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On the Moments of the Sum-of-Digits Function

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Applications of Fibonacci Numbers

Abstract

In a recent paper in The Fibonacci Quarterly R.E. Kennedy and C.N. Cooper [12] dealt with the second moment of the sum-of-digits function in the decimal number system, explicitly stating the question for higher moments as an open problem. We consider this question in the sequel.

These authors were supported by the Austrian Science Foundation Project Nr. P8274-PHY.

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References

  1. Apostol, T.M. Introduction to Analytic Number Theory, Spring Verlag, Berlin-Heidelberg-New York, 1984.

    Google Scholar 

  2. Brillhart, J., Erdös, P. and Morton, P. “On Sums of Rudin-Shapiro coefficients II.” Pacific J. Math., Vol. 707(1983): pp. 39–69.

    Article  Google Scholar 

  3. Coquet, J. “A Summation Formula Related to the Binary Digits.” Invent. Math., Vol. 73 (1983): pp. 107–115.

    Article  MathSciNet  MATH  Google Scholar 

  4. Coquet, J. “Power Sums of Digital Sums.” J. Number Th., Vol. 22 (1986): pp. 161–176.

    Article  MathSciNet  MATH  Google Scholar 

  5. Delange, H. Sur la fonction sommatoire de la fonction “Somme des Chiffres.” Enseignement Math. (2), Vol. 21 (1975): pp. 31–47.

    MathSciNet  MATH  Google Scholar 

  6. Doetsch, G. Handbuch der Laplace Transformation. Birkhäuser Verlag, Basel, 1950.

    MATH  Google Scholar 

  7. Flajolet, P., Grabner, P.J., Kirschenhofer, P., Prodinger, H. and Tichy, R.F. “Mellin Transform and Asymptotics: Digital Sums.” Theor. Comput. Sci., to appear.

    Google Scholar 

  8. Flajolet, P. and Ramshaw, L. “A Note on Gray Code and Odd-Even Merge.” SIAM J. Comput., Vol. 9 (1980): pp. 142–158.

    Article  MathSciNet  MATH  Google Scholar 

  9. Flajolet, P., Raoult, J.C. and Vuillemin, J. “The Number of Registers Required for Evaluating Arithmetic Expressions.” Theor. Comput. Sci., Vol. 9 (1979): pp. 99–125.

    Article  MathSciNet  MATH  Google Scholar 

  10. Flajolet, P. and Vitter, J. “Average-Case Analysis of Algorithms and Data Structures,” Handbook of Theoretical Computer Science Vol. A “Algorithms and Complexity.” North-Holland, (1990): pp. 431–524.

    Google Scholar 

  11. Grabner, P.J. and Tichy, R.F. “α-Expansions, Linear Recurrences and the Sum-of-Digits Function.” Manuscripia Math., Vol. 70 (1991): pp. 311–324.

    Article  MathSciNet  MATH  Google Scholar 

  12. Kennedy, R.E. and Cooper, C.N. “An Extension of a Theorem by Cheo and Yien Concerning Digital Sums.” The Fibonacci Quarterly, Vol. 29 (1991): pp. 145–149.

    MathSciNet  MATH  Google Scholar 

  13. Kirschenhofer, P. “Subblock Occurrences in the q-ary Representation of n.” SI AM J. Alg. Discr. Math., Vol. 4 (1983): pp. 231–236.

    Article  MathSciNet  MATH  Google Scholar 

  14. Kirschenhofer, P. “On the Variance of the Sum of Digits Function.” Number-Theoretic Analysis, Lecture Notes in Mathematics 1452 (E. Hlawka and R.F. Tichy, editors.) (1990): pp. 112–116.

    Google Scholar 

  15. Kirschenhofer, P. and Prodinger, H. “Subblock Occurrences in Positional Number Systems and Gray Code Representation.” J. Inf. Opt. Sci., Vol. 5 (1984): pp. 29–42.

    MathSciNet  MATH  Google Scholar 

  16. Kirschenhofer, P., Prodinger, H. and Tichy, R.F. “Über die Ziffernsumme natürlicher Zahlen und verwandte Probleme.” Zahlentheoretische Analysis, Lecture Notes in Mathematics 1114 (E. Hlawka, editor) (1985): pp. 55–65.

    Google Scholar 

  17. Kirschenhofer, P. and Tichy, R.F. “On the distribution of digits in Cantor representations of integers.” J. Number Th., Vol. 18 (1984): pp. 121–134.

    Article  MathSciNet  MATH  Google Scholar 

  18. Mauclaire, J.-L. and Murata, L. “On q-Additive Functions.” Proc. Japan Acad., Vol. 59, Series A (1987): pp. 441–444.

    MathSciNet  Google Scholar 

  19. Prodinger, H. “Generalizing the sum of digits function.” SIAM J. Alg. Discr. Math., Vol. 5(1982): pp. 35–42.

    MathSciNet  Google Scholar 

  20. Riordan, J. An Introduction to Combinatorial Theory. J. Wiley, New York, 1958.

    Google Scholar 

  21. Stolarsky, K.B. “Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity.” SIAMJ. Appl. Math., Vol. 32 (1977): pp. 717–730.

    Article  MathSciNet  MATH  Google Scholar 

  22. Trollope, H. “An explicit expression for binary digital Sums.” Math. Mag., Vol. 41 (1968): pp. 21–25.

    Article  MathSciNet  MATH  Google Scholar 

  23. Whittaker, E.T. and Watson, G.N. A Course in Modern Analysis, Cambridge University Press, 1927.

    Google Scholar 

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Authors
  1. P. J. Grabner
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  2. P. Kirschenhofer
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  3. H. Prodinger
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  4. R. F. Tichy
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Editor information

Editors and Affiliations

  1. Computer Science Department, South Dakota State University, Box 2201, Brookings, South Dakota, 57007-0194, USA

    Gerald E. Bergum

  2. Ministry of Education, Government House Z 50, Nicosia, Cyprus

    Andreas N. Philippou

  3. Department of Mathematics, Statistics and Computer Science, University of New England, Armidale, New South Wales, 2351, Australia

    Alwyn F. Horadam

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© 1993 Springer Science+Business Media Dordrecht

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Grabner, P.J., Kirschenhofer, P., Prodinger, H., Tichy, R.F. (1993). On the Moments of the Sum-of-Digits Function. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2058-6_25

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  • DOI: https://doi.org/10.1007/978-94-011-2058-6_25

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4912-2

  • Online ISBN: 978-94-011-2058-6

  • eBook Packages: Springer Book Archive

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