A new class of higher order hypergeometric Bernoulli polynomials associated with Hermite polynomials
DOI:
https://doi.org/10.5269/bspm.51845Abstract
In this paper, we introduce new class of higher order hypergeometric Hermite-Bernoulli numbers and polynomials. We shall provide several properties of higher order hypergeometric Hermite-Bernoulli
polynomials including summation formulae, sums of products identity, recurrence relations.
References
1. Arakawa, T., Ibukiyama, T., Kaneko, M., Bernoulli numbers and zeta functions. With an appendix by Don Zagier, Springer Monographs in Mathematics. Springer, Tokyo, 2014. https://doi.org/10.1007/978-4-431-54919-2
2. Aoki, M., Komatsu, T., Panda, G. K., Several properties of hypergeometric Bernoulli numbers, Journal of Inequalities and Applications, (2019) 2019:113. https://doi.org/10.1186/s13660-019-2066-y
3. Bell, E. T., Exponential polynomials, Ann. of Math., 35, 258-277, (1934). https://doi.org/10.2307/1968431
4. Dattoli, G., Lorenzutta, S., and Cesarano, C., Finite sums and generalized forms of Bernoulli polynomials, Rendiconti di Mathematica, 19, 385-391, (1999).
5. Howard, F. T., A sequence of numbers related to the exponential function, Duke Math. J., 34, 599-615, (1967). https://doi.org/10.1215/S0012-7094-67-03465-5
6. Howard, F. T., Some sequences of rational numbers related to the exponential function, Duke Math. J., 34, 701-716, (1967). https://doi.org/10.1215/S0012-7094-67-03473-4
7. Hu, S., and Kim, M. S., On hypergeometric Bernoulli numbers and polynomials, Acta Math. Hungar., 154, 134-146. (2018). https://doi.org/10.1007/s10474-017-0767-6
8. Hu, S., and Komatsu, T., Explicit expressions for the related numbers of higher order Appell polynomials, Quaestiones Mathematicae, (2019), 1-11, https://doi.org/10.2989/16073606.2019.1596174
9. Iwasawa, K., Lectures on p-Adic L-Functions, Ann. of Math. Stud., vol. 74, Princeton Univ. Press, Princeton, 1972. https://doi.org/10.1515/9781400881703
10. Kamano, K., Sums of products of hypergeometric Bernoulli numbers, J. Number Theory, 130, 2259-2271, (2010). https://doi.org/10.1016/j.jnt.2010.04.005
11. Kurokawa, N., Kurihara, M., Saito, N., Number theory. 3. Iwasawa theory and modular forms. Translated from the Japanese by Masato Kuwata. Translations of Mathematical Monographs, 242. Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI, 2012. https://doi.org/10.1090/mmono/242
12. Kim, M. S., and Hu, S. A., p-adic view of multiple sums of powers, Int. J. Number Theory, 7, 2273-2288, (2011). https://doi.org/10.1142/S1793042111005027
13. Nguyen, H. D., and Cheong, L. G., New convolution identities for hypergeometric Bernoulli polynomials, J. Number Theory, 137, 201-221, (2014). https://doi.org/10.1016/j.jnt.2013.11.008
14. N¨orlund, M. E., Differenzenrechnung, Springer-Verlag, Berlin, 1924. https://doi.org/10.1007/978-3-642-50824-0
15. Hassen, A., and Nguyen, H. D., Hypergeometric Bernoulli polynomials and Appell sequences, International Journal of Number Theory, 4(5), 767-774, (2008). https://doi.org/10.1142/S1793042108001754
16. Pathan, M. A., and Khan, W. A., Some implicit summation formulas and symmetric identities for the generalized Hermite-Bernoulli polynomials, Mediterr. J. Math., 12, 679-695, (2015). https://doi.org/10.1007/s00009-014-0423-0
17. Pathan, M. A., and Khan, W. A., A new class of generalized polynomials associated with Hermite and Euler polynomials, Mediterr. J. Math., 13, 913-928, (2016). https://doi.org/10.1007/s00009-015-0551-1
18. Pathan, M. A., and Khan, W. A., Some implicit summation formulas and symmetric identities for the generalized Hermite-Euler polynomials, East-West J. Maths., 16(1), 92-109, (2014).
