Skip to main content

Method of Coefficients: an algebraic characterization and recent applications

  • Conference paper
  • First Online:
Advances in Combinatorial Mathematics

Abstract

The article is devoted to the algebraic-logical foundations of the analytic approach to summation problems in various fields of mathematics and its applications. Here we present the foundations of the method of coefficients developed by the author in late 1970’s and its recent applications to several well-known problems.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramov S.A. and Tsarev S.P. (1997). Peripheral factorization of linear ordinary operators, Programming & Computer Software, No. 1, p. 59–67.

    Google Scholar 

  2. Aizenberg L.A. and Yuzhakov A.P. (1979). Integral representation and residues in multidimensional complex analysis. Nauka, Novosibirsk (in Russian).

    Google Scholar 

  3. Andrews G.E. (1970). On the foundations of combinatorial theory. IV. Finite vector space and Eulerian generating functions. Stud. Appl. Math. 49, 239–258.

    Google Scholar 

  4. Antipova I.A. (2001). Mellin transforms for superposition of the general algebraic functions. Proc. Intern. Conf. “Mathematical models and methods of their investigations”, vol. 1, Krasn. State Univ., Krasnoyarsk, 31–35 (in Russian).

    Google Scholar 

  5. Arnold V.I. (2005). Dynamics, statistic and projective geometry of Galois fields. Publ. MCCME, M. (in Russian).

    Google Scholar 

  6. Barnabei M., Brini A. and Nicoletti G. (1982). Recursive matrices and umbral calculus. J. Algebra 75, 546–573.

    Article  MATH  MathSciNet  Google Scholar 

  7. Balser W. (1994). From divergent power series to analytic functions. Theory and application of multisummable power series. Lecture Notes in Mathematics, 1582, Springer-Verlag, Berlin.

    Google Scholar 

  8. Bateman G. and Erdélyi A. (1955). Higher transcendental functions, vol. 3: Chapter 19. Mc Graw-Hill Comp., New York.

    MATH  Google Scholar 

  9. Belousov V.D. (1967). Foundations of the theory of quasigroups and loops. Nauka, Moscow, 223 pages (in Russian).

    Google Scholar 

  10. Bertozzi A. and McKenna J. (1993). Multidimensional residues, generating functions, and their application to queueing netwoks. SIAM Review 35: 2, 239–268.

    Article  MATH  MathSciNet  Google Scholar 

  11. Campello de Souza R.M., de Oliveira H.M. and Silva D. (2002). The Z transforms over Finite Fields. Intern. Telecom. Symp. – ITS2002, Natal, Brasil, 6 pages.

    Google Scholar 

  12. Cao Z.Q., Kim K.H. and Roush F.W. (1984). Incline Algebra and Applications. John Wiley, New York.

    MATH  Google Scholar 

  13. Cuninghame-Green R.A. (1979). Minimax Algebra. Lect. Notes in Economics and Mathematical Systems 166, Springer, Berlin.

    MATH  Google Scholar 

  14. Carlitz L. (1932). The arithmetic of polynomials in a Galois Field. Amer. J. Math. 54, 39–50.

    Article  MathSciNet  Google Scholar 

  15. Cartan H. (1961). Théorie élémentaire des fonctions analytiques d’une on plusieurs variables complexes. Hermann, Paris.

    Google Scholar 

  16. Chamberland M. and Dilcher K. (2006). Divisibility properties of a class of binomial sums. J. Number Theory 120, 349–371.

    Article  MATH  MathSciNet  Google Scholar 

  17. Chen W.Y.C., Qin J., Reidys C.M. and Zeilberger D. (2008). Efficient counting and asymptotic of k-noncrossing tangled-diagrams. Electron. J. Combin. 16 (2009), no. 1, Research Paper 37.

    Google Scholar 

  18. Cheng S.E. (2003). Generating function proofs of identities and congruences. PhD thesis, Michigan State Univ., Michigan, 86 pages.

    Google Scholar 

  19. Consul P.C. and Famoye F. (2006). Lagrangian probability distributions. Birkhäuser Boston Inc., Boston, MA.

    MATH  Google Scholar 

  20. Cooke D.J. and Bez H.E. (1984). Computer mathematics. Cambridge Univ. Press, Cambridge.

    MATH  Google Scholar 

  21. Dickson L.E. (1966). History of the Theory of Numbers, vol. 1, Chelsea Publishing Co., New York.

    Google Scholar 

  22. Dingle R.B. (1973). Asymptotic expansions: their derivation and interpretation. Acad. Press, New York.

    MATH  Google Scholar 

  23. Deng Y. (2006). A class of combinatorial identities. Discrete Math. 306, 2234–2240.

    Article  MATH  MathSciNet  Google Scholar 

  24. Doubilet P., Rota G.-C. and Stanley R. (1972). On the foundations of combinatorial theory. VI: The idea of generating function. In: Proc. Sixth Berkeley Sympos. on Math. Stat. and Prob. (1970/71): vol. II. Prob. Theory, Univ. California Press, Berkeley, CA, 267–318.

