Journal of Function Spaces

Journal of Function Spaces / 2021 / Article
Special Issue

Approximation Methods: Theory and Applications

View this Special Issue

Research Article | Open Access

Volume 2021 |Article ID 6660517 | https://doi.org/10.1155/2021/6660517

Waseem A. Khan, Rifaqat Ali, Khaled Ahmad Hassan Alzobydi, Naeem Ahmed, "A New Family of Degenerate Poly-Genocchi Polynomials with Its Certain Properties", Journal of Function Spaces, vol. 2021, Article ID 6660517, 8 pages, 2021. https://doi.org/10.1155/2021/6660517

A New Family of Degenerate Poly-Genocchi Polynomials with Its Certain Properties

Academic Editor: Tuncer Acar
Received27 Nov 2020
Revised07 Feb 2021
Accepted22 May 2021
Published15 Jun 2021

Abstract

In this paper, we introduce a new type of degenerate Genocchi polynomials and numbers, which are called degenerate poly-Genocchi polynomials and numbers, by using the degenerate polylogarithm function, and we derive several properties of these polynomials systematically. Then, we also consider the degenerate unipoly-Genocchi polynomials attached to an arithmetic function, by using the degenerate polylogarithm function, and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

1. Introduction

In [1, 2], Carlitz initiated a study of degenerate versions of some special polynomials and numbers, namely, the degenerate Bernoulli and Euler polynomials and numbers. Kim et al. [35] have studied the degenerate versions of special numbers and polynomials actively. These ideas provide a powerful tool in order to define special numbers and polynomials of their degenerate versions. The notion of degenerate version forms a special class of polynomials because of their great applicability. Despite the applicability of special functions in classical analysis and statistics, they also arise in communication systems, quantum mechanics, nonlinear wave propagation, electric circuit theory, electromagnetic theory, etc. In particular, Genocchi numbers have been extensively studied in many different contexts in such branches of mathematics as, for instance, elementary number theory, complex analytic number theory, differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), -adic analytic number theory (-adic -functions), and quantum physics (quantum groups). The works of Genocchi numbers and their combinatorial relations have received much attention [611]. In the paper, we focus on a new type of degenerate poly-Genocchi polynomial and numbers.

The aim of this paper is to introduce a degenerate version of the poly-Genocchi polynomials and numbers, the so-called new type of degenerate poly-Genocchi polynomials and numbers, constructing from the degenerate polylogarithm function. We derive some explicit expressions and identities for those numbers and polynomials.

The classical Euler polynomials and the classical Genocchi polynomials are, respectively, defined by the following generating functions (see [1222]):

In the case when , and are, respectively, called the Euler numbers and Genocchi numbers.

The degenerate exponential function [23, 24] is defined by

Note that

In [1, 2], Carlitz introduced the degenerate Bernoulli and degenerate Euler polynomials defined by

In the case when , are called the degenerate Bernoulli numbers and are called the degenerate Euler numbers.

Let be the degenerate falling factorial sequence given by with the assumption .

In [5], Kim et al. considered the degenerate Genocchi polynomials given by

In the case when , are called the degenerate Genocchi numbers.

For , the polylogarithm function is defined by a power series in , which is also a Dirichlet series in (see [25, 26]):

This definition is valid for arbitrary complex order and for all complex arguments with it can be extended to by analytic continuation.

It is noticed that

For , Kim and Kim [3] defined the degenerate version of the logarithm function, denoted by , as follows (see [4]): being the inverse of the degenerate version of the exponential function as has been shown below:

It is noteworthy to mention that

The degenerate polylogarithm function [3] is defined by Kim and Kim to be

It is clear that (see [27, 28])

From (11) and (14), we get

Very recently, Kim and Kim [3] introduced the new type of degenerate version of the Bernoulli polynomials and numbers, by using the degenerate polylogarithm function as follows:

When , are called the new type of degenerate poly-Bernoulli numbers.

