Date of Award
Doctor of Philosophy (PhD)
Calkin, Neil J.
Goddard , Wayne
James , Kevin
Matthews , Gretchen
In a note in the American Mathematical Monthly in 1960, Strodt mentions a way to prove both the Euler-Maclaurin summation formula and the Boole summation formula using operators. In a 2009 article in the Monthly, Borwein, Calkin, and Manna expand on this idea. Therein, they define Strodt operators and Strodt polynomials and show that the classical Bernoulli polynomials and Euler polynomials are examples of Strodt polynomials.
It is well known that both Bernoulli polynomials and Euler polynomials on a fixed interval are asymptotically sinusoidal. Borwein, Calkin, and Manna show that a similar result holds for the uniform Strodt polynomials. We extend this idea to other generalized families of Strodt polynomials. We state and prove several theorems which make explicit the asymptotic behavior of these families. We also explain our experiments and give examples of Strodt polynomials with unknown, non-sinusoidal asymptotics.
We also consider an identity for sums of Hurwitz class numbers and we state a theorem arising from taking subsums of this identity. The Hurwitz class numbers can be defined by counting classes of binary quadratic forms. They can also be related to isomorphism classes of elliptic curves over a finite field via Deuring's Theorem. We use a combinatorial proof to prove this theorem.
Flowers, Timothy, "Asymptotics of Families of Polynomials and Sums of Hurwitz Class Numbers" (2009). All Dissertations. 406.