numbers of the second kind , respectively, are given by the following (see [11, 12, 17–20]):
For each positive integer , it is well known that (see [11, 12, 17–20])
For any nonzero real number , the degenerate exponential function is defined by (see [1, 21–27])
Note that
A study of degenerate versions of some special numbers and polynomials was initiated by Carlitz who found interesting relationships connected with important numbers in combinatorics, Bernoulli polynomials, and Eulerian polynomials (see [28]). In the past decades, the study of degenerate versions of various special polynomials or numbers has been studied by many researchers (see [1, 2, 21–27, 29–32]).
By using (16), the higher-order degenerate Bernoulli polynomials are defined as follows (see [1, 10, 12, 30, 33, 34]):
When , are called the higher-order degenerate Bernoulli numbers. In addition, when , we denote .
On the other hand, Kim and Kim defined called the degenerate logarithm function as the compositional inverse function of satisfying . Then, we have (see [1, 10, 22, 24, 32])
By using (19), the degenerate Bernoulli polynomials of the second kind are defined by the generating function to be (see [16])
In the special case , are called the Bernoulli numbers of the second kind.
As degenerate version of the Stirling numbers of the first and second kind in (14), the degenerate Stirling numbers of the first kind and the degenerate Stirling numbers of the second kind are, respectively, introduced by Kim–Kim (see [1, 2, 21, 22, 24, 26–30, 35, 36]) as follows:
Let . Sinceby Theorem 1, we obtainand thus, we know thatwhere , is the falling factorial sequences. In the similar way, we also know that
The aim of this paper is to find some new and interesting identities related to the Changhee polynomials and some special polynomials by using -Sheffer sequences and -differential operators. In more detail, we find the coefficients which are also polynomials or numbers when the -th Changhee polynomial is expressed as a linear combination of some degenerate special polynomials by using the -Sheffer sequences and -differential operators (see Theorems 2–10), and by using the -Sheffer sequences and the linear combinations of those polynomials (see Theorems 5–8 and 10), and derive the inversion formulas of these identities.
2. Changhee Polynomials Arising from -Sheffer Sequences
In this section, we find some relationships between the Changhee polynomials and some special polynomials arising from -Sheffer sequences.
The Changhee polynomials are given by
By (24) and (26), we obtainand, by (27), we have
By (28), we compute the first few Changhee polynomials as follows:
In addition, graphs for some Changhee polynomials are shown in Figure 1.
Note thatwhere , , .
Theorem 2. For each nonnegative integer , we haveAs the inversion formula of (31), we have
Proof. Let . Sinceby Theorem 1 and (30), we haveConversely, we assume that . By (6) and (17), we obtain
Sinceand thus we know that
Theorem 3. For each , we haveAs the inversion formula of (38), we have
Proof. Let . Sinceby Theorem 1, we obtainConversely, let . By (8), (12), (15) and (37), we obtainOn the other hand, by Theorem 1, we haveand hence our proofs are completed.
The degenerate Euler polynomials are defined by the generating function to be (see [28])When , are called the degenerate Euler numbers.
By (44), we know thatand so
Theorem 4. For each , we haveAs the inversion formula of (47), we have
Proof. Let . Sinceby Theorem 1, we obtainConversely, we assume that . Then,On the other hand, by (11) and (46), we obtainand so our proofs are completed.
By (46), we compute the first few degenerate Euler polynomials as follows:Although , it is difficult to find , through Figure 2. But by Theorem 4, we see that , .
The degenerate Daehee polynomials are defined by the generating function to beIn the special case of , are called the degenerate Daehee numbers (see [32, 37]).
Note thatand by (24), we have
Theorem 5. For each nonnegative integer , we haveAs the inversion formula of (57), we have
Proof. Let . Sinceby Theorem 1, we haveOn the other hand, by (11) and (28), we obtainConversely, we assume that . Then, by Theorem 1 and (6), we obtainOn the other hand, by (11) and (56), we obtainThe degenerate Bell polynomials are defined by the generating function to be (see [1, 23])Note thatand thusIn addition, we know that
Theorem 6. For each nonnegative integer , we haveAs the inversion formula of (69), we have
Proof. Let . Sinceby Theorem 1and (67), we obtainConversely, we assume that . By (68), we obtainOn the other hand, by (11) and (66), we haveand thus our proofs are completed.
The unsigned Lah number counts the number of ways a set of elements can be partitioned into nonempty linearly ordered subsets and has the explicit formula (see [1, 20, 23, 38, 39])By (75), we can derive the generating function of to be (see [1, 20, 23, 38, 39])Recently, Kim–Kim introduced the degenerate Lah–Bell polynomials as follows (see [1, 20]):In the special case of , are called Lah–Bell numbers. Note that -th Lah–Bell number is the number of ways a set of elements can be partitioned into nonempty linearly ordered subsets. By (77), we can derive the following:and thus, we obtain
Theorem 7. For each nonnegative integer , we haveAs the inversion formula of (80), we have
Proof. Let . Sinceby (11) and (28), we obtainConversely, we assume that . Then,On the other hand, by (11) and (79), we obtainand so our proofs are completed.
By the definition of the degenerate Bernoulli polynomials of the second kind, we note thatand thus, we obtain
Theorem 8. For each , we haveAs the inversion formula of (88), we have
Proof. Let . Sinceby Theorem 1, we obtainConversely, we assume that . Then,On the other hand, by (11) and (87), we obtainand thus our proofs are completed.
The Mittag–Leffler polynomials are defined by the generating function to be see ([14, 34, 36])
Theorem 9. For each nonnegative integer , we haveAs the inversion formula of (95), we have
Proof. Let . Sinceby Theorem 1 and (30), we obtainConversely, we assume that . Then, by (76), we obtainand so our proofs are completed.
The degenerate Frobenius–Euler polynomials of order are defined by the generating function to be (see [1, 2, 31, 40])In the special case , are called the degenerate Frobenius–Euler numbers of order . By the definition of the Frobenius–Euler polynomials of order , we note thatand thus we haveFurthermore, we see that
Theorem 10. For each nonnegative integer , we haveAs the inversion formula of (104), we have
Proof. Let . Sinceby Theorem 1, we obtainConversely, we assume that . Then, by Theorem 1 and (103), we haveIn addition, by (11) and (102), we obtain
3. Conclusion
In this paper, we studied the Changhee polynomials related to the lambda falling factorial (Theorem 2), the degenerate Bernoulli polynomials (Theorem 3), the degenerate Euler polynomials (Theorem 4), the degenerate Daehee polynomials (Theorem 5), the degenerate Bell polynomials (Theorem 6), the degenerate Lah–Bell polynomials (Theorem 7), the degenerate Bernoulli polynomials of the second kind (Theorem 8), the Mittag–Leffler polynomials (Theorem 9), and the degenerate Frobenius–Euler polynomials (Theorem 10) by finding the coefficients which are also polynomials or numbers when the -th Changhee polynomial is expressed as a linear combination of those degenerate special polynomials by using the -Sheffer sequences and -differential operators. In addition, we derive the inversion formulas of these identities.
Umbral calculus 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, and physics. It is being actively applied in various fields by researchers. As one of our future projects, we would like to continue to study degenerate versions of certain special polynomials and numbers by using -Umbral calculus. [41].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
TK and JWP conceived the framework and structured the whole manuscript. TK and JWP wrote the paper. BMK and TAR checked the results of the manuscript. All authors read and approved the final paper.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) NRF-2020R1F1A1A01075658.