    Google Scholar 

  25. Egorychev G.P. (1977). Integral representation and the computation of combinatorial sums. Novosibirsk, Nauka (in Russian); English: Transl. of Math. Monographs 59, AMS, 1984, 2-nd Ed. in 1989.

    Google Scholar 

  26. Egorychev G.P. (2000). Algorithms of integral representation of combinatorial sums and their applications. Proc. of 12-th Intern. Conf. on Formal Power Series and Algebraic Combinatorics (FPSAC 2000), Moscow, Russia, June 2000, 15–29.

    Google Scholar 

  27. Egorychev G.P. (2004). Solution of the Margenstein-Matiyasevich’s question in 3x+1 problem. Preprint ISBN 5-7636-0632-9, Krasnoyarsk State Technical Univ., Krasnoyarsk, 12 pages (in Russian).

    Google Scholar 

  28. Egorychev G.P. and Zima E.V. (2004). The characteristic function in 3x+1 problem. Proc. Intern. School-Seminare “Synthesis and Complexity of Management Systems”, Math. Inst. of Sib. Branch of Russian Acad. Nauk, Novosibirsk, 34–40 (in Russian).

    Google Scholar 

  29. Egorychev G.P. and Zima E.V. (2005). Decomposition and group theoretic characterization of pairs of inverse relations of the Riordan type. Acta Appl. Math. 85, 93–109.

    Article  MATH  MathSciNet  Google Scholar 

  30. Egorychev G.P. (2008). Discrete Mathematics. Permanents. Sib. Federal Univ., Krasnoyarsk, 272 pages (in Russian).

    Google Scholar 

  31. Egorychev G.P. and Zima E.V. (2008). Integral representation and algorithms for closed form summation. Handbook of Algebra, vol.5, (ed. M. Hazewinkel), Elsevier, 459–529.

    Google Scholar 

  32. Egorychev G.P. (2008). Method of coefficients: an algebraic characterization and recent applications. Issues of VII Intern. School-conf. of Theory Group (Cheljabinsk, Russia, August 3–9, 2008), Inst. Math. and Mech. Ural. Otdel. RAN, Ekaterinburg, 2 pages.

    Google Scholar 

  33. Egorychev G.P. and Zima E.V. (2008). Collatz conjecture from the integral representation point of view. Inst. Math. and Mech. Ural. Otdel. RAN, 12 pages (to appear).

    Google Scholar 

  34. Evgrafov M.A. (1968). Analytical functions. Nauka, M.

    Google Scholar 

  35. Evgrafov M.A. (1986). Series and integral representations. Itogi Nauki i Techniki. Sovr. Problems Mat., Fund. Napr. 13, VINITI, M., 5–92 (in Russian).

    Google Scholar 

  36. Flajolet P. and Salvy B. (1998). Euler sums and contour integral representations. Experiment. Math. 7, 15–35.

    MATH  MathSciNet  Google Scholar 

  37. Flajolet P. and Sedgewick R. (2007). Analytic Combinatorics. Cambridge University Press, Cambridge, 2009.

    Google Scholar 

  38. Gerhard J., Giesbrecht M., Storjohann A. and Zima E. (2003). Shiftless decomposition and polynomial-time rational summation. Proc. of ISSAC 2003, ACM Press, 119–126.

    Google Scholar 

  39. Gessel I.M. (1980). A noncommutative generalization and q-analog of the Lagrange inversion formula. Trans. Amer. Math. Soc. 257, 455–482.

    MATH  MathSciNet  Google Scholar 

  40. Gessel I.M. (1997). Generating functions and generalized Dedekind sums. Elec. J. Comb., 4, Wilf Festschrift, R11.

    Google Scholar 

  41. Greene D.H. and Knuth D.E. (1981). Mathematics for the analysis of algorithms. Birkhäuser, Boston.

    MATH  Google Scholar 

  42. Gosper R.W. (1978). Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 40–42.

    Article  MATH  MathSciNet  Google Scholar 

  43. Gould H.W. (1972). Combinatorial identities. A standardized set of tables listing 500 binomial coefficient summations. Morgantown, W.Va.

