IDENTITIES AND RELATIONS ON THE HERMITE-BASED TANGENT POLYNOMIALS

Abstract

In this note, we introduce and investigate the Hermite-based Tangent numbers and polynomials, Hermite-based modified degenerate-Tangent polynomials, polyTangent polynomials. We give some identities and relations for these polynomials. Keywords: Bernoulli polynomials and numbers, Stirling numbers of the second kind, Tangent polynomials and numbers, polylogarithm function, Degenerate Bernoulli and Genocchi polynomials.

References

1. [1] Carlitz, L., (1979), Degenerate Stirling Bernoulli and Eulerian numbers, Util. Math., 15, pp. 51-88.
2. [2] Dolgy, D. V., Kim, T., Known, H.-In and Seo, J. J., (2016), On the modified degenerate Bernoulli polynomials, Advanced Studies in Contempt. Math., 26, pp. 203-209.
3. [3] Hamahata, Y., (2014), Poly-Euler polynomials and Arakawa-Kaneko type zeta functions, Functiones et Approximation Commentari Math., 51(1), pp. 7-22.
4. [4] Kaneko, M., (1999), Poly-Bernoulli numbers, J. Th´eor. Nr. Bordx, 9, pp. 199-206.
5. [5] Khan, S., Yasmin, G., Khan, R., and Hassan, N. A., (2009), Hermite-based Appell polynomials: Properties and applications, J. of Math. Anal. and Appl., 351, pp. 756-764.
6. [6] Kim, T., Kim, D. S., and Known, H.-In, (2016), Some identities relating to degenerate Bernoulli polynomials, Filomat, (30)4, pp. 905-912.
7. [7] Kim, T., Jang, V. S. and Seo, J. J., (2014), A note on poly-Genocchi numbers and polynomials, Appl. Math. Sci., 8 (96), pp. 4775-4781.
8. [8] Known, H.-In, Kim, T. and Seo, J. J., (2016), Modified degenerate Euler polynomials, Advanced Studies in Contemp. Math., 26(1), pp. 1-9.

9. [9] Liu, H. and Wang, W., (2009), Some identities on the Bernoulli, Euler and Genocchi polynomials via power sum and alternate power sums, Discrete Math., 309, pp. 3346-3363.
10. [10] Luo, M.-Q., (2009), The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integral Trans. Spec. Func., 20, pp. 337-391.
11. [11] Ozarslan, M. A., (2013), Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials, Advances in Diff. Equa., 2013.2013:116.
12. [12] Ryoo, C. S., (2013), A note on the Tangent numbers and polynomials, Adv. Stud. Theo. Phys., 7(9), pp. 447-454.
13. [13] Ryoo, C. S., (2017), On degenerate Carlitz’s type q-Tangent numbers and polynomials associated with p-adic q-integral on Zp, App. Math. Sci., 11(48), pp. 2367-2375.
14. [14] Ryoo, C. S., (2017), Symmetric identities for degenerate (h, q)-Tangent polynomials associated with the p-adic integral on Zp, Int. J. of Math. Anal., 11(8), pp. 353-362.
15. [15] Ryoo, C. S. and Agarwal, R. P., (2017), Some identities involving q-poly-Tangent numbers and polynomials and distribution of their zeros, Advences in Diff. Equa., 2017.2013.
16. [16] Son, J.-W. and Kim, M.-S., (1996), On poly-Eulerian numbers, Bull. Korean Math. Soc., 36, pp. 47-61.
17. [17] Shin, H. and Zeng, J., (2010), The q-Tangent and q-secant numbers via continued fractions, European J. of Combinatorics, 31(7), pp. 1689-1
18. [18] Srivastava, H. M., (2011), Some generalization and basic (or −q) extension of the Bernoulli, Euler and Genocchi polynomials, App. Math. and Infor. Sci., 5(3), pp. 390-444.
19. [19] Srivastava, H. M. and Manocha, H. L., (1984), A treatise on generating functions, John-Willey and Soc., (1984), New York-Toronto.
20. [20] Young, P. T., (2008), Degenerate Bernoulli polynomials, generalized factorial sums and their application, J. of Number Theory, 128, pp. 738-758.