The degenerate Stirling numbers of the first kind [24] are defined by

It is clear that calling the Stirling numbers of the first kind given by (see [29, 30])

The degenerate Stirling numbers of the second kind [31] are given by (see [2, 1322, 2532])

Note here that standing for the Stirling numbers of the second kind given by means of the following generating function (see [18, 1238]):

This paper is organized as follows. In Section 1, we recall some necessary stuffs that are needed throughout this paper. These include the degenerate exponential functions, the degenerate Genocchi polynomials, the degenerate Euler polynomials, and the degenerate Stirling numbers of the first and second kinds. In Section 2, we introduce the new type of degenerate poly-Genocchi polynomials by making use of the degenerate polylogarithm function. We express those polynomials in terms of the degenerate Genocchi polynomials and the degenerate Stirling numbers of the first kind and also of the degenerate Euler polynomials and the Stirling numbers of the first kind. We represent the generating function of the degenerate poly-Genocchi numbers by iterated integrals from which we obtain an expression of those numbers in terms of the degenerate Bernoulli numbers of the second kind. In Section 3, we introduce the new type of degenerate unipoly-Genocchi polynomials by making use of the degenerate polylogarithm function. We express those polynomials in terms of the degenerate Genocchi polynomials and the degenerate Stirling numbers of the first kind and also of the degenerate Euler polynomials and the Stirling numbers of the first kind and second kind.

2. New Type of Degenerate Genocchi Numbers and Polynomials

In this section, we define the new type of degenerate Genocchi numbers and polynomials by using the degenerate polylogarithm function which is called the degenerate poly-Genocchi polynomials as follows.

For , we define the new type of degenerate Genocchi numbers, which are called the degenerate poly-Genocchi numbers, as

Note that

Thus, we have (see [6])

Now, we consider the new type of degenerate Genocchi polynomials which are called the degenerate poly-Genocchi polynomials defined by

In the case when , . Using equation (27), we see

Therefore, by equation (28), we obtain the following theorem.

Theorem 1. Let be a nonnegative integer. Then, From (27), we note that

Therefore, by equations (30) and (31), we get the following theorem.

Theorem 2. Let be a nonnegative integer. Then, Using equations (27) and (6), we see

By equations (33) and (34), we obtain the following theorem.

Theorem 3. Let be a nonnegative integer. Then, From (27), we have For in (36) and using [3] (Eq. (27)), we get

Therefore, by equation (37), we get the following theorem.

Theorem 4. Let be a nonnegative integer. Then, In general, by equation (37), we see

By equation (39), we obtain the following theorem.

Theorem 5. Let and , we have From (27), we observe that On the other hand,

Therefore, by equations (41) and (42), we get the following theorem.

Theorem 6. Let and . Then, From equations (27) and (14), we see

By comparing the coefficients on both sides of (44), we obtain the following theorem.

Theorem 7. For , we have where is Kronecker’s symbol.
Note that

3. Degenerate Unipoly-Genocchi Numbers and Polynomials

Let be any arithmetic function which is a real or complex valued function defined on the set of positive integers . Kim and Kim [29] defined the unipoly function attached to polynomials by

Moreover (see [25]), is the ordinary polylogarithm function.

In [8], Lee and Kim defined the degenerate unipoly function attached to polynomials as follows:

It is worthy to note that is the degenerate polylogarithm function.

Now, we define the degenerate unipoly-Genocchi polynomials attached to polynomials by

In the case when , are called the degenerate unipoly-Genocchi numbers attached to .

From (51), we see

Thus, by (52), we have

From (51), we have

Therefore, by equation (54), we get the following theorem.

Theorem 8. Let be a nonnegative integer. Then, Using equations (49) and (51), we see

By, equations (51) and (56), we obtain the following theorem.

Theorem 9. Let be a nonnegative integer. Then, From (6), (49), and (51), we get

Therefore, by (58), we obtain the following theorem.

