Skip to main content
SpringerLink
Log in

A (p, q)-Analog of Poly-Euler Polynomials and Some Related Polynomials

  • Published:
Ukrainian Mathematical Journal Aims and scope

We introduce a (p, q)-analog of the poly-Euler polynomials and numbers by using the (p, q)-polylogarithm function. These new sequences are generalizations of the poly-Euler numbers and polynomials. We present several combinatorial identities and properties of these new polynomials and also show some relations with (p, q)-poly-Bernoulli polynomials and (p, q)-poly-Cauchy polynomials. The (p, q)-analogs generalize the well-known concept of q-analog.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. E. Andrews, R. Askey, and R. Roy, “Special functions,” Encyclopedia Math. Appl., 71, Cambridge Univ. Press, Cambridge (1999).

  2. S. Araci, U. Duran, and M. Acikgoz, “(𝜌, q)-Volkenborn integration,” J. Number Theory, 171, 18–30 (2017).

    Article  MathSciNet  Google Scholar 

  3. A. Bayad and Y. Hamahata, “Polylogarithms and poly-Bernoulli polynomials,” Kyushu J. Math., 65, 15–24 (2011).

    Article  MathSciNet  Google Scholar 

  4. C. Brewbaker, “A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues,” Integers, 8, article A02 (2008).

  5. I. M. Burban and A. U. Klimyk, “P,Q-differentiation, P,Q-integration, and P,Q-hypergeometric functions related to quantum groups,” Integral Transforms Spec. Funct., 2, No. 1, 15–36 (1994).

    Article  MathSciNet  Google Scholar 

  6. L. Carlitz, “Weighted Stirling numbers of the first kind and second kind-I,” Fibonacci Quart., 18, 147–162 (1980).

    MathSciNet  MATH  Google Scholar 

  7. M. Cenkci and T. Komatsu, “Poly-Bernoulli numbers and polynomials with a q parameter,” J. Number Theory, 152, 38–54 (2015).

    Article  MathSciNet  Google Scholar 

  8. M. Cenkci and T. Young, “Generalizations of poly-Bernoulli and poly-Cauchy numbers,” Europ. J. Math., 1, No. 5, 799–828 (2015).

    Article  MathSciNet  Google Scholar 

  9. R. Chakrabarti and R. Jagannathan, “A (p, q)-oscillator realization of two-parameter quantum algebras,” J. Phys. A, 24, No. 13, 711–718 (1991).

    Article  MathSciNet  Google Scholar 

  10. L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, Reidel, Dordrecht, The Netherlands (1974).

  11. R. B. Corcino, “On P, Q-binomial coefficients,” Integers, 8, No. 1, Article A29 (2008).

  12. Y. Hamahata, “Poly-Euler polynomials and Arakawa–Kaneko type zeta functions,” Funct. Approx. Comment. Math., 51, No. 1, 7–22 (2014).

    Article  MathSciNet  Google Scholar 

  13. M. N. Hounkonnou, J. Désiré, and B. Kyemba, “ℛ(p, q) calculus: differentiation and integration,” SUT J. Math., 49, No. 2, 145–167 (2013).

  14. R. Jagannathan and R. Sridhar, “(p, q)-Rogers–Szegő polynomial and the (p, q)-oscillator,” in: The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, Springer (2010), pp. 491–501.

  15. H. Jolany, R. B. Corcino, and T. Komatsu, “More properties on multi-poly-Euler polynomials,” Bol. Soc. Mat. Mex. (3), 21, 149–162 (2015).

  16. H. Jolany, M. Aliabadi, R. B. Corcino, and M. R. Darafsheh, “A note on multi-poly-Euler numbers and Bernoulli polynomials,” Gen. Math., 20, No. 2-3, 122–134 (2012).

  17. M. Kaneko, “Poly-Bernoulli numbers,” J. Théor. Nombres Bordeaux, 9, 199–206 (1997).

    Article  MathSciNet  Google Scholar 

  18. T. Kim, T. Komatsu, S.-H. Lee, and J. -J. Seo, “q-Poly-Cauchy numbers associated with Jackson integral,” J. Comput. Anal. Appl., 18, No. 4, 685–698 (2015).

  19. T. Komatsu, “Poly-Cauchy numbers,” Kyushu J. Math., 67, 143–153 (2013).

    Article  MathSciNet  Google Scholar 

  20. T. Komatsu, “Poly-Cauchy numbers with a q parameter,” Ramanujan J., 31, 353–371 (2013).

    Article  MathSciNet  Google Scholar 

  21. T. Komatsu, “q-Poly-Bernoulli numbers and q-poly-Cauchy numbers with a parameter by Jackson’s integrals,” Indag. Math. (N.S.), 27, 100–111 (2016).

