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SOME IDENTITIES OF DEGENERATE POLY-q-BERNOULLI AND POLY-q-EULER POLYNOMIALS ARISING FROM λ -q-SHEFFER SEQUENCES

김혜경|김태균|김대산
장전수학회 (Jangjeon Mathematical Society)|2022.07|Advanced Studies in Contemporary Mathematics|제32권 제3호|pp. 283-302

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참고문헌

키워드

q-Bernoulli polynomialsq-exponential functionsq-umbral calculusλ -umbral calculusThe q-Euler polynomials
주제분류수학

초록

Recently, we introduced the λ -q-umbral calculus which centers around the λ -q-Sheffer sequences and the degenerate q-Sheffer sequences by introducing λ -q-linear functionals and λ - q-differential operators, respectively, instead of λ -linear functionals and λ -differential operators. In this paper, we introduce the degenerate poly-q-Bernoulli polynomials and degenerate poly-q- Euler polynomials and numbers by using the q-polylogarithm function. These new sequences are q-analogues of the degenerate poly-Bernoulli polynomials and degenerate poly-Euler polynomials, respectively. We give interesting combinatorial identities and properties of these new polynomials by using λ -q-umbral calculus.

Abstract

Recently, we introduced the λ -q-umbral calculus which centers around the λ -q-Sheffer sequences and the degenerate q-Sheffer sequences by introducing λ -q-linear functionals and λ - q-differential operators, respectively, instead of λ -linear functionals and λ -differential operators. In this paper, we introduce the degenerate poly-q-Bernoulli polynomials and degenerate poly-q- Euler polynomials and numbers by using the q-polylogarithm function. These new sequences are q-analogues of the degenerate poly-Bernoulli polynomials and degenerate poly-Euler polynomials, respectively. We give interesting combinatorial identities and properties of these new polynomials by using λ -q-umbral calculus.

참고문헌 (42)

