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p-ADIC q-HIGHER-ORDER HARDY-TYPE SUMS

  • SIMSEK YILMAZ (Akdeniz University Faculty of Art and Science Department of Mathematics)
  • Published : 2006.01.01

Abstract

The goal of this paper is to define p-adic Hardy sums and p-adic q-higher-order Hardy-type sums. By using these sums and p-adic q-higher-order Dedekind sums, we construct p-adic continuous functions for an odd prime. These functions contain padic q-analogue of higher-order Hardy-type sums. By using an invariant p-adic q-integral on $\mathbb{Z}_p$, we give fundamental properties of these sums. We also establish relations between p-adic Hardy sums, Bernoulli functions, trigonometric functions and Lambert series.

References

  1. T. M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J. 17 (1950), 147-157 https://doi.org/10.1215/S0012-7094-50-01716-9
  2. T. M. Apostol, Theorems on Generalized Dedekind Sums, Pacific J. Math. 2 (1952), 1-9 https://doi.org/10.2140/pjm.1952.2.1
  3. B. C. Berndt and L. A. Goldberg, Analytic properties of arithmetic sums arising in the theory of the classical theta functions, SIAM J. Math. Anal. 15 (1984), no. 1, 143-150 https://doi.org/10.1137/0515011
  4. M. Cenkci, M. Can, and V. Kurt, Generalized Hardy sums, (preprint)
  5. U. Dieter, Cotangent sums, a futher generalization of Dedekind sums, J. Number Theory 18 (1984), 289-305 https://doi.org/10.1016/0022-314X(84)90063-5
  6. T. Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory 76 (1999), no. 2, 320-329 https://doi.org/10.1006/jnth.1999.2373
  7. T. Kim, On p-adic q- Bernoulli numbers, J. Korean Math. Soc. 37 (2000), no. 1, 21-30
  8. T. Kim, A note on p-adic q-Dedekind sums, C. R. Acad. Bulgare Soc. 54 (2001), no. 10, 37-42
  9. T. Kim, q-Volkenborn integration, Russ. J. Math Phys. 9 (2002), no. 3, 288-299
  10. T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), no. 3, 261-267
  11. T. Kim, Analytic continuation of multiple q-zeta functions and their values at negative integers, Russ. J. Math Phys. 11 (2004), 71-76
  12. T. Kim, p-adic q-integrals associated with the Changhee-Barnes' q-Bernoulli Polynomials, Integral Transform. Spec. Funct. 15 (2004), no. 5, 415-420 https://doi.org/10.1080/10652460410001672960
  13. T. Kim and H. S. Kim, Remark on p-adic q-Bernoulli numbers, Algebraic number theory (Hapcheon/Saga, 1996). Adv. Stud. Contemp. Math. (Pusan) 1 (1999), 127-136
  14. A. Kudo, On p-adic Dedekind Sums, Nagoya Math. J. 144 (1996), 155-170 https://doi.org/10.1017/S0027763000006048
  15. H. Rademacher and A. Whiteman, Theorems on Dedekind sums, Amer. J. Math. 63 (1941), 377-407 https://doi.org/10.2307/2371532
  16. K. H. Rosen and W. M. Snyder, p-adic Dedekind Sums, J. Reine Angew. Math. 361 (1985), 23-26
  17. Y. Simsek, Theorems on three-term relations for Hardy sums, Turkish J. Math. 22 (1998), no. 2, 153-162
  18. Y. Simsek, Relation between theta-function Hardy sums Eisenstein and Lambert series in the transformation formula of log $\eta_{g,h}$(z), J. Number Theory 99 (2003), no. 2, 338-360 https://doi.org/10.1016/S0022-314X(02)00072-0
  19. Y. Simsek, On generalized Hardy Sums $S_5$(h, k), Ukrain. Mat. Zh. 56 (2004), no. 10, 1434-1440; translation in Ukrainian Math. J. 56 (2004), no. 10, 1712-1719 (2005)
  20. Y. Simsek and Y. Cetinkaya, On p-adic Hardy sums, XV. National Mathe-matic Symposium (XV. Ulusal Matematik Sempozyumu), 4-7 September, Mer-sin(Turkey) (2002), 105-112
  21. R. Sitaramachandrarao, Dedekind and Hardy Sums, Acta Arith. 48 (1987), no. 4, 325-340 https://doi.org/10.4064/aa-48-4-325-340

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  3. Transformation formulas of a character analogue of $$\log \theta _{2}(z)$$logθ2(z) pp.1572-9303, 2018, https://doi.org/10.1007/s11139-018-0042-7