Relationships between Mahler Expansion and Higher Order q-Daehee Polynomials

Year 2021, Volume: 9 Issue: 2, 337 - 341, 15.10.2021

Ugur Duran , Mehmet Acikgoz

Abstract

In this paper, we derive multifarious relationships among the two types of higher order q-Daehee polynomials and p-adic gamma function via Mahler theorem. Also, we compute some weighted p-adic q-integrals of the derivative of p-adic gamma function related to the Stirling numbers of the both kinds and the q-Bernoulli polynomials of order k.

Keywords

q-numbers, p-adic numbers, p-adic gamma function, Mahler expansion, Bernoulli polynomials, Daehee polynomials, Stirling numbers of the first kind, Stirling numbers of the second kind

References

• [1] S. Araci, E. Agyüz, M. Acikgoz, On a q-analog of some numbers and polynomials, J. Inequal. Appl., 19, 2015 doi:10.1186/s13660-014-0542-y.
• [2] Y.K. Cho, T. Kim, T. Mansour, S.-H. Rim, Higher order q-Daehee polynomials, J. Comput. Anal. Appl., 19 (1), 2015,167-173.
• [3] J. Diamond, The p-adic log gamma function and p-adic Euler constant, Trans. Amer. Math. Soc., 233, 1977, 321-337.
• [4] U. Duran, M. Acikgoz, On p-adic gamma function related to q-Daehee polynomials and numbers, accepted for publicationin Proyecciones J. Math., 2018.
• [5] B. N. Guo, F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, Anal. Appl., 19 (1), 2015,167-173.
• [6] Ö. Ç. Havara and H. Menken, On the Volkenborn integral of the q-extension of the p-adic gamma function, J. Math. Anal.,8 (2), 2017, 64-72.
• [7] Ö. Ç. Havare and H. Menken, The Volkenborn integral of the p-adic gamma function, Int. J. Adv. Appl. Funct., 5 (2),2018, 56-59.
• [8] Y. S. Kim, q-analogues of p-adic gamma functions and p-adic Euler constants, Comm. Korean Math. Soc., 13 (4), 1998,735–741.
• [9] T. Kim, S.-H. Lee, T. Mansour and J.-J. Seo, A note on q-Daehee polynomials and numbers, Adv. Stud. Contemp. Math.,24 (2), 2014, 155-160.
• [10] D. S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci., 7 (120), 2013, 5969-5976.
• [11] N. Koblitz, p-adic numbers, p-adic analysis, and Zeta functions (Springer-Verlag, New York Inc, 1977).
• [12] K. Mahler, An interpolation series for continuous functions of a p-adic variable, J. Reine Angew. Math., 199, 1958, 23-34.
• [13] H. Menken, A. Körükçü, Some properties of the q-extension of the p-adic gamma function, Abst. Appl. Anal., Volume2013, Article ID 176470, 4 pages.
• [14] Y. Morita, A p-adic analogue of the gamma-function, J. Fac. Sci., 22 (2), 1975, 255-266.
• [15] H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp.Math., 18 (1), 2009, 41-48.
• [16] A. M. Robert, A course in p-adic analysis (Springer-Verlag New York, Inc., 2000).
• [17] Y. Simsek, Apostol type Daehee numbers and polynomials, Adv. Stud. Contemp. Math.,