Identities of Degenerate Poly-Changhee Polynomials Arising from <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 9.63935 12.533" width="9.63935pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-233" d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"/></g></svg>-</span>Sheffer Sequences

Yun, Sang Jo; Park, Jin-Woo

doi:https://doi.org/10.1155/2022/1482534

Journal of Mathematics

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1482534 | https://doi.org/10.1155/2022/1482534

Identities of Degenerate Poly-Changhee Polynomials Arising from -Sheffer Sequences

Sang Jo Yun¹and Jin-Woo Park²

Academic Editor: Barbara Martinucci

Received18 Jan 2022

Accepted12 Jun 2022

Published14 Jul 2022

Abstract

In the 1970s, Gian-Carlo Rota constructed the umbral calculus for investigating the properties of special functions, and by Kim-Kim, umbral calculus is generalized called -umbral calculus. In this paper, we find some important relationships between degenerate Changhee polynomials and some important special polynomials by expressing the Changhee polynomial as a linear combination of some special polynomials. In addition, we derive some interesting identities related to degenerate poly-Changhee polynomials and some important special functions by using -umbral calculus.

1. Introduction

Let be an odd prime number, and let , and denote the ring of -adic rational integer, the field of -adic rational numbers, and the complication of the algebraic closure of , respectively. The -adic norm is normally defined by . As is known, the -adic invariant integral on is defined by Kim to bewhere is a continuous functions on (see [1]).

By using equation (1), the generation function of the Changhee polynomials is defined to be(see [2–4]). When , are called the Changhee numbers.

As one of the special polynomials, the Changhee polynomials and numbers are related closely some important special polynomials in the combinatorics, applied mathematics, mathematical physics, engineering, or economics. In [3], the authors found symmetric identities for that polynomials by using the fermionic -adic integral on , and some properties and zeros of these polynomials on the locally constant space were studied in [5]. In [4, 6], the authors generalized these polynomials to ring of -adic integer . Kim-Kim defined different Changhee polynomials named type 2 Changhee polynomials in [7] and found some relationships between those polynomials and Changhee polynomials, the Stirling numbers of the first and second kind, Euler polynomials and type 2 Euler polynomials. Moreover, in [2, 4, 8, 9], the authors studied the properties of degenerate version of Changhee polynomials, -Changhee polynomials, and Appell-type -Changhee polynomials, respectively.

For , the Stirling numbers of the first kind and the Stirling numbers of the second kind , respectively, are given bywhere and is the falling factorial sequences.

For each positive integer , it is well known (see [10]) that

For any nonzero real number , the degenerate exponential function is defined to be(see [7, 11, 12]) and let be the compositional inverse function of satisfying . Then, we have(see [12–16] where are -falling factorial of .

By using the degenerate exponential function, Carlitz defined the degenerate Bernoulli polynomials in [11, 17], and in recent years, studies of degenerate versions of some special polynomials have been increased by many researchers with their interests not only in combinatorial properties but also its applications (see [8, 9, 12–16, 18–21]).

As a degenerate version of the Stirling numbers of the first and second kind in equation (3), the degenerate Stirling numbers of the first kind and the degenerate Stirling numbers of the second kind are, respectively, introduced by Kim-Kim (see [8, 14–16, 19]) as follows:

The degenerate polyexponential functions are defined by Kim-Kim as(see [12, 18]). In particular, if , then, .

As a generalization of the degenerate exponential function, the degenerate polyexponential function is used by many researchers. In [15], Kim-Kim defined degenerate poly-Bell polynomials and provided explicit representations and combinatorial identities for these polynomials. Kurt defined the degenerate poly-Euler numbers and polynomials arising from the degenerate polyexponential functions and derived explicit relations for these numbers and polynomials (see [16]). In [14], Kim-Jang defined the degenerate poly-Genocchi polynomials and found some interesting identities of those polynomials.

Let be the field of complex numbers,and let

Let be the vector space of all linear functionals on .

Then, each gives rise to the linear functional on , called -linear functional given by , which is defined byand by linear extension (see [19]). From (11), we havewhere is Kronecker’s symbol.

For each and each , Kim-Kim defined the differential operator on in [19] byand 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 [22]).

Let be a delta series and let be an invertible series. Then, there exists a unique sequence of polynomials satisfying the orthogonality conditions:(see [19]). Here, is called the -Sheffer sequence for , which is denoted by . The sequence is the -Sheffer sequence for if and only if(see [19]) for all , where is the compositional inverse of such that .

Let and let . Then, by (16), we haveand thus, we know that

The following theorem is very useful tools in the -umbral calculus which is found by Kim and Kim (see [19]).