Theorem 10. Let be a nonnegative integer and . Then, From (51), we have

By equation (60), we get the following theorem.

Theorem 11. Let be a nonnegative integer and . Then,

4. Conclusion

In this article, we introduced degenerate poly-Genocchi polynomials and numbers by using the degenerate polylogarithm function and derived several properties on the degenerate poly-Genocchi numbers. We represented the generating function of the degenerate poly-Genocchi numbers by iterated integrals in Theorems 46 and explicit degenerate poly-Genocchi polynomials in terms of the Euler polynomials and degenerate Stirling numbers of the second kind in Theorem 3. We also represented those numbers in terms of the degenerate Stirling numbers of the second kind in Theorem 7. In the last section, we defined the degenerate unipoly-Genocchi polynomials by using degenerate polylogarithm function and obtained the identity degenerate unipoly-Genocchi polynomials in terms of the degenerate Genocchi polynomials and degenerate Stirling numbers of the second kind in Theorem 8, the degenerate Euler polynomials and the degenerate Stirling numbers of the second kind in Theorem 9, the degenerate Bernoulli and degenerate Stirling numbers of the second kind in Theorem 10, and the degenerate unipoly-Genocchi numbers and Stirling numbers of the second kind in Theorem 11. It is important that the study of the degenerate version is widely applied not only to numerical theory and combinatorial theory but also to symmetric identity, differential equations, and probability theory. In particular, many symmetric identities have been studied for degenerate versions of many special polynomials [1, 3, 12, 23, 2932]. Genocchi numbers have been also extensively studied in many different branches of mathematics. The works of Genocchi numbers and their combinatorial relations have received much attention [69]. With this in mind, as a future project, we would like to continue to study degenerate versions of certain special polynomials and numbers and their applications to physics, economics, and engineering as well as mathematics.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

Acknowledgments

The authors would like to express the gratitude to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for providing funding research group under the research grant number R G P.1/162/42.

