An elementary algorithm for the automatic derivation and proof of tensor product identities via computer algebra

ABSTRACT

Tensor product identities in two variables are quite common in mathematics: Exponential, logarithmic, trigonometric, and hyperbolic functions all satisfy tensor product identities, and the binomial theorem is a familiar example of a tensor product identity for polynomial functions. This article presents a new elementary technique which can derive and prove all of these identities---automatically! This unified approach is based on the author's recent research on uniqueness theory for dual asymptotic expansions and remainder theory for Taylor interpolation on two lines. These results provide a simple iterative algorithm which derives a tensor product from a closed-form expression using the author's asymptotic splitting operator, and a simple hyperbolic eigenfunction criterion which proves that the two forms are identically equal. The author has implemented these methods as a complete derivation and proof system in the Maple 8 computer algebra system. The Maple code, which is surprisingly brief, is included in its entirety. The article also includes numerous examples which illustrate a variety of novel techniques for deriving and proving tensor product identities using this simple but effective system. (Maple is a registered trademark of Waterloo Maple Inc.

REFERENCES

• 1: F. W. Chapman. Theory and Applications of Dual Asymptotic Expansions. Master's thesis, Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada, May 1998. Supervised by Professor Keith O. Geddes. Available through University Microfilms International. 178 pages.
• 2: F. W. Chapman. Generalized Orthogonal Series for Natural Tensor Product Interpolation. Doctoral thesis, Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada, May 2003. Supervised by Professor Keith O. Geddes. Available through University Microfilms International. 300+ pages.
• 3: Frédéric Chyzak , Bruno Salvy, Non-commmutative elimination in ore algebras proves multivariate identities, Journal of Symbolic Computation, v.26 n.2, p.187-227, Aug. 1998 [doi>10.1006/jsco.1998.0207]
• 4: Maple 8 Getting Started Guide. Webcom Ltd., Toronto, 2002. Prepared by Waterloo Maple Inc. and based in part on the work of B. W. Char.
• 5: Maple 8 Learning Guide. Webcom Ltd., Toronto, 2002. Prepared by Waterloo Maple Inc. and based in part on the work of B. W. Char.
• 6: M. B. Monagan, K. O. Geddes, K. M. Heal, G. Labahn, S. M. Vorkoetter, J. McCarron, and P. DeMarco. Maple 8 Advanced Programming Guide. Webcom Ltd., Toronto, 2002.
• 7: M. B. Monagan, K. O. Geddes, K. M. Heal, G. Labahn, S. M. Vorkoetter, J. McCarron, and P. DeMarco. Maple 8 Introductory Programming Guide. Webcom Ltd., Toronto, 2002.
• 8: Doron Zeilberger, A holonomic systems approach to special functions identities, Journal of Computational and Applied Mathematics, v.32 n.3, p.321-368, Nov. 27, 1990 [doi>10.1016/0377-0427(90)90042-X]

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION

Additional Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Algorithm design and analysis

General Terms:
Algorithms, Experimentation, Theory

Keywords:
Taylor interpolation, asymptotic splitting operator, automatic derivation, automatic proof, binormal forms, bivariate identities, characteristic cauchy problem, computer algebra, differential equations, dual asymptotic expansions, homogeneous hyperbolic eigenproblem, hyperbolic eigenfunction, lie groups, maple, symmetries, tensor products