Summer Al Hamdani

316 Van Vleck Hall,

480 Lincoln Dr,

Madison, WI 53705

I am a mathematics Ph.D. student at University of Wisconsin-Madison, with intent to declare my doctoral minor in statistics. My primary interests of the mathematical variety include, though are not limited to, harmonic analysis, analytic number theory, probability theory, and applied mathematics/statistics. Nowadays, I am more of a casual reader of medical physics - my interests are in imaging and diagnostic modalities.

Before starting the doctoral program, I completed my undergraduate studies in Mathematics and Biomedical Physics along with a Master’s in Mathematics at California State University, Fresno (go Bulldogs!). My Master’s thesis studied the zero distribution of Chebyshev polynomials of the second kind with binomial-type coefficients. Afterwards, I briefly worked in institutional research and data science. Further details can be found on my CV.

The best way to reach me is by email: alhamdani at math dot wisc dot edu. I usually try to respond within 24 ± 12 hours on business days.

news

Jan 22, 2023	I updated this webpage! How exciting.

selected publications

Zeros of a binomial combination of Chebyshev polynomials

Summer Al Hamdani, and Khang Tran

International Journal of Number Theory, 2022

Abs arXiv HTML

For 0 < α< 1, we study the zeros of the sequence of polynomials { P_m(z) }_m=0^∞generated by the reciprocal of (1 − t)^α(1 − 2zt + t^2), expanded as a power series in t. Equivalently, this sequence is obtained from a linear combination of Chebyshev polynomials whose coefficients have a binomial form. We show that the number of zeros of P_m(z) outside the interval (−1, 1) is bounded by a constant independent of m.
Zero distribution of binomial combinations of Chebyshev polynomials of the second kind

Summer Al Hamdani

Master’s thesis, California State University, Fresno, May 2021

Abs PDF

For α∈\mathbbR such that 0 < α≤1, we can construct a sequence of real polynomials { P_m(z) }_m=0^∞that is generated by 1/(1-t)^α(1-2zt+t^2) = \sum_m=0^∞P_m(z)t^m . The sequence { P_m(z) }_m=0^∞is a binomial combination of the well-known Chebyshev polynomials of the second kind, which have all real zeros on the interval (−1, 1). We prove that there exists a constant C (independent of m) such that the number of zeros of P_m(z) outside of the interval (−1, 1) is at most C for all m ∈\mathbbN.