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A note on the \(p\)-adic gamma function and \(q\)-Changhee polynomials. (English) Zbl 1427.11136

Summary: In the present work, we consider the fermionic \(p\)-adic \(q\)-integral of \(p\)-adic gamma function and the derivative of \(p\)-adic gamma function by using their Mahler expansions. The relationship between the \(p\)-adic gamma function and \(q\)-Changhee numbers is obtained. A new representation is given for the \(p\)-adic Euler constant. Also, we study on the relationship between \(q\)-Changhee polynomials and \(p\)-adic Euler constant using the fermionic \(p\)-adic \(q\)-integral techniques the idea that the \(q\)-Changhee polynomial.

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
33E50 Special functions in characteristic \(p\) (gamma functions, etc.)
05A19 Combinatorial identities, bijective combinatorics
26C05 Real polynomials: analytic properties, etc.
11S05 Polynomials
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[1] Araci, S.; Erdal, D.; Seo, J. J., A study on the fermionic p-adic q-integral representation on \(\mathbb{Z}_p\) associated with weighted q-Bernstein and q-Genocchi polynomials, Abstr. Appl. Anal., 2011, 1-10 (2011) · Zbl 1269.11020
[2] Barsky, D., On Morita’s p-adic gamma function, Math. Proc. Cambridge Philos. Soc., 89, 23-27 (1981) · Zbl 0455.12012
[3] Diamond, J., The p-adic log gamma function and p-adic Euler constants, Trans. Amer. Math. Soc., 233, 321-337 (1977) · Zbl 0382.12008
[4] B. Dwork, A note on the p-adic gamma function, Study group on ultrametric analysis, 9th year: 1981/82, Marseille, (1982), Inst. Henri Poincaré, Paris, 1-10 (1983) · Zbl 0512.12013
[5] Kim, T., q-Volkenborn integration, Russ. J. Math. Phys., 9, 288-299 (2002) · Zbl 1092.11045
[6] Kim, T., q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., 14, 15-27 (2007) · Zbl 1158.11009
[7] Kim, T., Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on \(\mathbb{Z}_p\), Russ. J. Math. Phys., 16, 484-491 (2009) · Zbl 1192.05011
[8] Kim, T., Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on \(\mathbb{Z}_p\), Russ. J. Math. Phys., 16, 93-96 (2009) · Zbl 1200.11089
[9] Kim, D. S.; Kim, T., Daehee numbers and polynomials, Appl. Math. Sci. (Ruse), 7, 5969-5976 (2013)
[10] Kim, T.; Kim, D. S.; Mansour, T.; Rim, S.-H.; M. Schork, Umbral calculus and Sheffer sequences of polynomials, J. Math. Phys., 54, 1-15 (2013) · Zbl 1285.05015
[11] Kim, T.; Kwon, H.-I.; J. J. Seo, Degenerate q-Changhee polynomials, J. Nonlinear Sci. Appl., 9, 2389-2393 (2016) · Zbl 1362.11030
[12] Kim, T.; Mansour, T.; Rim, S.-H.; Seo, J. J., A note on q-Changhee Polynomials and Numbers, Adv. Studies Theor. Phys., 8, 35-41 (2014)
[13] Mahler, K., An interpolation series for continuous functions of a p-adic variable, J. Reine Angew. Math., 199, 23-34 (1958) · Zbl 0080.03504
[14] Morita, Y., A p-adic analogue of the \(\Gamma \)-function, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 22, 255-266 (1975) · Zbl 0308.12003
[15] A. M. Robert, A course in p-adic analysis, Graduate Texts in Mathematics, Springer-Verlag, New York (2000) · Zbl 0947.11035
[16] Schikhof, W. H., Ultrametric calculus, An introduction to p-adic analysis, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1984) · Zbl 0553.26006
[17] Vladimirov, V. S.; Volovich, I. V., Superanalysis, I, Differential calculus, (Russian) Teoret. Mat. Fiz., 59, 3-27 (1984) · Zbl 0552.46023
[18] Volovich, I. V., Number theory as the ultimate physical theory, p-Adic Numbers Ultrametric Anal. Appl., 2, 77-87 (2010) · Zbl 1258.81074
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