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Binomial prolate spheroidal functions, Pascal matrices, and arithmetic of elliptic curves
1Department of Mathematics, California State University Fullerton, Fullerton, CA 92831.
Summary
Researchers studied the symmetric Pascal matrix and its eigenvectors, called binomial prolates. They derived an explicit formula for the generating function of a specific eigenvector, revealing connections to Legendre elliptic curves over finite fields.
Area of Science:
- Mathematics
- Linear Algebra
- Number Theory
Background:
- The symmetric Pascal matrix is a generalized discrete time and band-limiting operator for the binomial transform.
- Its eigenvectors, termed binomial prolates, are generalized discrete prolate spheroidal wave functions.
- These binomial prolates' generating functions are also generalized prolate spheroidal functions.
Purpose of the Study:
- To derive an explicit formula for the generating function of an eigenvector of the symmetric Pascal matrix with eigenvalue 1 for even, positive integers N.
- To investigate the behavior of this generating function modulo an odd prime p.
- To explore the relationship between the generating function and periods of Legendre elliptic curves over finite fields.
Main Methods:
- Derivation of an explicit formula for the generating function of a specific eigenvector of the symmetric Pascal matrix.
- Modular arithmetic analysis to study the generating function modulo p.
- Connections to the theory of elliptic curves and p-adic analysis.
Main Results:
- An explicit formula for the generating function of an eigenvector (with eigenvalue 1) of the symmetric Pascal matrix is obtained for even N.
- When N is congruent to 0 modulo an odd prime p, the generating function is equivalent modulo p to the number of points on the Legendre elliptic curve y^2 = x^3 + x over the finite field F_p.
- For N congruent to 0 modulo p, the generating function is the square of a period of the Legendre elliptic curve modulo p in the open p-adic unit disk.
Conclusions:
- The study establishes a novel explicit formula for the generating function of binomial prolates.
- It reveals a significant connection between the symmetric Pascal matrix, its eigenvectors, and the arithmetic of Legendre elliptic curves over finite fields.
- The findings extend to p-adic analysis, linking generating functions to periods of elliptic curves in the p-adic domain.
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