    Google Scholar 

  44. Goulden I.P. and Jackson D.M. (1983). Combinatorial enumeration. John Wiley, New York.

    MATH  Google Scholar 

  45. Han H.S.W. and Reidys C.M. (2008). Pseudoknot RNA structures with arc-length ≥4. J. Comput. Biol. 15, no. 9, 1195–1208.

    Article  MathSciNet  Google Scholar 

  46. Hardy G.H. (1949). Divergent series. Clarendon Press, Oxford.

    MATH  Google Scholar 

  47. Henrici P. (1991). Applied and computational complex analysis. John Wiley, New York.

    MATH  Google Scholar 

  48. Howe R. (1974). The Fourier transform and germs of characters (case of Gl n over p-adic field). Math. Ann. 208, 305–322.

    Article  MATH  MathSciNet  Google Scholar 

  49. Huang I-Ch. (1997). Applications of residues to combinatorial identities. Proc. Amer. Math. Soc. 125: 4, 1011–1017.

    Article  MATH  MathSciNet  Google Scholar 

  50. Huang I-Ch. (1998). Reversion of power series by residues. Comm. Algebra, 26, 803–812.

    Article  MATH  MathSciNet  Google Scholar 

  51. Huang I-Ch. (2002). Inverse relations and Schauder bases. J. Combin. Theory Series A 97, 203–224.

    Article  MATH  Google Scholar 

  52. Joni S.A. (1978). Lagrange inversion in higher dimensions and umbral operators. Lin. and Mult. Algebra 6, 111–121.

    Article  MATH  MathSciNet  Google Scholar 

  53. Krattenthaler Ch. (1984). A new q-Lagrange formula and some applications. Proc. Amer. Math. Soc. 90, 338–344.

    MATH  MathSciNet  Google Scholar 

  54. Krattenthaler Ch. (1988). Operator methods and Lagrange inversion: a unified approach to Lagrange formulas. Trans. Amer. Math. Soc. 305, 431–465.

    Article  MATH  MathSciNet  Google Scholar 

  55. Krattenthaler Ch. (1996). A new matrix inverse. Proc. Amer. Math. Soc. 124, 47–59.

    Article  MATH  MathSciNet  Google Scholar 

  56. Krattenthaler Ch. and Schlosser M. (1999). A new multidimensional matrix inverse with applications to multiple q-series. Discrete Math. 204, 249–279.

    Article  MATH  MathSciNet  Google Scholar 

  57. Krivokolesko V.P. and Tsikh A.K. (2005). Integral representations in linearly convex polyhedra. Sib. Math. Journal 46: 3, 579–593 (in Russian).

    MathSciNet  Google Scholar 

  58. Krivokolesko V.P. (2008). About an integral representation, 59 pages (to appear).

    Google Scholar 

  59. Kurosh A.G. (1973). Lecture of general algebra. Nauka, M. (in Russian).

    Google Scholar 

  60. Lagarias J.E. (1997). The 3x+1 Problem and its Generalizations, In: Borwein J. et al. (Eds.), Organic mathematics. Proc. workshop Simon Fraser Univ., Barnaby, Canada, Dec. 12-14, 1995; AMS, Providence, RI, 305–334.

    Google Scholar 

  61. Leinartas E.K. (1989). The Hadamard multidimensional composition and sums with linear constraints on summation indices. Sib. Mat. J. 30: 4, 102–107 (in Russian).

    MathSciNet  Google Scholar 

  62. Leinartas E.K. (2006). Integral methods in multiple theory of power series and difference equations. PhD thesis, Krasnoyarsk State Univ., Krasnoyarsk, 156 pages (in Russian).

    Google Scholar 

  63. Leont’ev A.F. (1980). The sequences of exponential polynomials. Nauka, M. (in Russian).

    Google Scholar 

  64. Leont’ev V.K. (2006). On the roots of random polynomials over a finite field. Math. Zametki, 80: 2, 300–304 (in Russian).

    MATH  MathSciNet  Google Scholar 

  65. Liu, Y. (1999). Enumerative theory of maps. Mathematics and its Applications, 468. Kluwer Academic Publishers, Dordrecht; Science Press, Beijing.

    MATH  Google Scholar 

  66. López B., Marco J.M. and Parcet J. (2006) Taylor series and the Askey – Wilson operator and classical summation formulas. Proc. Amer. Math. Soc. 134: 8, 2259–2270.

