A note on partially degenerate Legendre–Genocchi polynomials

N. U. Khan, T. Kim and T. Usman
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 2, Pages 76-90
DOI: 10.7546/nntdm.2019.25.2.76-90
Full paper (PDF, 195 Kb)

Details

Authors and affiliations

N. U. Khan
Department of Applied Mathematics, Faculty of Engineering and Technology
Aligarh Muslim University, Aligarh-202002, India

T. Kim
Department of Mathematics, College of Natural Science
Kwangwoon University, Seoul 139-704, S. Korea

T. Usman
Department of Applied Mathematics, Faculty of Engineering and Technology
Aligarh Muslim University, Aligarh-202002, India

Abstract

In the past years, many researchers have worked on degenerate polynomials and a variety of its extentions and variants can be found in literature. Following up, in this article, we firstly introduce the partially degenerate Legendre–Genocchi polynomials, and further define a new generalization of degenerate Legendre–Genocchi polynomials. From our generalization, we establish some implicit summation formulae and symmetry identities by the generating function of partially degenerate Legendre–Genocchi polynomials. Eventually, it can be found that some recently demonstrated summations and identities stated in the article, are special cases of our results.

Keywords

• Legendre polynomials
• Partially degenerate Genocchi polynomials
• Partially degen-erate Legendre–Genocchi polynomials
• Summation formula
• Symmetric identities

2010 Mathematics Subject Classification

• 33B15
• 33C10
• 33C15

References

1. Dattoli, G., Lorenzutta, S., & Cesarano, C. (1999). Finite sums and generalized forms of Bernoulli polynomials,Rend. Mat. Appl., 19, 385–391.
2. Dattoli, G., Ricci, P. E., & Casarano, C. (2001). A note on Legendre polynomials, Int. J.Nonlinear Sci. Numer. Simul., 2 (4), 65–370.
3. Jang, L.C., Kwon, H. I., Lee, J. G., & Ryoo, C. S. (2015). On the generalized partially degenerate Genocchi polynomials, Global J. Pure Appl. Math., 11, 4789–4799.
4. Khan, N. U., & Usman, T. (2017). A new class of Laguerre poly-Bernoulli numbers and polynomials,Adv. Stud. Contemp. Math., 27 (2), 229–241.
5. Khan, N. U., & Usman, T. (2016). A new class of Laguerre-based Generalized Apostol Polynomials, Fasc. Math., 57, 67–89.
6. Khan, N. U., & Usman, T. (2016). A new class of Laguerre-based Poly-Euler and Multi Poly-Euler Polynomials, J. Ana. Num.Theor., 4 (2), 113–120.88
7. Khan, N. U., Usman, T., & Aman, M. (2017). Generating functions for Legendre-based poly-Bernoulli numbers and polynomials, Honam Mathematical J., 39, (2), 217–231.
8. Khan, N. U., Usman, T., & Aman, M. (2017). Certain Generating funtion of generalized Apostol type Legendre-based polynomials,Note Mat., 37 (2), 21–43.
9. Khan, N. U., Usman, T., & Choi, J. (2017). Certain generating function of Hermite–Bernoulli–Laguerre polynomials, Far East J. Math. Sci., 101 (4), 893–908.
10. Khan, N.U., Usman, T., & Choi, J. (2017). A New generalization of Apostol type Laguerre–Genocchi polynomials, C. R. Acad. Sci. Paris, Ser. I, 355, 607–617.
11. Khan, N. U., Usman, T., & Choi, J. (2018). A new class of generalized polynomials, Turkish J. Math., 42, 1366–1379.
12. Khan, N. U., Usman, T., & Choi, J. (2019). A new class of generalized polynomials associated with Laguerre and Bernoulli polynomials, Turkish J. Math., 43, 486–497.
13. Khan, N. U., Usman, T., & Khan, W. A. A new class of Laguerre-based generalized Hermite-Euler polynomials and its properties, Kragujevac J. Math.(In press)
14. Kim, D. S., & Kim, T. (2013). Daehee numbers and polynomials, Appl. Math. Sci., 7, 5969–5976.
15. Kim, D. S., & Kim, T. (2015). Some identities of degenerate special polynomials, OpenMath., 13, 380–389.
16. Kim, D. S., Kim, T., Lee, S. H., & Seo, J. J. (2013). A note on the lambda-Daehee polynomials,Int. J. Math. Anal., 7, 3069–3080.
17. Kim, D. S., Lee, S. H., Mansour, T., & Seo, J. J. (2014). A note onq-Daehee polynomials and numbers, Adv. Stud. Contemp. Math., 24, 155–160.
18. Kim, T., & Seo J. J (2015). A note on partially degenerate Bernoulli numbers and polynomials, J. Math. Anal., 6, 1–6.
19. Lim, D. (2015). Degenerate, partially degenerate and totally degenerate Daehee numbers and polynomials, Adv. Difference Equ., 2015:287, 14 pages, DOI: 10.1186/s13662-015-0624-2.
20. Lim, D. (2016). Some identities of Degenerate Genocchi polynomials,Bull. Korean Math.Soc., 53, 569–579.
21. Park, J. W., & Kwon, J. (2015). A note on the degenerate High Order Daehee polynomials, Global J. Applied Mathematical Sciences, 9, 4635–4642.
22. Pathan, M. A., & Khan, W. A. (2015). Some implicit summation formulas and symmetric identities for the generalized Hermite–Bernoulli polynomials, Mediterr. J. Math., 12, 679–695.
23. Qi, F., Dolgy, D. V. Kim, T., & Ryoo, C. S. (2015). On the partially degenerate Bernoulli polynomials of the first kind, Global J. Pure Appl. Math., 11, 2407–2412.
24. Rainville, E. D. (1960). Special functions, The Macmillan Company, New York.
25. Ryoo, C. S., Kim, T., Choi, J., & Lee, B. (2011). On the generalized q-Genocchi numbers and polynomials of higher order, Adv. Difference Equ., Volume 2011, Article ID 424809, 8 pages.
26. Srivastava H. M., & Manocha, H. L. (1984). A Treatise on Generating Functions, Ellis Horwood Limited, New York.
27. Tuenter, H. J. H. (2011). A symmetry power sum of polynomials and Bernoulli numbers, Amer. Math. Monthly, 108, 258–26