Skip to main content
SpringerLink
Log in

A Survey on p-Adic Integrals

  • Chapter
  • First Online:
Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

Abstract

The p-adic numbers are a counterintuitive arithmetic system and were firstly introduced circa end of the nineteenth century. In conjunction with the introduction of these numbers, many mathematicians and physicists started to develop new scientific tools using their available, useful, and applicable properties. Several effects of these researches have emerged in mathematics and physics such as p-adic analysis, string theory, p-adic quantum mechanics, quantum field theory, representation theory, algebraic geometry, complex systems, dynamical systems, and genetic codes. One of the important tools of the mentioned advancements is the p-adic integrals. Intense research activities in such an area like p-adic integrals are principally motivated by their significance in p-adic analysis. Recently, p-adic integrals and its diverse extensions have been studied and investigated in detail by many mathematicians. This chapter considers and investigates multifarious extensions of the p-adic integrals elaborately. q-Analogues with diverse extensions of p-adic integrals are also considered such as the weighted p-adic q-integral on \( \mathbb {Z} _{p}\). The two types of the weighted q-Boole polynomials and numbers are introduced and investigated in detail. As several special polynomials and numbers can be derived from the p-adic integrals, some generalized and classical q-polynomials and numbers are obtained from the aforesaid extensions of p-adic integrals. Finally, the importance of these extensions is analyzed.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. Araci, D. Erdal, J.J. Seo, A study on the fermionic p-adic q-integral representation on associated with weighted q -Bernstein and q-Genocchi polynomials. Abst. Appl. Anal. 2011 (2011). Article ID 649248. https://doi.org/10.1155/2011/649248

  2. S. Araci, E. Ağyüz, M. Acikgoz, On a q-analog of some numbers and polynomials. J. Inequal. Appl. 19 (2015). https://doi.org/10.1186/s13660-014-0542-y

  3. S. Araci, U. Duran, M. Acikgoz, Symmetric identities involving q-Frobenius-Euler polynomials under Sym (5). Turkish J. Anal. Number Theory 3(3), 90–93 (2015)

    Article  Google Scholar 

  4. S. Araci, M. Acikgoz, U. Duran, On weighted q-Daehee numbers and polynomials, Indag. Math. 30(2), 365–374 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. D.V. Dolgy, T. Kim, S.-H. Rim, J.-J. Seo, A note on Changhee polynomials and numbers with q-parameters. Int. J. Math. Anal. 8(26), 1255–1264 (2014)

    Article  Google Scholar 

  6. D.V. Dolgy, T. Kim, S.-H. Rim, S.-H. Lee, Some symmetric identities for h-extension of q-Euler polynomials under third dihedral group D 3. Int. J. Math. Anal. 8(48), 2369–2374.3 (2014)

    Article  Google Scholar 

  7. U. Duran, M. Acikgoz, On applications for Mahler expansion associated with p-adic q-integrals. Int. J. Number Theory 15(1), 67–84 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. F.G. Gurkan, M. Acikgoz, E. Agyuz, A study on the new mixed-type polynomials related to Boole polynomials. Afr. Mat. 28(1–2), 279–290 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ö.Ç. Havara, H. Menken, On the Volkenborn integral of the q-extension of the p-adic gamma function. J. Math. Anal. 8(2), 64–72 (2017)

    MathSciNet  Google Scholar 

  10. L.-C. Jang, T. Kim, q-Genocchi numbers and polynomials associated with fermionic p-adic invariant integrals on \( \mathbb {Z} _{p}\). Abstr. Appl. Anal. 2008, 8 pp. (2008). Article ID 232187

    Google Scholar 

  11. Y.S. Jang, T. Kim, S.-H. Rim, J.-J. Seo, Symmetry identities for the generalized higher-order q-Bernoulli polynomials under S 3. Int. J. Math. Anal. 8(38), 1873–1879 (2014)

    Article  Google Scholar 

  12. T. Kim, q-Volkenborn integration. Russ. J. Math. Phys. 9, 288–299 (2002)

    Google Scholar 

  13. T. Kim, A note on q-Volkenborn integration. Proc. Jangjeon Math. Soc. 8(1), 13–17 (2005)

    MathSciNet  MATH  Google Scholar 

  14. D.S. Kim, A note on p-adic invariant integral in the rings of p -adic integers. Adv. Stud. Contemp. Math. 13, 95–99 (2006)