    Article  MATH  MathSciNet  Google Scholar 

  67. Lushnikov A.A. (2005). Exact kinetics of the sol-gel transition. Phys. Rev. E 71, 0406129-1–0406129-10.

    Article  Google Scholar 

  68. Lushnikov A.A. (2006). Gelation in coagulating systems. Phys. D 222, 37–53.

    Article  MATH  MathSciNet  Google Scholar 

  69. MacMahon P.A. (1915–1916). Combinatory analysis. Vol. I, II. Cambridge Univ. Press.

    Google Scholar 

  70. Mandelbrojt S. (1973). Séries de Dirichlet. Principes et méthodes. Mir, Moscow (in Russian).

    Google Scholar 

  71. Margenstern M. and Matiyasevich Y. (1999). A binomial representation of the 3x+1 problem, Acta Arith. 91, 367–378.

    MATH  MathSciNet  Google Scholar 

  72. Marichev O.I. (1983). Handbook of integral transforms of higher transcendental functions. Theory and algorithmic tables. Ellis Horwood Limited.

    Google Scholar 

  73. Materov E.N. and Yuzhakov A.P. (2000). The Bott formula for toric varieties and some combinatorial identities. Complex analysis and differential operators, Krasnoyarsk, 85–92.

    Google Scholar 

  74. Materov E.N. (2002). The Bott formula for toric varieties. Mosc. Math. J. 2, no. 1, 161–182, 200.

    MATH  MathSciNet  Google Scholar 

  75. Merlini D., Sprugnoli R. and Verri M.C. (2007). The method of coefficients. Amer. Math. Monthly 114, 40–57.

    MATH  MathSciNet  Google Scholar 

  76. Mitrinović D.S. and Kečkić J.D. (1984). Cauchy method of residues. Theory and applications. Vol. I, II. Kluwer Acad Press.

    Google Scholar 

  77. Morrison K.E. (2006). Integer Sequences and Matrices Over Finite Fields. J. Integer Sequences 9, Article 06.2.1, 28 pages.

    Google Scholar 

  78. Odlyzko A.M. (1995). Asymptotic enumeration methods. Handbook of combinatorics, Vol. 1, 2, 1063–1229, Elsevier, Amsterdam.

    Google Scholar 

  79. Paule P. (1990). Computer Algebra Algorithmen für q-Reihen und kombinatorische Identitaten. RISC Linz, No 90-02.0, .25 pages.

    Google Scholar 

  80. Paule P. (1995). Greatest factorial factorization and symbolic summation. J. Symbolic Comput. 20, 235–268.

    Article  MATH  MathSciNet  Google Scholar 

  81. Petkovšek M., Wilf H.S. and Zeilberger D. (1996). A=B. A K Peters, Wellesley, MA.

    Google Scholar 

  82. Plamenevskii B.A. (1986). Algebras of pseudo-differential operators. Nauka, M., 256 pages (in Russian).

    Google Scholar 

  83. Pollard J.M. (1971). The Fast Fourier Transform in a Finite Field. Math. Comp. 25: 365–374.

    Article  MATH  MathSciNet  Google Scholar 

  84. Pólya G. (1937). Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math. 68, 145–254.

    Article  MATH  Google Scholar 

  85. Postnikov M.M. (1963). Foundations of Galois theory. Fizmatlit, Moscow (in Russian).

    Google Scholar 

  86. Prudnikov A.P., Brychkov Yu.A. and Marichev O.M. (1988). Integrals and Rings. Special functions. Vol. 1. John Wiley, Berlin.

    Google Scholar 

  87. Rademacher H. (1973). Topics in Analytic Number Theory. Springer Verlag, New York.

    MATH  Google Scholar 

  88. Ramakrishan D. and Valenza R.J. (1999). Fourier analysis on number fields. Graduate Texts in Mathematics, 186. Springer-Verlag, New York.

    Google Scholar 

  89. Riordan J. (1968). Combinatorial identities. John Wiley.

    Google Scholar 

  90. Roman S. (1984). The umbral calculus. Pure and Applied Mathematics, 111. Acad. Press, New York.

    MATH  Google Scholar 

  91. Rota G.-C. (1964). On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrsch. Verw. Gebiete 2, 340–368.

    Article  MATH  MathSciNet  Google Scholar 

  92. Sadykov T.M. (2009). Hypergeometric functions of many complex variables. PhD thesis, Siberian Federal Univ., Krasnoyarsk, 261 pages (in Russian).

    Google Scholar 

  93. Samko S.G., Kilbas A.A. and Marichev O.I. (1993). Fractional integrals and derivatives. Theory and applications. Gordon and Breach, Yverdon.

    MATH  Google Scholar 

  94. Schwatt I.J. (1962). An introduction to the operations with series. Second edition, Chelsea Publishing Co., New York.

