Richard Blecksmith

Some infinite product identities

In this paper we derive the power series expansions of four infinite products of the form 1I (1Xn) 7 (1 + Xn), nES1 nES2 where the index sets Si and S2 are specified with respect to a modulus (Theorems 1, 3, and 4). We also establish a useful formula for expanding the product of two Jacobi triple products (Theorem 2). Finally, we give nonexistence results for identities of two forms.

A computer-assisted investigation of Ramanujan pairs

Etude par ordinateur des paires {ai} et {bj} de Ramanujan liees par la relation πi≥1(1−x au i) −1 =1+∈j≥1 x bj /(1−x)(1−x 2 )...(1−x j ). Etablissement de nouvelles identites

The calculus of differences: Effects of a psychosocial, cultural, and pedagogical intervention in an all women’s university calculus class

The purpose of this study was to investigate the ways in which a multi-layered women’s calculus course influenced the participants’ learning of mathematics. This study, conducted in a state university in the Midwestern region of the United States, revealed not only that women in this particular section of calculus were likely to select careers that involved mathematics, but that the focus on peer support, psychosocial issues such as self-confidence, and pedagogy helped the young women overcome gender barriers, as well as barriers of class, poverty, and race. In this article we provide some of the relevant quantitative statistics and relate the stories of two particular women through excerpts from interviews, student artefacts, and participant observation data. We selected these young women because they faced multiple barriers to success in Calculus I and might not have completed the course or taken additional mathematics courses without the support structures that were fundamental to the course.

Women in Calculus: The Effects of a Supportive Setting

The purpose of this study was to investigate a multi-faceted womens calculus course designed to retain women in advanced mathematics courses. With this research, we wanted to find out, first, in what ways students were influenced by participation in the course and, second, in what ways these influences affected their mathematics learning or willingness to take additional mathematics courses. Findings from this study demonstrate that students formed a supportive group of individuals who valued being in an all womens mathematics class and took on roles that facilitated their success. In regard to learning mathematics, the students developed confidence, became comfortable in asking questions about mathematics, valued mathematics for understanding, and continued to take mathematics in order to open up career possibilities. This article describes the findings through excerpts from student and instructor interviews, student artifacts, participant observation data, and descriptive statistics.

Linear quintuple-product identities

In the first part of this paper, series and product representations of four single-variable triple products T0, T1, T2, T3 and four single-variable quintuple products Q0, Q1, Q2, Q3 are defined. Reduced forms and reduction formulas for these eight functions are given, along with formulas which connect them. The second part of the paper contains a systematic computer search for linear trinomial Q identities. The complete set of such families is found to consist of two 2-parameter families, which are proved using the formulas in the first part of the paper.

On the mod 2 reciprocation of infinite modular-part products and the parity of certain partition functions

An infinite modular-part (MP) product is defined to be a product of the form Π n∈S (1−x n ), where S={n∈Z + / n≡r 1 ,...,r t (mod m)}. A complete method is developed which determines if a given MP product has an MP reciprocal modulo 2 and finds it if it does. Next, a graph-theoretic interpretation of this method is made from which a streamlined algorithm is derived for deciding whether the given MP product is such a reciprocal. This algorithm is applied to the single- variable Jacobi triple product and the quintuple product

The completion of Euler's factoring formula

In this paper we derive a formula for a nontrivial factorization of an odd, composite integer N that has been expressed in two different ways as mx2 + ny2. This derivation is based on an approach that Euler used in a special case in 1778. We also modify this formula to handle the case when N is expressed in two different ways as mx2 −ny2. This latter factorization, however, may sometimes be trivial.

Using Conic Sections to Factor Integers

Abstract This paper explores the factorization of an odd, composite integer N that has been expressed in two different ways as mx2 ± ny2. The negative case mx2 — ny2 = N turns out to be quite different from the positive case mx2 + ny2 = N because it deals with a hyperbola instead of an ellipse. Of particular interest in the negative case is that Pell-connected representations produce trivial factorizations.

Proving Balanced T 2 and Q 2 Identities Using Modular Forms

Abstract In our paper “Algorithms for Finding and Proving Balanced Q 2 Identities” we pointed out that the proof algorithm in that paper had not been able to prove 1,417 of the 97,306 tentative identities which the papers search algorithm had found. To deal with this problem, we give here a more powerful proof algorithm based on a familiar technique in the theory of modular forms. The first six sections of this paper establish the connection with modular forms. As it happens, a given balanced T 2 identity in x is mapped directly to a modular form identity of weight 1 on a particular congruence subgroup. The map itself consists of simply multiplying each term of the T 2 identity by the same power of x, written xĪ . The proof algorithm, which is then applied to this identity, consists basically of the following steps: Move the terms of the identity onto the left side of the equation and expand this side into a power series up to a predetermined degree L. If the first L + 1 terms of this expansion are zero, then the identity is true. When this algorithm was applied to the 1,417 recalcitrant identities, it showed they were all true, as hoped.

Algorithms for finding and proving balanced Q2 identities

This paper has three major parts: (1) A search algorithm is presented for finding mod 2 identities involving certain sums of products of the the familiar quintuple product in one variable, called balanced Q identities. The method involves row reduction of large matrices whose columns contain the mod series coefficients of the Q terms. The fact that the search is finite is based on an invariant conjecture. These mod 2 identities are lifted to integer equations by forcing the appropriate signs. All such tentative Q identities can be written as T 2 identities, where T is the familiar triple product of Jacobi. (2) A proof algorithm is presented, where the truth of a tentative T 2 identity is ascertained by expressing that identity as a linear combination of several (true) identities obtained by correctly choosing the values in a fundamental six parameter formula. (3) A massive implementation of both the search algorithm and the proof algorithm is discussed, including data, statistics, pitfalls, and an elaboration of some of the new identities found, including several infinite parametric families of four term Q identities. –Dedicated to our friend Ron Graham