A generalization of Krull-Webster's theory to higher order convex functions: multiple Γ\Gamma-type functions

We provide uniqueness and existence results for the eventually $p$-convex and eventually $p$-concave solutions to the difference equation $\Delta f=g$ on the open half-line $(0,\infty)$, where $p$ is a given nonnegative integer and $g$ is a given function satisfying the asymptotic property that the sequence $n\mapsto\Delta^p g(n)$ converges to zero. These solutions, that we call $\log\Gamma_p$-type functions, include various special functions such as the polygamma functions, the logarithm of the Barnes $G$-function, and the Hurwitz zeta function. Our results generalize to any nonnegative integer $p$ the special case when $p=1$ obtained by Krull and Webster, who both generalized Bohr-Mollerup-Artin's characterization of the gamma function. We also follow and generalize Webster's approach and provide for $\log\Gamma_p$-type functions analogues of Euler's infinite product, Weierstrass' infinite product, Gauss' limit, Gauss' multiplication formula, Legendre's duplication formula, Euler's constant, Stirling's constant, Stirling's formula, Wallis's product formula, and Raabe's formula for the gamma function. We also introduce and discuss analogues of Binet's function, Burnside's formula, Fontana-Mascheroni's series, Euler's reflection formula, and Gauss' digamma theorem. Lastly, we apply our results to several special functions, including the Hurwitz zeta function and the generalized Stieltjes constants, and show through these examples how powerful is our theory to produce formulas and identities almost systematically

Reconceptualizing Mathematics Education

This dissertation is to explore theoretically mathematics education in the United States and the need for reconcepualizing mathematics education. Mathematics education needs reconceptualizing because students know very little mathematics by the time they graduate from high school. Mathematics has become a subject to be feared and dreaded for centuries. High school teachers blame middle school teachers, middle school teachers blame elementary teachers, and elementary teachers blame parents for their students\u27 lack of preparedness in mathematics. Elementary teachers express frustration in teaching mathematics because of their own lack of content knowledge and lack of preparation for the mathematics component of their profession. Regardless of who is to blame, most students entering high school are not prepared to problem solve nor are they interested in mathematics except as the dreaded requirement needed to graduate. Because I have been involved in mathematics education for more than three decades, I have seen many programs come and go. I have seen different types of pedagogy be the in way to teach mathematics. Naturally, technology has influenced mathematics education tremendously in the last decade. Unfortunately, many mathematics educators use technology as a crutch instead of using it to enhance mathematics education. Mathematics education in the United States has been debated for over three centuries. The debate is ongoing. Standardized testing has become a way of life in schools today. Teachers are expected to tell the students exactly what they are supposed to know in mathematics. Standardized tests do not allow students to be creative or struggle in their quest for knowledge because teachers must make sure they have covered the material for the test. The No Child Left Behind Act of 2001 (NCLBA) adds to the problem of mathematics education. The shortage of mathematics teachers throughout the nation is acute. Compliance with the NCLBA requires more mathematics teachers than can possibly be found. My purpose in writing this dissertation is to convey my thoughts and ideas about how the study of mathematics developed, how mathematics education progressed throughout history how mathematics education is progressing today, and how mathematics education will progress in the future. In my opinion, teacher preparation of elementary and middle school teachers will be a very strong component in the reconceptualization of mathematics education. Mathematics teachers at all levels should be grounded in a history of mathematics and be cognizant of the development of mathematics education throughout the relatively short history of America. Furthermore, a dialogue must be implemented and maintained between mathematics educators at all levels. With the implementation of this dialogue, mathematics education will become a subject of intrigue and beauty and will no longer remain the subject to be feared and dreaded

University catalog, 2016-2017

The catalog is a comprehensive reference for your academic studies. It includes a list of all degree programs offered at MU, including bachelors, masters, specialists, doctorates, minors, certificates, and emphasis areas. It details the university wide requirements, the curricular requirements for each program, and in some cases provides a sample plan of study. The catalog includes a complete listing and description of approved courses. It also provides information on academic policies, contact information for supporting offices, and a complete listing of faculty members. -- Page 3

University catalog, 2018-19

Welcome to the University of Missouri 2018-2019 catalog! We are pleased to provide an interactive and searchable catalog online. The catalog is a comprehensive reference for your academic studies. It includes a list of all degree programs offered at MU, including bachelors, masters, specialists, doctorates, minors, certificates, and emphasis areas. It details the university wide requirements, the curricular requirements for each program, and in some cases provides a sample plan of study. The catalog includes a complete listing and description of approved courses. It also provides information on academic policies, contact information for supporting offices, and a complete listing of faculty members. Information in the catalog is current as of May 2018.--Page 17