    Google Scholar 

  95. Shabat B.V. (1969). An introduction to the complex analysis, Nauka, M. (in Russian).

    Google Scholar 

  96. Shapiro L.W., Getu S., Woan W.J. and Woodson L.C. (1991). The Riordan group. Discrete Appl. Math. 34, 229–239.

    Article  MATH  MathSciNet  Google Scholar 

  97. Sloane N.J.A. and Plouffe S. (1995). The encyclopedia of integer sequences. Acad. Press, San Diego.

    MATH  Google Scholar 

  98. Sprugnoli R. (2006). An introduction to mathematical methods in combinatorics. Dipartimento di Sistemi e Informatica Viale Morgagni, 65 – Firenze (Italy), 100 pages.

    Google Scholar 

  99. Stanley R.P. (1997, 1999). Enumerative combinatorics: Vol. I, II. Cambridge Univ. Press, Cambridge.

    Google Scholar 

  100. Stepanenko V.A. (2003). On the solution of the system of n algebraic equations with n variables with the help of hypergeometric functions. Vestnik Krasnoyarsk State Univ. 2, Krasnoyarsk, 35–48 (in Russian).

    Google Scholar 

  101. Stepanenko V.A. (2005). Systems of algebraic equations, hypergeometric functions and integrals of several rational differentials. PhD thesis, Krasnoyarsk State Univ., Krasnoyarsk, 81 pages (in Russian).

    Google Scholar 

  102. Stepanenko V.A. (2008). Further chapters of mathematical analysis. Siberian Federal Univ., Krasnoyarsk, 176 pages (in press).

    Google Scholar 

  103. Sun Y. (2004). The statistic “number of udu’s” in Dyck paths, Discrete Math. 287, 177–186.

    Article  MATH  MathSciNet  Google Scholar 

  104. Sun Z.W. and Davis M. (2007). Combinatorial congruences modulo prime powers. Trans. Amer. Math. Soc. 359: 11, 5525–5553.

    Article  MATH  MathSciNet  Google Scholar 

  105. Tefera A. (2002). MultInt, a MAPLE Package for Multiple Integration by the WZ Method. J. Symbolic Comput. 34, 329–353.

    Article  MATH  MathSciNet  Google Scholar 

  106. Titchmarsh E.C. (1951). The theory of the Riemann Zeta-function, Oxford.

    Google Scholar 

  107. Tsikh A.K. (1992). Multidimensional residues and their applications. Translations of Mathematical Monographs, 103, AMS, Providence, RI.

    MATH  Google Scholar 

  108. Ufnarovsky V.A. (1995). Combinatorial and asymptotic methods in algebra. Algebra, VI, 1–196, Encyclopaedia Math. Sci., 57, Springer, Berlin.

    Google Scholar 

  109. Wang W. and Wang T. (2009). Identities on Bell polynomials and Sheffer sequences. Discrete Math., 309, no. 6, 1637–1648.

    Article  MATH  MathSciNet  Google Scholar 

  110. Whiteman A.L. (1953). Finite Fourier Series and equations in finite fields. Trans. Amer. Math. Soc. 74, 78–98.

    MATH  MathSciNet  Google Scholar 

  111. Wilf H.S. (1989). The “Snake-Oil” method for proving combinatorial identities. Surveys in combinatorics, London Math. Soc., Lecture Note Ser. 141, Cambridge Univ. Press, Cambridge, 208–217.

    Google Scholar 

  112. Wirsching G.J. (1998). The dynamic system generated by the 3n+1 function. Lecture Notes in Math. 1681, Springer-Verlag, Berlin.

    Book  Google Scholar 

  113. Woodcock C.F. (1996). Special p-adic analytic functions and Fourier transforms. J. Theory Numbers 60, 393–408.

    Article  MATH  MathSciNet  Google Scholar 

  114. Xin G. (2005). A residue theorem for Malcev – Neumann series, Adv. in Appl. Math. 35, 271–293.

    Article  MATH  MathSciNet  Google Scholar 

  115. Xin G. (2004). The ring of Malcev – Neumann series and the residue theorem, PhD thesis, Brandeis University, Waltham, MA, USA.

    Google Scholar 

  116. Zeilberger D. (1991). The method of creative telescoping. J. Symbolic Comput. 11, 195–204.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Georgy P. Egorychev .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Egorychev, G.P. (2009). Method of Coefficients: an algebraic characterization and recent applications. In: Kotsireas, I., Zima, E. (eds) Advances in Combinatorial Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03562-3_1

Download citation

Publish with us

Policies and ethics