Chicago 430 S. Michigan Ave.Chicago, IL 60605(312) 341-3500
Schaumburg 1400 N. Roosevelt Blvd.Schaumburg, IL 60173(847) 619-7300
The first Roosevelt Lectures in Mathematics will be held April 18-19, 2014. There will be three lectures of two hours each:one on Friday afternoon and the other two on Saturday morning and Saturday afternoon.
Wabash Building room 1017
420 S. Wabash Avenue, Chicago, IL 60605
Part I 2:30 pm -3:30 pm
Part II 3:45 pm -4:45 pm
We will start with n! and the binomial theorem, and use them to motivate and develop the gamma function as an extension of n!
and the beta function integral as a consequence of the binomial theorem. [Some but not all of this will likely be known by many who come. The next will be new to many, but it will be done in a way that students should have little trouble following most of it.]
About 175 years ago a problem was posed which led in one solution to 1(1+q)(1+q+q^2)...(1+q+...+q^(n-1)) as an
extension of n!. About 200 years ago a similar extension of the binomial theorem was found. We will see how the first
part of this talk extends to this setting. There will be a surprise in that some of the work in the more general setting will be easier than in the classical case.
Part I 9:30 am- 10:30 am
Part II 11:00 am - 12:00 pm
Classical Fourier series using sines and cosines are treated in a number of undergraduate courses, both in mathematics and in some applied areas. What is usually not mentioned is that cos(nt) is a polynomial of degree n in cos(t) and sin(n+1)t/sin(t) is also a polynomial of degree n in cos(t). The polynomials in these two cases are called Chebyshev polynomials. The classical orthogonal polynomials generalize these two sets of polynomials, and their weight functions are the integrands in the gamma and beta integrals discussed earlier, and exp(-x^2). Some properties of these polynomials will be discussed and some uses of them will be described. One is an old problem of Friedrichs and Levy which Szego solved: Show that the power series coefficients of 1/f'(1) are positive when f(x)=(x-r)(x-s)(x-t). Various extensions will also be described and some indications of what can be done with them will be mentioned.
Part I 2:00 pm-3:00 pm
Part II 3:15 pm- 4:15 pm
One useful set of problems deals with inequalities for various orthogonal polynomials and sums of them. Some of the techniques used to prove such results will be motivated and used, and some open problems will be described.