  22. T. Komatsu, K. Liptai, and L. Szalay, “Some relationships between poly-Cauchy type numbers and poly-Bernoulli type numbers,” East-West J. Math., 14, No. 2, 114–120 (2012).

    MathSciNet  MATH  Google Scholar 

  23. T. Komatsu and F. Luca, “Some relationships between poly-Cauchy numbers and poly-Bernoulli numbers,” Ann. Math. Inform., 41, 99–105 (2013).

    MathSciNet  MATH  Google Scholar 

  24. T. Komatsu and L. Szalay, “Shifted poly-Cauchy numbers,” Lith. Math. J., 54, No. 2, 166–181 (2014).

    Article  MathSciNet  Google Scholar 

  25. T. Komatsu and J. L. Ramírez, “Generalized poly-Cauchy and poly-Bernoulli numbers by using incomplete r-Stirling numbers,” Aequationes Math., 91, 1055–1071 (2017).

    Article  MathSciNet  Google Scholar 

  26. T. Komatsu and J. L. Ramírez, “Incomplete poly-Bernoulli numbers and incomplete poly-Cauchy numbers associated to the q-Hurwitz–Lerch Zeta function,” Mediterran. J. Math., 14, No. 3, 1–19 (2017).

    Article  MathSciNet  Google Scholar 

  27. T. Komatsu, J. L. Ramírez, and V. Sirvent, “Multi-poly-Bernoulli numbers and polynomials with a q parameter,” Lith. Math. J., 54, No. 4, 490–505 (2017).

    Article  MathSciNet  Google Scholar 

  28. G. D. Liu and H. M. Srivastava, “Explicit formulas for the Nőrlund polynomials \( {B}_n^{(x)} \) and \( {b}_n^{(x)} \),” Comput. Math. Appl., 51, No. 9-10, 1377–1384 (2006).

    Article  MathSciNet  Google Scholar 

  29. T. Mansour, “Identities for sums of a q-analogue of polylogarithm functions,” Lett. Math. Phys., 87, 1–18 (2009).

    Article  MathSciNet  Google Scholar 

  30. T. Mansour, J. L. Ramírez, and M. Shattuck, “A generalization of the r-Whitney numbers of the second kind,” J. Comb., 8, No. 1, 29–55 (2017).

    MathSciNet  MATH  Google Scholar 

  31. M. Nishizawa, “Ur,s(gl4)-symmetry for (r, s)-hypergeometric series,” J. Comput. Appl. Math., 160, No. 1-2, 233–239 (2003).

    Article  MathSciNet  Google Scholar 

  32. Y. Ohno and Y. Sasaki, “On the parity of poly-Euler numbers,” RIMS Kôkyûroku Bessatsu, 32, 271–278 (2012).

    MathSciNet  MATH  Google Scholar 

  33. J. L. Ramírez and M. Shattuck, “Generalized r-Whitney numbers of the first kind,” Ann. Math. Inform., 46, 175–193 (2016).

    MathSciNet  MATH  Google Scholar 

  34. J. L. Ramírez and M. Shattuck, “(p, q)-Analogue of the r-Whitney–Lah numbers,” J. Integer Seq., 19, Article 16.5.6 (2016).

  35. S. Roman, The Umbral Calculus, Dover (1984).

  36. V. Sahai and S. Srivastava, “On irreducible p, q-representations of gl(2),” J. Comput. Appl. Math., 160, No. 1-2, 271–281 (2003).

  37. V. Sahai and S. Yadav, “Representations of two parameter quantum algebras and p, q-special functions,” J. Math. Anal. Appl., 335, No. 1, 268–279 (2007).

    Article  MathSciNet  Google Scholar 

  38. Y. F. Smirnov and R. F. Wehrhahn, “The Clebsch–Gordan coefficients for the two-parameter quantum algebra SUp,q(2) in the Lowdin–Shapiro approach,” J. Phys. A, 25, No. 21, 5563–5576 (1992).

  39. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, Amsterdam (2012).

  40. M. Wachs and D. White, “p, q-Stirling numbers and set partition statistics,” J. Combin. Theory, Ser. A, 56, 27–46 (1991).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Sciences, Zhejiang Sci-Tech University, Hangzhou, China

    T. Komatsu

  2. Universidad Nacional de Colombia, Bogotá, Colombia

    J. L. Ramírez

  3. Universidad Católica del Norte, Antofagasta, Chile

    V. F. Sirvent

Authors
  1. T. Komatsu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. L. Ramírez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. F. Sirvent
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. L. Ramírez.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 467–482, April, 2020.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Komatsu, T., Ramírez, J.L. & Sirvent, V.F. A (p, q)-Analog of Poly-Euler Polynomials and Some Related Polynomials. Ukr Math J 72, 536–554 (2020). https://doi.org/10.1007/s11253-020-01799-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-020-01799-6

Search

Navigation