Taiwanese Journal of Mathematics

q-Extensions of Some Relationships Between the Bernoulli and Euler Polynomials

Qiu-Ming Luo and H. M. Srivastava

Full-text: Open access

Abstract

The main object of this paper is to give $q$-extensions of several explicit relationships of H. M. Srivastava and Á. Pintér [Appl. Math. Lett. 17 (2004), 375-380] between the Bernoulii and Euler polynomials. We also derive several other formulas in series of Carlitz's $q$-Stirling numbers of the second kind.

Article information

Source
Taiwanese J. Math., Volume 15, Number 1 (2011), 241-257.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406173

Digital Object Identifier
doi:10.11650/twjm/1500406173

Mathematical Reviews number (MathSciNet)
MR2780283

Zentralblatt MATH identifier
1238.05025

Subjects
Primary: 05A30: $q$-calculus and related topics [See also 33Dxx] 33E20: Other functions defined by series and integrals
Secondary: 11B68: Bernoulli and Euler numbers and polynomials 11B73: Bell and Stirling numbers

Keywords
Bernoulli polynomials Euler polynomials $q$-Bernoulli polynomials $q$-Euler polynomials $q$-Bernoulli numbers $q$-Euler numbers Kronecker symbol difference equations addition theorems recurrence relationships $q$-Stirling numbers

Citation

Luo, Qiu-Ming; Srivastava, H. M. q-Extensions of Some Relationships Between the Bernoulli and Euler Polynomials. Taiwanese J. Math. 15 (2011), no. 1, 241--257. doi:10.11650/twjm/1500406173. https://projecteuclid.org/euclid.twjm/1500406173


Export citation

References

  • M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions withFormulas$,$ Graphs$,$ and Mathematical Tables, Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., 1964.
  • G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, London and New York, 1999.
  • L. Carlitz, $q$-Bernoulli numbers and polynomials, Duke Math. J., 15 (1948), 987-1000.
  • L. Carlitz, $q$-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., 76 (1954), 332-350.
  • L. Carlitz, Expansions of $q$-Bernoulli numbers, Duke Math. J., 25 (1958), 355-364.
  • M. Cenkci and M. Can, Some results on $q$-analogue of the Lerch Zeta function, Adv. Stud. Contemp. Math., 12 (2006), 213-223.
  • M. Cenkci, M. Can and V. Kurt, $q$-extensions of Genocchi numbers, J. Korean Math. Soc., 43 (2006), 183-198.
  • M. Cenkci, V. Kurt, S. H. Rim and Y. Simsek, On $(i,q)$ Bernoulli and Euler numbers, Appl. Math. Lett., 21 (2008), 706-711.
  • G.-S. Cheon, A note on the Bernoulli and Euler polynomials, Appl. Math. Lett., 16 (2003), 365-368.
  • J. Choi, P. J. Anderson and H. M. Srivastava, Some $q$-extensions of theApostol-Bernoulli and the Apostol-Euler polynomials of order $n$, and the multiple Hurwitz Zeta function, Appl. Math. Comput., 199 (2008), 723-737.
  • J. Choi, P. J. Anderson and H. M. Srivastava, Carlitz's $q$-Bernoulli and $q$-Euler numbers and polynomials and a class of $q$-Hurwitz zeta functions, Appl. Math. Comput., 215 (2009), 1185-1208.
  • L. Comtet, Advanced Combinatorics$:$ The Art of Finite and Infinite Expansions (Translated from the French by J. W. Nienhuys), Reidel Publishing Company,Dordrecht and Boston, 1974.
  • A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953.
  • M. Garg, K. Jain and H. M. Srivastava, Some relationships between the generalizedApostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions, IntegralTransform. Spec. Funct. 17 (2006), 803-815.
  • G. Gasper and M. Rahman, Basic Hypergeometric Series, Second Edition, Cambridge University Press, Cambridge, London and New York, 2004.
  • T. Kim, Some formulae for the $q$-Bernoulli and Euler polynomial of higher order, J. Math. Anal. Appl., 273 (2002), 236-242.
  • T. Kim, $q$-Generalized Euler numbers and polynomials, Russian J. Math. Phys. 13 (2006), 293-298.
  • T. Kim, On the $q$-Extension of Euler numbers and Genocchi numbers, J. Math. Anal. Appl., 326 (2007), 1458-1465.
  • T. Kim, $q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russian J. Math. Phys., 15 (2008), 51-57.
  • T. Kim, The modified $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math., 16 (2008), 161-170.
  • T. Kim, L. C. Jang and H. K. Pak, A note on $q$-Euler numbers and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci., 77 (2001), 139-141.
  • T. Kim, Y.-H. Kim and K.-W. Hwang, On the $q$-extensions of the Bernoulli and Euler numbers. related identities and Lerch zeta function, Proc. Jangjeon Math. Soc., 12 (2009), 77-92.
  • T. Kim, S.-H. Rim, Y. Simsek and D. Kim, On the analogs of Bernoulli and Euler numbers, related identities and zeta and $L$-functions, J. Korean Math. Soc., 45 (2008), 435-453.
  • S.-D. Lin, H. M. Srivastava and P.-Y. Wang, Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions, Integral Transform. Spec. Funct., 17 (2006), 817-827.
  • W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third Enlarged Edition, Springer-Verlag, Berlin, Heidelberg and New York, 1966.
  • H. Ozden and Y. Simsek, A new extension of $q$-Euler numbers and polynomials related to their interpolation functions, Appl. Math. Lett., 21 (2008), 934-939.
  • C. S. Ryoo, J. J. Seo and T. Kim, A note on generalized twisted $q$-Euler numbers and polynomials, J. Comput. Anal. Appl., 10 (2008), 483-493.
  • Y. Simsek, $q$-Analogue of the twisted $l$-series and $q$-twisted Euler numbers, J. Number Theory, 110 (2005), 267-278.
  • Y. Simsek, Twisted $(h,q)$-Bernoulli numbers and polynomials related to twisted $(h,q)$-zeta function and $L$-function, J. Math. Anal. Appl., 324 (2006), 790-804.
  • Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math., 16 (2008), 251-278.
  • H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc., 129 (2006), 77-84.
  • H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
  • H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto.
  • H. M. Srivastava and Á. Pintér, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17 (2004), 375-380.
  • H. M. Srivastava, T. Kim and Y. Simsek, $q$-Bernoulli numbers and polynomials associated with multiple $q$-Zeta functions and basic $L$-series, Russian J. Math. Phys., 12 (2005), 241-268.
  • E. T. Whittaker and G. N. Watson, A Course of Modern Analysis$:$ An Introduction to the General Theory of Infinite Processes and of Analytic Functions$;$ With an Account of the Principal Transcendental Functions, Fourth edition, Cambridge University Press, Cambridge, London and New York, 1963.