Digits and Beyond

  • Helmut Prodinger
Conference paper
Part of the Trends in Mathematics book series (TM)


This is a survey about digits from a personal point of view. Counting the occurrences of digits (and, more generally, subblocks) is discussed in the context of various positional number systems. The methods to achieve this are Delange's elementary method and Flajolet’s idea to use the Mellin-Perron summation (or integral) formula.

Then we move to problems from Theoretical Computer Science (register function of binary trees, number of exchanges in Baxter’s odd-questions) are also discussed. An open problem from physicists Yekutieli and Mandelbrot can also be treated in that fashion.

Furthermore, we consider representations of numbers where some digits are forbidden. As a representative example, we discuss the Cantor distribution and its moments, asymptotically analyzed by Mellin transforms. Other problems in this context lead to sums involving Bernoulli numbers which can be attacked by analytic Depoissonization.

Very briefly we mention carry propagation, mergesort parameters and jump interpolation search trees.


Binary Tree Dirichlet Series Gray Code Probability Letter Exponential Generate Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    F. Bassino and H. Prodinger. (q 6)-numeration systems with missing digits. In preparation 2002.Google Scholar
  2. [2]
    W. M. Chen, H. K. Hwang, and G. H. Chen. The cost distribution of queuemergesort, optimal mergesorts, and power-of-2 rules. J. Algorithms 30:423–448, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    H. Delange. Sur la fonction sommatoire de la fonction somme des chiffres. Enseignement Mathématique 21:31–47, 1975.MathSciNetzbMATHGoogle Scholar
  4. [4]
    L. Devroye and P. Kruszewski. A note on the Horton—Strahler number for random trees. Inform. Process. Lett. 56:95–99, 1995.MathSciNetCrossRefGoogle Scholar
  5. [5]
    L. Devroye and P. Kruszewski. On the Horton—Strahler number for random tries. RAIRO Inform. Théor. Appl. 30:443–456, 1996.MathSciNetGoogle Scholar
  6. [6]
    H. G. Diamond and B. Reznick. Problems and solutions. American Mathematical Monthly 106:175–176, 1999.MathSciNetCrossRefGoogle Scholar
  7. [7]
    G. Doetsch. Handbuch der Laplace-Transformation 3 volumes. Birkhäuser Verlag, Basel, 1972.zbMATHGoogle Scholar
  8. [8]
    P. Flajolet and M. Golin. Mellin transforms and asymptotics. The mergesort recurrence. Acta Inform. 31:673–696, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science 144:3–58, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger, and R. Tichy. Mellin transforms and asymptotics: Digital sums Theoretical Computer Science 123:291–314, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    P. Flajolet and A. Odlyzko. Singularity analysis of generating functions. SIAM Journal on Discrete Mathematics 3:216–240, 1990.MathSciNetCrossRefGoogle Scholar
  12. [12]
    P. Flajolet and H. Prodinger. Register allocation for unary—binary trees. SIAM J. Comput. 15:629–640, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    P. Flajolet and L. Ramshaw. A note on Gray code and odd—even merge. SIAM J. Comput. 9(1):142–158, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    P. Flajolet, J.-C. Raoult, and J. Vuillemin. The number of registers required for evaluating arithmetic expressions. Theoretical Computer Science 9:99–125, 1979.MathSciNetCrossRefGoogle Scholar
  15. [15]
    P.Flajolet and R. Sedgewick. Mellin transforms and asymptotics: Finite differences and Rice’s integrals. Theoretical Computer Science 144:101–124, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    P. Grabner, C. Heuberger, and H. Prodinger. Subblock occurrences in signed digit representations. Submitted. Google Scholar
  17. [17]
    P. Grabner, P. Kirschenhofer, and H. Prodinger. The sum—of—digits function for complex bases. J. London Math. Soc. 57:20–40, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    P.Grabner and H. Prodinger. Asymptotic analysis of the moments of the Cantor distribution. Statistics and Probability Letters 26:243–248, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete mathematics. A foundation for computer science. Addison-Wesley, second edition, 1994.Google Scholar
  20. [20]
    U. Güntzer and M. Paul. Jump interpolation search trees and symmetric binary numbers. Information Processing Letters 26:193–204, 1987/88.Google Scholar
  21. [21]
    C. Heuberger and H. Prodinger. Carry propagation in signed digit representations. submitted 2001.Google Scholar
  22. [22]
