Dokazování kombinatorických identit pomocí formálních mocninných řad

• Thesis title in Czech: Dokazování kombinatorických identit pomocí formálních mocninných řad
• Thesis title in English: Proving combinatorial identities via formal power series
• Academic year of topic announcement: 2022/2023
• Thesis type: Bachelor's thesis
• Thesis language: čeština
• Department: Department of Algebra (32-KA)
• Supervisor: doc. Mgr. Vítězslav Kala, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 25.03.2023
• Date of assignment: 25.03.2023
• Confirmed by Study dept. on: 03.04.2023
• Advisors: Dr. rer. nat. Siu Hang Man, Ph.D.

Guidelines

Formal power series are a powerful technique in the study of problems in combinatorics and number theory. By considering the generating functions which encode combinatorial information, one can convert difficult problems into elegant formulae. The student will cover the basics about formal power series (e.g. from [Hir], [Sam]); he will give full details of the proofs and provide more examples. He will then give detailed proofs to classical theorems concerning formal power series, such as Jacobi’s triple product formula and Lagrange-Jacobi four-square theorem [HW]. The student may further potentially consider the application of the technique in the study of integer partitions [And].

References

[And] Andrews, G. (1984). The Theory of Partitions. Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press.
[Hir] Hirschhorn, M.D. (2017). The Power of q - A Personal Journey. Developments in Mathematics, vol. 49. Springer, Cham.
[HW] Hardy, G.H., Wright, E.M. (2008). An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford.
[Sam] Sambale, B. An Invitation to Formal Power Series. Jahresber. Dtsch. Math. Ver. 125, 3–69 (2023).