  1. Ac¸ikg¨oz, M., "A new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials", Adv. Difference Equ, pp.9~ (2010)[DOI]
  2. Ac¸ikg¨oz, M., "Applications of q-umbral calculus to modified apostol type q-Bernoulli polynomials", J. Math. Statistics, 14(1), pp.7~15 (2018)
  3. Abdelmejid BAYAD, "POLYLOGARITHMS AND POLY-BERNOULLI POLYNOMIALS", Kyushu Journal of Mathematics, 65(1), pp.15~24 (2011), Faculty of Mathematics, Kyushu University[DOI]
  4. Bayad, A., "Identities involving values of Bernstein, q-Bernstein, q-Bernoulli, and q-Euler polynomials", Russ. J. Math. Phys, 18(2), pp.133~143 (2011)
  5. Carlitz, L, "Degenerate Stirling, Bernoulli and Eulerian numbers", Utilitas Math, 15, pp.51~88 (1979)
  6. Carlitz, L, "q-Bernoulli numbers and polynomials", Duke Math. J, 15, pp.987~1000 (1948)
  7. Mehmet Cenkci, "Poly-Bernoulli numbers and polynomials with a q parameter", Journal of Number Theory, 152, pp.38~54 (2015), Elsevier BV[DOI]
  8. R. Dere, "Applications of umbral algebra to some special polynomials", Advanced Studies in Contemporary Mathematics, 22(3), pp.433~438 (2012), 장전수학회[DOI]
  9. Duran, U., "Construction of the type 2 poly-Frobenius Genocchi polynomials with their certain applications", Adv. Differ. Equ, (2020) (2020)
  10. Hamahata, Y., "Poly-Euler polynomials and Arakawa-Kaneko type zeta functions", Functiones et Approximatio, 51(1), pp.7~22 (2014)[DOI]
  11. Hardy, G.H., "On a class a functions", Proc. Lond. Math. Soc, 3(2), pp.441~460 (1905)
  12. Ihrig, E. C., "A q-umbral calculus", J. Math. Analy. Applic, 84, pp.178~207 (1981)
  13. Ismail, M. E. H., "Inverse operators, q-fractional integrals, and q-Bernoulli polynomials", J. Approx. Theory, 114(2), pp.269~307 (2002)
  14. Jang, L-C., "Study of Degenerate Poly-Bernoulli Polynomials by l -umbral Calculus", CMES, 129(1), pp.393~408 (2021)[DOI]
  15. Kim, D. S., "Degenerate Bernstein polynomials", RACSAM, 113, pp.2913~2920 (2019)
  16. Kim, D. S., "A note on a new type of degenerate Bernoulli numbers", Russ. J. Math. Phys, 27(2), pp.227~235 (2020)
  17. Kim, D. S., "Degenerate Sheffer sequence and l -Sheffer sequence", J. Math. Anal. Appl, 493(1), pp.124~521 (2021)
  18. Kim, D. S., "q-Bernoulli polynomials and q-umbral calculus", Sci. China Math, 57, pp.1867~1874 (2014)
  19. Kim, D. S., "Some identities of q-Euler polynomials arising from q-umbral calculus", J. Ineq. Appl, 1(2014), pp.12~ (2014)[DOI]
  20. Kim, D. S., "Fullly degenerate poly-Bernoulli polynomials with a q parameter", Filomat, 30(4), pp.1029~1035 (2016)[DOI]
  21. Kim, H. K., "Poly-central factorial sequences and poly-central-Bell polynomials.", Adv. Difference Equ, 505(2021), pp.12~ (2021)[DOI]
  22. 김택연, "A note on degenerate stirling polynomials of the second kind", Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집), 20(3), pp.319~332 (2017), 장전수학회[DOI]
  23. Kim, T., "q-Bernoulli numbers associated with q-Stirling numbers", Adv. Difference Equ, 2008 (2008)
  24. Kim, T., "q-Generalized Euler numbers and polynomials", Russ. J. Math. Phys., 13, pp.293~298 (2006)
  25. Kim, T., "q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients", Russ. J. Math. Phys., 15(1), pp.51~57 (2008)
  26. Kim, T., "egenerate poly-Bell polynomials and numbers", Adv. Difference Equ, 361, pp.12~ (2021)[DOI]
  27. Kim, T., "On q-derangement numbers and poynomials. to appear in Fractals", ~
  28. Kim, T., "l -q-Sheffer sequences and its applications", ~
  29. Kim, T., "Degenerate binomial coefficients and degenerate hypergeometric functions", Adv. Difference Equ, 115 (2020)[DOI]
  30. Kim, T., "A note on degenerate gamma random variables", Revista Edu, 38894, pp.39~44 (2020)
  31. Kim, T., "A study on l -Sheffer sequences by other l -Sheffer sequences", J. Nonlin. Convex Analy, 23(2), pp.321~336 (2022)
  32. Kim, T., "Degenerate poly-Bernoulli numbers and polynomials", Informatica, 31(3), pp.1~7 (2020)
  33. Kim, T., "Degenerate Bell polynomials associated with umbral calculus", J. Inequal. Appl, pp.15~ (2020)
  34. Young Rok Kim, "A note on q-analogue of poly-Euler polynomials and Arakawa-Kaneko type zeta function", Journal of Applied Mathematics and Informatics, 38(5), pp.611~623 (2020), 한국전산응용수학회
  35. Komatsu, T., "A (p,q)-analogue of poly-Euler polynomials and some related polynomials", arXiv:1604.0378v1, ~ (2016)
  36. Kupershmidt, B. O., "Reflection symmetries of q-Bernoulli polynomials", J. Nonlinear Math. Phys, 12(suppl. 1), pp.412~422 (2005)
  37. Lee, D. S., "Type 2 degenerate poly-Euler polynomials", Symmetry, 1011, pp.15~ (2020)
  38. Mahmudov, N. I., "On a class of q-Bernoulli and q-Euler polynomials", Adv. Difference Equ, 108 (2013)[DOI]
  39. M. Mechacha, "Some results for q-poly-Bernoulli polynomials with a parameter", Miskolc Mathematical Notes, 19(1), pp.355~ (2018), Mathematical Notes[DOI]
  40. Roman, S., "More on the umbral calculus, with emphasis on the q-umbral calculus", J. Math. Anal. Appl, 107, pp.222~254 (1985)
  41. Roman, S., "The umbral calculus", Adv. Math, 27, pp.95~188 (1978)
  42. Takao Komatsu, "A (p,q)-analogue of poly-Euler polynomials and some related polynomials", arXiv:1604.03787v1, ~ (2016)

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