    C. Heuberger and H. Prodinger. On minimal expansions in redundant number systems: Algorithms and quantitative analysis. Computing 66:377–393, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    J. R. M. Hosking. Moments of order statistics of the Cantor distribution. Statistics and Probability Letters 19:161–165, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    P. Jacquet and W. Szpankowski. Analytical de-Poissonization and its applications. Theoretical Computer Science 201:1–62, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    R. Kemp. The average number of registers needed to evaluate a binary tree optimally. Acta Inform. 11:363–372, 1978/79.Google Scholar
  26. [26]
    P. Kirschenhofer and H. Prodinger. Subblock occurrences in positional number systems and Gray code representation. Journal of Information and Optimization Sciences 5:29–42, 1984 . MathSciNetzbMATHGoogle Scholar
  27. [27]
    A. Knopfmacher and H. Prodinger. Exact and asymptotic formule for average values of order statistics of the Cantor distribution. Statistics and Probability Letters 27:189–194, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    D. E. Knuth . The Art of Computer Programming volume 3: Sorting and Searching. Addison-Wesley 1973. Second edition, 1998.Google Scholar
  29. [29]
    D. E. Knuth. The average time for carry propagation. Indagationes Mathematicce 40:238–242, 1978.MathSciNetGoogle Scholar
  30. [30]
    P. Kruszewski. A note on the Horton-Strahler number for random binary search trees. Inform. Process. Lett. 69:47–51, 1999.MathSciNetCrossRefGoogle Scholar
  31. [31]
    F. R. Lad and W. F. C. Taylor. The moments of the Cantor distribution. Statistics and Probability Letters 13:307–310, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    A. Meir, J. W. Moon, and J. R. Pounder. On the order of random channel networks. SIAM J. Algebraic Discrete Methods 1:25–33, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    J. W. Moon. An extension of Horton’s law of stream numbers. Math. Colloq. Univ. Cape Town 12:1–15 (1980), 1978/79.Google Scholar
  34. [34]
    J. W. Moon. On Horton’s law for random channel networks. Ann. Discrete Math. 8:117–121, 1980. Combinatorics 79 (Proc. Colloq., Univ. Montréal, Montreal, Que., 1979), Part I.Google Scholar
  35. [35]
    M. Nebel. On the Horton-Strahler number for combinatorial tries. Theor. Inform. Appl. 34:279–296, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    A. H. Osbaldestin and P. Shiu. A correlated digital sum problem associated with sums of three squares. Bull. London Math. Soc. 21:369–374, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    W Panny and H. Prodinger. Bottom-up mergesort—a detailed analysis. Algorithmica 14:340–354, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    H. Prodinger. Some analytic techniques for the investigation of the asymptotic behaviour of tree parameters. EATCS Bulletin 47:180–199, 1992.MathSciNetzbMATHGoogle Scholar
  39. [39]
    H. Prodinger. On a problem of Yekutieli and Mandelbrot about the bifurcation ratio of binary trees. Theoretical Computer Science 181:181–194, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    H. Prodinger. On binary representations of integers with digits —1, 0, 1. Integers pages A8, 14 pp. (electronic), 2000.Google Scholar
  41. [41]
    H. Prodinger. On Cantor’s singular moments. Southwest J. Pure Appl. Math. 2000(1):27–29 (electronic), 2000.MathSciNetzbMATHGoogle Scholar
  42. [42]
    G. Reitwiesner. Binary arithmetic. Vol. 1 of Advances in Computers Academic Press pages 231–308, 1960.Google Scholar
  43. [43]
    M. Riedel. Applications of the Mellin—Perron formula in number theory. Technical report, University of Toronto, 1996.Google Scholar
  44. [44]
    R. Sedgewick. Data movement in odd—even merging. SIAM J. Comput. 7:239–272, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    L. A. Shepp and S. P. Lloyd. Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121:340–357, 1966.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    P. Shiu. Counting sums of three squares. Bull. London Math. Soc. 20:203–208, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    J. Thuswaldner. Summatory functions of digital sums occurring in Cryptography. Periodica Math. Hungarica 38:111–130, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    E. T. Whittaker and G. N. Watson. A Course of Modern Analysis. Cambridge University Press, fourth edition, 1927. Reprinted 1973.Google Scholar
  49. [49]
    I. Yekutieli and B. Mandelbrot. Horton—Strahler ordering of random binary trees. J. Phys. A 27:285–293, 1994.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Helmut Prodinger
    • 1
  1. 1.The John Knopfmacher Centre for Applicable Analysis and Number Theory School of MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

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