References

  1. L. Carlitz, “Degenerate Stirling, Bernoulli and Eulerian numbers,” Utilitas Mathematica, vol. 15, pp. 51–88, 1979. View at: Google Scholar
  2. L. Carlitz, “A degenerate Staudt-Clausen theorem,” Archiv der Mathematik, vol. 7, no. 1, pp. 28–33, 1956. View at: Publisher Site | Google Scholar
  3. D. S. Kim and T. Kim, “A note on a new type of degenerate Bernoulli numbers,” Russian Journal of Mathematical Physics, vol. 27, no. 2, pp. 227–235, 2020. View at: Publisher Site | Google Scholar
  4. T. Kim, D. S. Kim, J. Kwon, and H. Lee, “Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials,” Advances in Difference Equations, vol. 2020, no. 1, 2020. View at: Publisher Site | Google Scholar
  5. T. Kim, D. S. Kim, J. Kwon, and H. Y. Kim, “A note on degenerate Genocchi and poly-Genocchi numbers and polynomials,” Journal of Inequalities and Applications, vol. 2020, no. 1, 2020. View at: Publisher Site | Google Scholar
  6. T. Kim, D. S. Kim, D. V. Dolgy, and J. Kwon, “Some identities on degenerate Genocchi and Euler numbers,” Informatica, vol. 31, no. 4, pp. 42–51, 2020. View at: Google Scholar
  7. H. K. Kim and L. C. Jang, “A note on degenerate poly-Genocchi numbers and polynomials,” Advances in Difference Equations, vol. 2020, no. 1, 2020. View at: Publisher Site | Google Scholar
  8. D. S. Lee and H.-Y. Kim, “On the new type of degenerate poly-Genocchi numbers and polynomials,” Advances in Difference Equations, vol. 2020, 2020. View at: Publisher Site | Google Scholar
  9. S. K. Sharma, W. A. Khan, S. Araci, and S. S. Ahmed, “New type of degenerate Daehee polynomials of the second kind,” Advances in Difference Equations, vol. 2020, no. 1, 2020. View at: Publisher Site | Google Scholar
  10. S. K. Sharma, “A note on degenerate poly-Genocchi polynomials,” International Journal of Advanced and Applied Sciences, vol. 7, no. 5, pp. 1–5, 2020. View at: Publisher Site | Google Scholar
  11. Y. Simsek, I. N. I. Cangul, V. Kurt, and D. S. Kim, “q-Genocchi numbers and polynomials associated with q-Genocchi-type l-functions,” Advances in Difference Equations, vol. 2008, Article ID 815750, 13 pages, 2008. View at: Publisher Site | Google Scholar
  12. S. Araci, M. Acikgoz, and A. Esi, “A note on the -Dedekind-type Daehee-Changhee sums with weight arising from modified -Genocchi polynomials with weight .,” Journal of Assam Academy of Mathematics, vol. 5, pp. 47–54, 2012. View at: Google Scholar
  13. D. Duran, M. Acikgoz, A. Esi, and S. Araci, “Some new symmetric identities involving -Genocchi polynomials under ,” Journal of Mathematical Analysis, vol. 6, no. 4, pp. 22–31, 2015. View at: Google Scholar
  14. D. Duran, M. Acikgoz, and S. Araci, “On -Genocchi polynomials with weighted and under symmetric group of degree .,” Advances and Applications in Mathematical Sciences, vol. 15, no. 7, pp. 205–214, 2016. View at: Google Scholar
  15. D. Duran, M. Acikgoz, and S. Araci, “Symmetric identities involving weighted -Genocchi polynomials under ,” Proceedings of the Jangjeon Mathematical Society, vol. 18, no. 4, pp. 455–465, 2015. View at: Google Scholar
  16. D. Duran, M. Acikgoz, A. Esi, and S. Araci, “Some new symmetric identities involving q Genocchi polynomials under S4,” Journal of Mathematical Analysis, vol. 6, no. 5, pp. 22–31, 2015. View at: Google Scholar
  17. U. Duran, M. Acikgoz, and S. Araci, “Construction of the type 2 poly-Frobenius-Genocchi polynomials with their certain applications,” Advances in Difference Equations, vol. 2020, no. 1, 2020. View at: Publisher Site | Google Scholar
  18. H. Haroon and W. A. Khan, “Degenerate Bernoulli numbers and polynomials associated with degenerate Hermite polynomials,” Korean Mathematical Society, vol. 33, no. 2, pp. 651–669, 2018. View at: Google Scholar
  19. W. A. Khan and H. Haroon, “Some symmetric identities for the generalized Bernoulli, Euler and Genocchi polynomials associated with Hermite polynomials,” Springer Plus, vol. 5, article 1920, 2016. View at: Publisher Site | Google Scholar
