<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 6 Issue 1" %>
Polynomial Sequences of Binomial Type and Path Integrals
Vladimir V. Kisil
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
Annals of Combinatorics 6 (1) p.45-56 March, 2002
AMS Subject Classification: 05A40, 05A15, 58D30, 81Q30, 81R30, 81S40
Polynomial sequences of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express as a path integral in the ``phase space'' . The Hamiltonian is and it produces a Schrödinger type equation for . This establishes a bridge between enumerative combinatorics and quantum field theory. It also provides an algorithm for parallel quantum computation.
Keywords: Feynman path integral, umbral calculus, polynomial sequence of binomial type, token, Schrodinger equation, propagator, wave function, cumulants, quantum computation


1. V.I. Arnold, Mathematical Methods of Classic Mechanics, Springer-Verlag, Berlin, 1991.

2. J. Cigler, Some remarks on Rota's umbral calculus, Nederl. Akad.Wetensch. Proc. Ser. A 81 (1978) 27每42.

3. Y.A. Daletski, Path integrals connected with operator evolutionary equations, Uspehi Mat. Nauk 17 (5(107)) (1962) 3每115.

4. A. Dynin, A rigorous path integral construction in any dimension, Lett. Math. Phys. 44 (1998) 317每329.

5. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integral, McGraw-Hill Book Company, New York, 1965.

6. I.M. Gel'fand and A.M. Jaglom, Integration in functional spaces and its applications in quantum physics, J. Math. Phys. 1 (1960) 48每69.

7. M. Henle, Binomial enumeration on dissects, Trans. Amer. Math. Soc. 202 (1975) 1每39.

8. V.V. Kisil, An algebraic construction common in combinatorics, analysis, and physics, preprint LEEDS-MATHS-PURE-2002-02, 2002, E-print arXiv:math.FA/0201012.

9. V.V. Kisil, Relativistic quantization and improved equation for a free relativistic particle, Phys. Essays 11 (1) (1998) 69每80.

10. V.V. Kisil, Wavelets in Banach spaces, Acta Appl. Math. 59 (1) (1999) 79每109.

11. V.V. Kisil, Umbral calculus and cancellative semigroup algebras, Z. Anal. Anwendungen 19 (2) (2000) 315每338.

12. D.E. Knuth, Convolution polynomials, preprint, http://www-cs-faculty.stanford.edu/knuth/papers/cp.tex.gz.

13. J.P.S. Kung, Ed., Gian-Carlo Rota on Combinatorics: Introductory Papers and Commentaries, Contemporary Mathematicians, Vol. 1, Birkhäuser Verlag, Boston, 1995.

14. Y.I. Manin, Classical computing, quantum computing, and Shor's factoring algorithm, Ast谷risque 266 (2000) Exp. No. 862, 5, 375每404, S谷minaire Bourbaki, Vol. 1998/99.

15. L. Mattner, What are cumulants? Doc. Math. 4 (1999) 601每622.

16. R. Mullin and G.-C. Rota, On the foundation of combinatorial theory (III): theory of binomial enumeration, In: Graph Theory and Its Applications, B. Harris, Ed., Academic Press, Inc., New York, 1970, pp. 167每213, reprinted in [13, pp. 118每147].

17. M. Reed and B. Simon, Fourier Analysis, Self-Adjointness, Methods of Modern Mathematical Physics, Vol. 2, Academic Press, New York, 1975.

18. S. Roman and G.-C. Rota, The umbral calculus, Adv. Math. 27 (1978) 95每188.

19. G.-C. Rota, The number of partitions of a set, Amer. Math. Monthly 71 (5) (1964) 498每504, reprinted in [20, pp. 1每6] and [13, pp. 111每117].

20. G.-C. Rota, Finite Operator Calculus, Academic Press, Inc., New York, 1975.

21. G.-C. Rota, D. Kahaner, and A. Odlyzko, Finite operator calculus, J. Math. Anal. Appl. 42 (3) (1973) 685每760, reprinted in [20, pp. 7每82].

22. L.H. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge, 2nd Edition, 1996.

23. S.V. Shabanov and J.R. Klauder, Path integral quantization and Riemannian-symplectic manifolds, Phys. Lett. B 435 (3/4) (1998) 343每349.

24. M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987.

25. B. Simon, Functional Integration and Quantum Physics, Academic Press Inc., New York, 1979.

26. M.E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, Vol. 34, Princeton University Press, Princeton, New Jersey, 1981.