    MathSciNet  MATH  Google Scholar 

  15. T. Kim, q-Euler numbers and polynomials associated with p -adic q-integrals. J. Nonlinear Math. Phys. 14(1), 15–27 (2007). https://doi.org/10.2991/jnmp.2007.14.1.3

  16. T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on \( \mathbb {Z} _{p}\). Russ. J. Math. Phys. 16, 484–491 (2009)

    Google Scholar 

  17. T. Kim, On the weighted q-Bernoulli numbers and polynomials. Adv. Stud. Contemp. Math. 21, 207–215 (2011)

    MathSciNet  MATH  Google Scholar 

  18. D.S. Kim, T. Kim, Daehee numbers and polynomials. Appl. Math. Sci. 7(120), 5969–5976 (2013)

    MathSciNet  Google Scholar 

  19. D.S. Kim, T. Kim, A note on Boole polynomials. Integral Transforms Spec. Funct. 25(8), 627–633 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. T. Kim, S.-H. Lee, T. Mansour, J.-J. Seo, A note on q -Daehee polynomials and numbers. Adv. Stud. Contemp. Math. 24(2), 155–160 (2014)

    MATH  Google Scholar 

  21. D.S. Kim, T. Kim, J.J. Seo, A note on q-analogue of Boole polynomials. Appl. Math. Inf. Sci. 9(6), 3153–3158 (2015)

    MathSciNet  Google Scholar 

  22. N. Koblitz, p-Adic Numbers,p-Adic Analysis, and Zeta Functions (Springer, New York, 1977)

    Book  MATH  Google Scholar 

  23. J. Kwon, J.-W. Park, A note on \(\left ( h,q\right ) \)-Boole polynomials. Adv. Differ. Equ. 2015, 198 (2015). https://doi.org/10.1186/s13662-015-0536-1

  24. H. Ozden, I.N. Cangul, Y. Sımsek, Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 18(1), 41–48 (2009)

    MathSciNet  MATH  Google Scholar 

  25. A.M. Robert, A Course inp-Adic Analysis (Springer, New York, 2000)

    Book  MATH  Google Scholar 

  26. C.S. Ryoo, Symmetric identities for Carlitz’s twisted q -Bernoulli polynomials associated with p-adic q-integral on \( \mathbb {Z} _{p}\). Appl. Math. Sci. 9(72), 3569–3575 (2015)

    Google Scholar 

  27. W.H. Schikhof, Ultrametric Calculus: An Introduction top-Adic Analysis. Cambridge Studies in Advanced Mathematics, vol. 4 (Cambridge University Press, Cambridge, 1984)

    Google Scholar 

  28. Y. Simsek, A. Yardimci, Applications on the Apostol-Daehee numbers and polynomials associated with special numbers, polynomials, and p -adic integrals. Adv. Differ. Equ. 308 (2016). https://doi.org/10.1186/s13662-016-1041-x

  29. A. Volkenborn, Ein p-adisches integral und seine Anwendungen I. Manuscripta Math. 7(4), 341–373 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Volkenborn, Ein p-adisches integral und seine Anwendungen II. Manuscripta Math. 12(1), 17–46 (1974)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of the Basic Concepts of Engineering, Faculty of Engineering and Natural Sciences, İskenderun Technical University, Hatay, Turkey

    Ugur Duran

  2. Department of Mathematics, Gauhati University, Guwahati, India

    Hemen Dutta

Authors
  1. Ugur Duran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hemen Dutta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ugur Duran .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  2. University of Nis, Faculty of Sciences and Mathematics, Aleksandrovac, Serbia

    Ljubiša D. R. Kočinac

  3. Dept. Mathematics and Statistics, University of Victoria, Victoria, BC, Canada

    Hari M. Srivastava

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Duran, U., Dutta, H. (2019). A Survey on p-Adic Integrals. In: Dutta, H., Kočinac, L.D.R., Srivastava, H.M. (eds) Current Trends in Mathematical Analysis and Its Interdisciplinary Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-15242-0_22

Download citation

Publish with us

Policies and ethics

Search

Navigation