19. Sun, Z. W., Introduction to Bernoulli and Euler polynomials, a lecture given in Taiwan on June 6, 2002.
20. Srivastava, H. M., and Manocha, H. L., A treatise on generating functions, Ellis Horwood Limited. Co. New York, 1984.
2. Aoki, M., Komatsu, T., Panda, G. K., Several properties of hypergeometric Bernoulli numbers, Journal of Inequalities and Applications, (2019) 2019:113. https://doi.org/10.1186/s13660-019-2066-y
3. Bell, E. T., Exponential polynomials, Ann. of Math., 35, 258-277, (1934). https://doi.org/10.2307/1968431
4. Dattoli, G., Lorenzutta, S., and Cesarano, C., Finite sums and generalized forms of Bernoulli polynomials, Rendiconti di Mathematica, 19, 385-391, (1999).
5. Howard, F. T., A sequence of numbers related to the exponential function, Duke Math. J., 34, 599-615, (1967). https://doi.org/10.1215/S0012-7094-67-03465-5
6. Howard, F. T., Some sequences of rational numbers related to the exponential function, Duke Math. J., 34, 701-716, (1967). https://doi.org/10.1215/S0012-7094-67-03473-4
7. Hu, S., and Kim, M. S., On hypergeometric Bernoulli numbers and polynomials, Acta Math. Hungar., 154, 134-146. (2018). https://doi.org/10.1007/s10474-017-0767-6
8. Hu, S., and Komatsu, T., Explicit expressions for the related numbers of higher order Appell polynomials, Quaestiones Mathematicae, (2019), 1-11, https://doi.org/10.2989/16073606.2019.1596174
9. Iwasawa, K., Lectures on p-Adic L-Functions, Ann. of Math. Stud., vol. 74, Princeton Univ. Press, Princeton, 1972. https://doi.org/10.1515/9781400881703
10. Kamano, K., Sums of products of hypergeometric Bernoulli numbers, J. Number Theory, 130, 2259-2271, (2010). https://doi.org/10.1016/j.jnt.2010.04.005
11. Kurokawa, N., Kurihara, M., Saito, N., Number theory. 3. Iwasawa theory and modular forms. Translated from the Japanese by Masato Kuwata. Translations of Mathematical Monographs, 242. Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI, 2012. https://doi.org/10.1090/mmono/242
12. Kim, M. S., and Hu, S. A., p-adic view of multiple sums of powers, Int. J. Number Theory, 7, 2273-2288, (2011). https://doi.org/10.1142/S1793042111005027
13. Nguyen, H. D., and Cheong, L. G., New convolution identities for hypergeometric Bernoulli polynomials, J. Number Theory, 137, 201-221, (2014). https://doi.org/10.1016/j.jnt.2013.11.008
14. N¨orlund, M. E., Differenzenrechnung, Springer-Verlag, Berlin, 1924. https://doi.org/10.1007/978-3-642-50824-0
15. Hassen, A., and Nguyen, H. D., Hypergeometric Bernoulli polynomials and Appell sequences, International Journal of Number Theory, 4(5), 767-774, (2008). https://doi.org/10.1142/S1793042108001754
16. Pathan, M. A., and Khan, W. A., Some implicit summation formulas and symmetric identities for the generalized Hermite-Bernoulli polynomials, Mediterr. J. Math., 12, 679-695, (2015). https://doi.org/10.1007/s00009-014-0423-0
17. Pathan, M. A., and Khan, W. A., A new class of generalized polynomials associated with Hermite and Euler polynomials, Mediterr. J. Math., 13, 913-928, (2016). https://doi.org/10.1007/s00009-015-0551-1
18. Pathan, M. A., and Khan, W. A., Some implicit summation formulas and symmetric identities for the generalized Hermite-Euler polynomials, East-West J. Maths., 16(1), 92-109, (2014).
19. Sun, Z. W., Introduction to Bernoulli and Euler polynomials, a lecture given in Taiwan on June 6, 2002.
20. Srivastava, H. M., and Manocha, H. L., A treatise on generating functions, Ellis Horwood Limited. Co. New York, 1984.
Downloads
Published
2022-02-04
Issue
Section
Proceedings
License
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
