Applying computer algebra systems to study Chaundy-Bullard identities for the vector partition function with weight

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Abstract

An algorithm for obtaining the Chaundy-Bullard identity for a vector partition function with weight that uses computer algebra methods is proposed. To automate this process in Maple, an algorithm was developed and implemented that calculates the values of the vector partition function with weight by finding non-negative solutions of systems of linear Diophantine equations that are used to form the identities involved. The algorithm’s input data is represented by the set of integer vectors that form a pointed lattice cone and by some point from this cone, and the Chaundy-Bullard identity for the vector partition function with weight is its output. The code involved is stored in the depository and is ready-to-use. An example demonstrating the algorithm’s operation is given.

About the authors

A. B. Leinartene

Siberian Federal University

Author for correspondence.
Email: aleina@mail.ru
Russian Federation, pr. Svobodny 79, Krasnoyarsk, 660041

A. P. Lyapin

Siberian Federal University

Email: aplyapin@sfu-kras.ru
Russian Federation, pr. Svobodny 79, Krasnoyarsk, 660041

References

  1. Abramov S.A., Barkatou M.A., van Hoeij M., Petkovsek M. Subanalytic Solutions of Linear Difference Equations and Multidimensional Hypergeometric Sequences // Journal of Symbolic Computation. 2011. № 46(11). P. 1205–1228.
  2. Abramov S.A., Petkovšek M., Ryabenko A.A. Hypergeometric Solutions of First-order Linear Difference Dystems with Rational-function Coefficients // Lecture Notes in Computer Science. 2015. № 9301. P. 1–14.
  3. Abramov S.A., Barkatou M.A., Petkovšek M. Linear difference operators with coefficients in the form of infinite sequences // Comput. Math. Math. Phys. 2021. № 61(10). P. 1582–1589.
  4. Abramov S.A., Ryabenko A.A., Khmelnov D.E. Regular solutions of linear ordinary differential equations and truncated series // Comput. Math. Math. Phys. 2020. № 60(1). P. 1–14.
  5. Kytmanov A.A., Lyapin A.P., Sadykov T.M. Evaluating the Rational Generating Function for the Solution of the Cauchy Problem for a Two-dimensional Difference Equation with Constant Coefficients // Programming and computer software. 2017. V. 43. № 2. P. 105–111.
  6. Kruchinin D., Kruchinin V., Shablya Y. Method for Obtaining Coefficients of Powers of Multivariate Generating Functions // Mathematics, 2023. № 11. P. 2859.
  7. Chandragiri S. Counting Lattice Paths by Using Difference Equations with Non-constant Coefficients // The Bulletin of Irkutsk State University. Series Mathematics. 2023. № 44. P. 55–70.
  8. Chaundy T.W., Bullard J.E. John Smith’s problem // Math. Gazette. 1960. V. 44. P. 253–260.
  9. Koornwinder T.H., Schlosser M.J. On an identity by Chaundy and Bullard. I // Indag. Math.(N.S.). 2008. № 19. P. 239–261.
  10. Krivokolesko V.P., Leinartas E.K. On identities with polynomial coefficients // Irkutsk Gos. Univ. Mat. 2012. № 5(3). P. 56–63 (in Russian).
  11. Leinartas E.K. Multidimensional Hadamard Composition And Sums With Linear Constraints On The Summation Indices // Sib. Math. J. 1989. № 30. P. 250–255.
  12. Koornwinder T.H., Schlosser M.J. On an identity by Chaundy and Bullard. II. More history // Indag. Math.(N.S.). 2013. № 24. P. 174–180.
  13. Herrmann O. On the approximation problem in nonrecursive digital filter design // IEEE Trans. Circuit Theory. 1971. V. 18. P. 411–413.
  14. Daubechies I. Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992.
  15. Vidunas R. Degenerate Gauss hypergeometric functions // Kyushu J. Math. № 61. 2007. P. 109–135.
  16. Zeilberger D. On an Identity of Daubechies // Amer. Math. Monthly. 1993. № 100. P. 487.
  17. Kouba O. A Chaundy-Bullard type identity involving the Pochhammer symbol // Indag. Math. New ser. 2023. № 34(1). P. 186–189.
  18. Zhang H. New proofs of Chaundy-Bullard identity in “the problem of points” // Math. Intell. 2016. № 38(1). P. 4–5.
  19. Aharonov D., Elias U. More on the identity of Chaundy and Bullard // J. Math. Anal. Appl. 2014. № 419(1). P. 422–427.
  20. Alzer H. On a combinatorial sum // Indag. Math. New Ser. 2015. № 26(3). P. 519–525.
  21. Brion M., Vergne M. Residue formulae, vector partition functions and lattice points in rational polytopes // J. American Math. Soc. 1997. V. 10. № 4. P. 797–833.
  22. Beck M., Gunnells P.E., Materov E. Weighted lattice point sums in lattice polytopes, unifying Dehn-Sommerville and Ehrhart-Macdonald // Discrete Comput. Geom. 2021. № 65(2). P. 365–384.
  23. Stanley R. Enumerative Combinatorics, V. 1. 1990.
  24. Pukhlikov A.V., Khovanskii A.G. The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes // St. Petersburg Mathematical Journal. 1993. № 4. P. 789–812.
  25. De Concini C., Procesi С., Vergne M. Vector partition functions and generalized Dahmen and Micchelli spaces // Transform. Groups. 2010. № 15(4). P. 751–773.
  26. Sturmfels B. On vector partition functions // Journal of Combinatorial Theory. Series A. 1995. № 72. P. 302–309.
  27. Lyapin A.P., Chandragiri S. Generating Functions For Vector Partition Functions And A Basic Recurrence Relation // Journal of Difference Equations & Applications. 2019. № 25(7). P. 1052–1061.
  28. Leinartas E.K., Nekrasova T.I. Constant Coefficient Linear Difference Equations On The Rational Cones Of The Integer Lattice // Siberian Math. J. 2016. № 57(1). P. 74–85.
  29. Lyapin A.P., Cuchta T. Sections of the generating series of a solution to the multidimensional difference equation // Bulletin of Irkutsk State University-Series mathematics. 2022. №. 42. P. 75–89
  30. Leinartas E.K. Multiple Laurent Series And Difference Equations // Siberian Mathematical Journal. 2004. № 45(2). P. 321–326.

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