Theorem 1. Let and let . Ifthen,

Let . Since

By equation (22), we getand thus, we know that

In the similar way, we also know that

In this paper, we find some interesting identities about degenerate poly-Changhee polynomials, higher-order degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Bell polynomials, degenerate Lah-Bell polynomials, degenerate Frobenius–Euler polynomials, and Mittag-Leffer polynomials by using -umbral calculus.

2. Degenerate Poly-Changhee Polynomials

The degenerate poly-Changhee polynomials are defined by the generating function:

In the special case , are called the degenerate poly-Changhee numbers.

By equation (17), we know that the -Sheffer sequences of degenerate poly-Changhee polynomials is

Theorem 2. For each non-negative integer , we have

Proof. Let . SinceBy Theorem 1, we getand thus, our theorem is proved.
Let . By (15) and (16),and soBy equations (24) and (27), we know thatand soNote thatwhere , , and are type 2 Bernoulli numbers which are defined by the generating function to be(see [7]) By equations (13) and (35), we know thatBy equation (36), we obtain the following theorem.

Theorem 3. For each , we have

Note that by Theorem 1 and equation (27), we get

Thus, by (39), we obtain the following theorem.

Theorem 4. For each non-negative integer , we haveBy using equation (17), the higher-order degenerate Bernoulli polynomials are defined by the generating function:(see [24, 25]). In the special case , are called the higher-order degenerate Bernoulli numbers.
By the definition of higher-order degenerate Bernoulli polynomials (41), we know thatIf we put , then, by the Theorem 1 and (42), we havewhere the higher-order degenerate Bernoulli numbers of the second kind are given.(see [26]). By equation (43), we obtain the following theorem.

Theorem 5. For each non-negative integer , we haveThe degenerate Euler polynomials are defined by the generating function:(see [24]). When , are called the degenerate Euler numbers.
By the definition of the degenerate Euler polynomials, we know thatLet . By Theorem 1 and (47), we get

Therefore, by equation (48), we obtain the following theorem.

Theorem 6. For each non-negative integer , we have

The fully degenerate Bell polynomials are defined as(see [12, 20]). By the definition of fully degenerate Bell polynomials, we note that

Note that

Let . By Theorem 1, equations (51), and (52), we getand thus, by equation (53), we obtain the following theorem.

Theorem 7. For each non-negative integer , we have

The degenerate Lah-Bell polynomials are defined by the generating function:(see [27]). In the special case , are called the Lah-Bell numbers.

By equation (55), we know the -Sheffer sequence of degenerate Lah-Bell polynomials is

Note that

Let . By Theorem 1, (56) and (57), we get

By equation (58), we obtain the following theorem.

Theorem 8. For each non-negative integer , we have

The degenerate Daehee polynomials are defined by the generating function:

By equation (60), we know that

Let . Then, by Theorem 1 and (61), we get

By equation (62), we obtain the following theorem.

Theorem 9. For each non-negative integer , we have

The degenerate Frobenius–Euler polynomials of order defined by the generating function to be(see [1, 19, 30]). In the special case , are called the degenerate Frobenius–Euler numbers of order .

By equation (64), we know the -Sheffer sequence of degenerate Frobenius–Euler polynomials is

Let . Then, by Theorem 1, equations (27) and (65), we haveand so by (66), we obtain the following theorem.

Theorem 10. For each non-negative integer , we have

The Mittag-Leffler polynomials are defined by the generating function to be

By the definition of Mittag-Leffler polynomials, we see that

Let . Then, by Theorem 1, equations (27) and (69), we haveand so by (70), we obtain the following theorem.

Theorem 11. For each non-negative integer , we have

3. Conclusion

In 1970s Gian-Carlo Rota constructed the important tools, called umbral calculus, for investigating the properties of special functions which was based on a linear functional, a linear differential operator, and adjoint (see [22]). In 2021, Kim-Kim generalized the umbral calculus to -umbral calculus, and -umbral calculus has been used as an important tool to study special functions by many researchers.

In this paper, we find some important relationships between degenerate Changhee polynomials and some important special polynomials by expressing the Changhee polynomial as a linear combination of higher-order degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Bell polynomials, degenerate Lah-Bell polynomials, degenerate Frobenius–Euler polynomials or Mittag-Leffer polynomials.

If the methods used in this paper are applied to expandable polynomials using polyexponential functions, many new and interesting identities can be found.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

J. W. P. conceived of the framework and structured the whole paper. S. J. Y. and J. W. P. wrote the paper. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) NRF-2020R1F1A1A01075658.

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Copyright © 2022 Sang Jo Yun and Jin-Woo Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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