On $q$-hypergeometric Bernoulli polynomials and numbers

Year 2021, Volume: 50 Issue: 5, 1251 - 1267, 15.10.2021

Salifou Mboutngam , Patrick Njıonou Sadjang

https://doi.org/10.15672/hujms.669940

Abstract

We introduce $q$-analogues of the hypergeometric Bernoulli polynomials in one and two real parameters and study several of their properties. Also we provide the inversion, the power representation, the multiplication and the addition formula for these polynomials. Classical results are recovered by limit transition.

Keywords

$q$-Bernoulli polynomials, $q$-Bernoulli numbers, $q$-hypergeometric Bernoulli polynomials, recurrence relation, $q$-difference equation

References

• [1] W.A. Al-Salam, q-Bernoulli numbers and polynomials, Math. Nachr. 17, 239–260, 1959.
• [2] W.A. Al-Salam, q-Appell polynomials, Ann. Mat. Pura Appl. 77 (4), 31–45, 1967.
• [3] P. Appell, Une classe de polynomes, Annalles scientifique, Ecole Normale Sup., Ser. 2, 9, 119–144, 1880.
• [4] F. Avram and M.S. Taqqu, Noncentral limit theorems and Appell polynomials, Ann. Probab. 15, 767–775, 1987.
• [5] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974.
• [6] K. Dilcher, Bernoulli numbers and confluent hypergeometric functions, in: Number Theory for the Millennium, I (Urbana, IL, 2000), 343–363, A K Peters, Natick, MA, 2002.
• [7] M. Foupouagnigni, Laguerre Hahn Orthogonal Polynomials with respect to the Hahn Operator, Fourth-order Difference Equation for the rth Associated and the Laguerre- Freud Equations for the Recurrence Coefficients [dissertation]. Porto Novo: Université Nationale du Benin, ISMP, 1998.
• [8] G. Gasper and M. Rahman, Basic hypergeometric Series, Encyclopedia Math. Appl. 35, Cambridge Univ. Press, Cambridge, 1990.
• [9] R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics, second ed., Addison-Wesley, Reading, 1994
• [10] A. Hassen and H.D. Nguyen, Hypergeometric Bernoulli polynomials and Appell sequences, Int. J. Number Theory 4, 767–774, 2008.
• [11] G. Hetyei, Enumeration by kernel positions for strongly Bernoulli type truncation games on words, J. Combin. Theory Ser. A, 117, 1107–1126, 2010.
• [12] F.T. Howard, A sequence of numbers related to the exponential function, Duke Math. J. 34, 599–616, 1967.
• [13] F.T. Howard, Some sequences of rational numbers related to the exponential function, Duke Math. J. 34, 701–716, 1967.
• [14] S. Hu and M.-S. Kim, On hypergeometric Bernoulli numbers and polynomials, Acta Math. Hungar. 154, 134–146, 2018.
• [15] V. Kac and P. Cheung, Quantum calculus, Springer, 2001.
• [16] T. Kim, q-Extension of the Euler Formula and Trigometric functions, Russ. J. Math. Phys. 13 (3), 275–278, 2007.
• [17] R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and their q-Analogues, Springer, Berlin, 2010.
• [18] N.I. Mahmudov and M.E. Keleshteri, On a class of generalized q-Bernoulli and q- Euler polynomials, Adv. Differ. Equ. 2013, Article number: 115, 2013.
• [19] H. Pan and Z.-W. Sun, New identities involving Bernoulli and Euler polynomials, J. Combin. Theory Ser. A 113, 156–175, 2006.
• [20] P. Tempesta, Formal groups, Bernoulli-type polynomials and L-series, C. R. Math. Acad. Sci. Paris 345, 303–306, 2007.
• [21] A. Urieles, M.J. Ortega, W. Ramirez and S. Vega, New result on the q-generalized Bernoulli polynomials of level m, Demonstr. Math. 52, 511–522, 2019.