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IDENTITIES AND RELATIONS ON THE HERMITE-BASED TANGENT POLYNOMIALS

Year 2020, Volume: 10 Issue: 2, 321 - 337, 01.03.2020

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] Carlitz, L., (1979), Degenerate Stirling Bernoulli and Eulerian numbers, Util. Math., 15, pp. 51-88.
  • [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] Hamahata, Y., (2014), Poly-Euler polynomials and Arakawa-Kaneko type zeta functions, Functiones et Approximation Commentari Math., 51(1), pp. 7-22.
  • [4] Kaneko, M., (1999), Poly-Bernoulli numbers, J. Th´eor. Nr. Bordx, 9, pp. 199-206.
  • [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] Kim, T., Kim, D. S., and Known, H.-In, (2016), Some identities relating to degenerate Bernoulli polynomials, Filomat, (30)4, pp. 905-912.
  • [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] 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] 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] 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] Ozarslan, M. A., (2013), Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials, Advances in Diff. Equa., 2013.2013:116.
  • [12] Ryoo, C. S., (2013), A note on the Tangent numbers and polynomials, Adv. Stud. Theo. Phys., 7(9), pp. 447-454.
  • [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] 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] 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] Son, J.-W. and Kim, M.-S., (1996), On poly-Eulerian numbers, Bull. Korean Math. Soc., 36, pp. 47-61.
  • [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] 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] Srivastava, H. M. and Manocha, H. L., (1984), A treatise on generating functions, John-Willey and Soc., (1984), New York-Toronto.
  • [20] Young, P. T., (2008), Degenerate Bernoulli polynomials, generalized factorial sums and their application, J. of Number Theory, 128, pp. 738-758.
Year 2020, Volume: 10 Issue: 2, 321 - 337, 01.03.2020

Abstract

References

  • [1] Carlitz, L., (1979), Degenerate Stirling Bernoulli and Eulerian numbers, Util. Math., 15, pp. 51-88.
  • [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] Hamahata, Y., (2014), Poly-Euler polynomials and Arakawa-Kaneko type zeta functions, Functiones et Approximation Commentari Math., 51(1), pp. 7-22.
  • [4] Kaneko, M., (1999), Poly-Bernoulli numbers, J. Th´eor. Nr. Bordx, 9, pp. 199-206.
  • [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] Kim, T., Kim, D. S., and Known, H.-In, (2016), Some identities relating to degenerate Bernoulli polynomials, Filomat, (30)4, pp. 905-912.
  • [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] 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] 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] 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] Ozarslan, M. A., (2013), Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials, Advances in Diff. Equa., 2013.2013:116.
  • [12] Ryoo, C. S., (2013), A note on the Tangent numbers and polynomials, Adv. Stud. Theo. Phys., 7(9), pp. 447-454.
  • [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] 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] 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] Son, J.-W. and Kim, M.-S., (1996), On poly-Eulerian numbers, Bull. Korean Math. Soc., 36, pp. 47-61.
  • [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] 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] Srivastava, H. M. and Manocha, H. L., (1984), A treatise on generating functions, John-Willey and Soc., (1984), New York-Toronto.
  • [20] Young, P. T., (2008), Degenerate Bernoulli polynomials, generalized factorial sums and their application, J. of Number Theory, 128, pp. 738-758.
There are 20 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

B. Kurt This is me

Publication Date March 1, 2020
Published in Issue Year 2020 Volume: 10 Issue: 2

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