Construction of Degenerate $q$-Daehee Polynomials with Weight $\alpha $ and its Applications

Year 2021, Volume: 4 Issue: 1, 25 - 32, 01.03.2021

Serkan Araci

https://doi.org/10.33401/fujma.837479

Abstract

The fundamental aim of the present paper is to deal with introducing a new family of Daehee polynomials which is called degenerate $q$-Daehee polynomials with weight $\alpha$ by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$. From this definition, we obtain some new summation formulae and properties. We also introduce the degenerate $q$-Daehee polynomials of higher order with weight $\alpha $ and obtain some new interesting identities.

Keywords

Weighted q-Bernoulli polynomials, Stirling numbers of first kind, Stirling numbers of second kind, Degenerate $q$-Daehee polynomials with weight $\alpha $

References

• [1] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel), 7 (1956), 28–33.
• [2] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math., 15 (1979), 51-88.
• [3] J. Kwon, Y. Kim, G. Sohn, J.-W. Park, On a q -analogue degenerate Carlitz’s type Daehee polynomials and numbers, J. Math. Computer Sci., 19 (2019), 136-142
• [4] G. Na, Y. Cho, J.-W. Park, On a degenerate q -Euler polynomials and numbers with weight, J. Math. Computer Sci., 20 (2020), 216-224
• [5] S. Aracı, U. Duran, M. Açıkgöz, On weighted q -Daehee polynomials with their applications, Indagationes Mathematicae, 30 (2) (2019), 365-374.
• [6] C.S. Ryoo, A note on the weighted q-Euler numbers and polynomials, Adv. Stud. Contemp. Math., 21 (2011), 47-54.
• [7] Y.K. Cho, T. Kim, T. Mansour, S.-H. Rim, Higher-order q-Daehee polynomials, J. Comput. Anal. Appl., 19(1) (2015), 167-173.
• [8] B.N. Guo, F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math., 255 (2014), 568-579.
• [9] S.-H. Rim, J. Jeong, On the modified q-Euler numbers of higher order with weight, Adv. Stud. Contemp. Math., 22(1) (2012), 93-98.
• [10] T. Kim, q-Volkenborn Integration, Russ. J. Math. Phys., 9(3) (2002), 288-299.
• [11] T. Kim, On the weighted q-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math., 21(2) (2011), 207-215.
• [12] T. Kim, On degenerate q-Bernoulli polynomials, Bull. Korean Math. Soc., 53(4) (2016), 1149-1156.
• [13] T. Kim, D.S. Kim, HY. Kim, J. Kwon, Degenerate Stirling polynomials of the second kind and some applications, Symmetry, 11 (8) (2019), 1046.
• [14] T. Kim, D.S. Kim, Degenerate Laplace transform and degenerate gamma function, Russ. J. Math. Phys., 24(2) (2017), 241-248.
• [15] T. Kim, A note on degenerate Stirling polynomials of the second kind, Proc. Jangjeon Math. Soc., 20(3) (2017), 319-331.
• [16] T. Kim, D.-W. Park, S.-H. Rim, On multivariate p -adic q-integrals, J. Phys. A: Math. Gen., 34 (2001), 7633-7638.
• [17] D.S. Kim, T. Kim, Daehee numbers and polynomials, Appl. Math. Sci., 7(120) (2013), 5969-5976.
• [18] D.S. Kim, T. Kim, H.I. Kwon, J.-J. Seo, Daehee polynomials with q-parameter, Adv. Stud. Theor. Phys., 8(13) (2014), 561-569.
• [19] T. Kim, S.-H. Lee, T. Mansour, J.-J. Seo, A note on q-Daehee polynomials and numbers, Adv. Stud. Contemp. Math., 24(2) (2014), 155-160.
• [20] T. Kim, L.-C. Jang, D.S. Kim, H.Y. Kim, Some identities on type 2 degenerate Bernoulli polynomials of the second kind, Symmetry, 12(4) (2020), 510.
• [21] L.-C. Jang, W. Kim, H.-I. Kwon, T. Kim, Degenerate Daehee polynomials and numbers of the third kind, J. Comput. Appl. Math. 364 (2020), 112343, 9 pp.
• [22] T. Kim, D.S. Kim, A note on type 2 Changhee and Daehee polynomials, RACSAM 113(3) (2019), 2783.
• [23] D. Lim, Degenerate, partially degenerate and totally degenerate Daehee numbers and polynomials, Adv. Difference Equ., 2015:287(2015).
• [24] E.J. Moon, J.-W. Park, S.-H. Rim, A note on the generalized q-Daehee number of higher order, Proc. Jangjeon Math. Soc., 17(4) (2014), 557-565.
• [25] H. Özden, I.N. Cangül, Y. Şimşek, Remarks on q -Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math., 18(1) (2009), 41-48.
• [26] J.-W. Park, On the q-analogue of Daehee numbers and polynomials, Proc. Jangjeon, Math. Soc., 19(3) (2016), 537-544.
• [27] G.E. Andrews, R. Askey, R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge 71 1999.
• [28] Y. Şimşek, Apostol type Daehee numbers and polynomials, Adv. Stud. Contemp. Math., 26(3) (2016), 555-566.
• [29] Y. Şimşek, A. Yardımcı, Applications on the Apostol-Daehee numbers and polynomials associated with special numbers, polynomials, and p-adic integrals, Adv. Difference Equ., 2016:308 (2016).
• [30] H.M. Srivastava, H.L. Manocha, A treatise on generating functions, Ellis Horwood Limited. Co. New York 1984.
• [31] H.M. Srivastava, A. Pinter, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17 (2004), 375-380.
• [32] H.M. Srivastava, B. Kurt, Y. Şimşek, Some families of Genocchi type polynomials and their interpolation functions, Integral Transforms Spec. Funct., 23 (2012), 919-938; see also Corrigendum, Integral Transforms Spec. Funct., 23 (2012), 939-940.