  20. W. A. Khan, “A new class of degenerate Frobenius-Euler Hermite polynomials,” Adv. Stud. Contemp. Math, vol. 30, no. 4, pp. 567–576, 2018. View at: Google Scholar
  21. W. A. Khan, I. A. Khan, and M. Ali, “Degenerate Hermite poly-Bernoulli numbers and polynomials with -parameter,” Studia Universitatis Babes-Bolyai, Mathematica, vol. 65, no. 1, pp. 3–15, 2020. View at: Publisher Site | Google Scholar
  22. W. A. Khan, I. A. Khan, and M. Ali, “A note on q-analogue of Hermite-poly-Bernoulli numbers and polynomials,” Mathematica Morvica, vol. 23, no. 2, pp. 1–16, 2019. View at: Publisher Site | Google Scholar
  23. T. Kim and D. S. Kim, “Degenerate polyexponential functions and degenerate Bell polynomials,” Journal of Mathematical Analysis and Applications, vol. 487, no. 2, p. 124017, 2020. View at: Publisher Site | Google Scholar
  24. T. Kim and D. S. Kim, “Degenerate Laplace transform and degenerate gamma function,” Russian Journal of Mathematical Physics, vol. 24, no. 2, pp. 241–248, 2017. View at: Publisher Site | Google Scholar
  25. M. Kaneko, “Poly-Bernoulli numbers,” Journal de Théorie des Nombres de Bordeaux, vol. 9, no. 1, pp. 221–228, 1997. View at: Publisher Site | Google Scholar
  26. W. A. Khan, “A note on degenerate Hermite-poly-Bernoulli numbers and polynomials,” J. Classical Anal, vol. 8, no. 1, pp. 65–76, 2016. View at: Google Scholar
  27. W. A. Khan, M. Acikgoz, and U. Duran, “Note on the type 2 degenerate multi-poly-Euler polynomials,” Symmetry, vol. 12, no. 10, p. 1691, 2020. View at: Publisher Site | Google Scholar
  28. W. A. Khan and M. Ahmad, “Partially degenerate poly-Bernoulli polynomials,” Adv. Stud. Contemp. Math, vol. 28, no. 3, pp. 487–496, 2018. View at: Google Scholar
  29. D. S. Kim and T. Kim, “A note on polyexponential and unipoly functions,” Russian Journal of Mathematical Physics, vol. 26, no. 1, pp. 40–49, 2019. View at: Publisher Site | Google Scholar
  30. D. S. Kim, T. Kim, and J. Seo, “A note on Changhee polynomials and numbers,” Advanced Studies in Theoretical Physics, vol. 7, no. 1, pp. 993–1003, 2013. View at: Publisher Site | Google Scholar
  31. T. Kim, “A note on degenerate Stirling polynomials of the second kind,” Proceedings of the Jangjeon Mathematical Society, vol. 20, no. 3, pp. 319–331, 2017. View at: Google Scholar
  32. T. Kim, L.-C. Jang, D. S. Kim, and H.-Y. Kim, “Some identities on type 2 degenerate Bernoulli polynomials of the second kind,” Symmetry, vol. 12, no. 4, p. 510, 2020. View at: Publisher Site | Google Scholar
  33. T. Kim, D. S. Kim, H. Y. Kim, and J. Kwon, “Some identities of degenerate Bell polynomials,” Mathematics, vol. 8, no. 1, p. 40, 2020. View at: Publisher Site | Google Scholar
  34. T. Kim, D. S. Kim, H. Y. Kim, and J. Kwon, “Degenerate Stirling polynomials of the second kind and some applications,” Symmetry, vol. 11, no. 8, article 1046, 2019. View at: Publisher Site | Google Scholar
  35. T. Kim, W. A. Khan, S. K. Sharma, and M. Ghayasuddin, “A note on parametric kinds of the degenerate poly-Bernoulli and poly-Genocchi polynomials,” Symmetry, vol. 12, no. 4, article 614, 2020. View at: Publisher Site | Google Scholar
  36. I. Kucukoglu, B. Simsek, and Y. Simsek, “An approach to negative hypergeometric distribution by generating function for special numbers and polynomials,” Turkish Journal of Mathematics, vol. 43, no. 5, pp. 2337–2353, 2019. View at: Publisher Site | Google Scholar
  37. N. Raza, U. Zainab, S. Araci, and A. Esi, “Identities involving 3-variable Hermite polynomials arising from umbral method,” Advances in Difference Equations, vol. 2020, 2020. View at: Publisher Site | Google Scholar
  38. S. K. Sharma, W. A. Khan, and C. S. Ryoo, “A parametric kind of the degenerate Fubini numbers and polynomials,” Mathematics, vol. 8, no. 3, p. 405, 2020. View at: Publisher Site | Google Scholar

Copyright © 2021 Waseem A. Khan et al. 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.

Related articles

No related content is available yet for this article.