A relation between Catalan Numbers and World Series type problems

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March clearly comes in like a lion in this month’s *Monthly*. We honor Pi Day with a special article by *Monthly* Associate Editor Jon Borwein and David Bailey: "Pi Day Is Upon Us Again and We Still Do Not Know if Pi Is Normal." This paper is based on an annual lecture that Professor Borwein gives at his home institution in Australia which has gained a “cult” following. We also honor Joan Leitzel, the winner of the 2014 Gung and Hu Award for distinguished service to mathematics.

In our Notes section, don’t miss Bill Johnston’s student-friendly proof that the weighted Hermite polynomials form a basis for $$L^2(\mathbb{R})$$. Mark Kozek reviews *Mathematics in Popular Culture: Essays on Appearances in Film, Fiction, 274 Games, Television and Other Media* by Jessica Sklar and Elizabeth Sklar. Need something to do over Spring Break? Take our Problem Section along for the ride. Stay tuned for the April edition when Andrew Simoson shows us when to expect the next transit of Venus. —*Scott Chapman*

Vol. 121, No. 3, pp.187-278.

To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.

Kenneth A. Ross and Ann E. Watkins

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.187

David H. Bailey and Jonathan Borwein

The digits of $$\pi$$ have intrigued both the public and research mathematicians from the beginning of time. This article briefly reviews the history of this venerable constant, and then describes some recent research on the question of whether $$\pi$$ is normal, or, in other words, whether its digits are statistically random in a specific sense.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.191 (pdf)

Stephan Berendonk

In this note we will show, by elementary means, that we can express the length of curves drawn with a spirograph in terms of the perimeter of certain ellipses. We will present the proof in the language of Euclidean geometry.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.207

Alec S. Cooper, Oleg Pikhurko, John R. Schmitt, and Gregory S.Warrington

In Martin Gardner’s October 1976 Mathematical Games column in *Scientific American*, he posed the following problem: “What is the smallest number of [queens] you can put on an [$$n\times n$$ chessboard] such that no [queen] can be added without creating three in a row, a column, or, except in the case when $$n$$ is congruent to 3 modulo 4, in which case one less may suffice.” We use the Combinatorial Nullstellensatz to prove that this number is at least $$n$$. A second, more elementary proof is also offered in the case that $$n$$ is even.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.213

Aleksander Maliszewski and Marcin Szyszkowski

We prove that every continuous function from a disk to the real line has a level set containing a connected component of diameter at least $$\sqrt{3}$$. We also show that if the disk is split into two sets—one open and the other closed—then one of them contains a component of diameter at least $$\sqrt{3}$$.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.222

Junesang Choi and Hari M. Srivastava

We first present the corrected expression for a certain widely-recorded generalized Goldbach–Euler series. The corrected forms are then shown to be connected with the problem of closed-form evaluation of series involving the Zeta functions, which happens to be an extensively-investigated subject since the time of Euler as (for example) in the classical three-century old-Goldbach theorem.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.229

B. Sury

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.236

Trevor Hyde

The $$m$$-clover is the plane curve defined by the polar equation $$r^{m/2}=\cos{\left(\frac{m}{2}\theta\right)}$$. In this article we extend a well-known derivation of the Wallis product to derive a generalized Wallis product for arc lengths of $$m$$-clovers.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.237

Mircea Merca

In this note, using the multisection series method, we establish the formulas for various power sums of sine or cosine functions.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.244

William Johnston

We present a student-friendly proof that the weighted Hermite polynomials form a complete orthonormal system (a basis) for the collection of $$L^{2}(\mathbb{R})$$ real-valued functions.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.249

Gerd Herzog

For $$d\times d$$ matrices $$A$$, $$B$$ and entire functions $$f$$, $$g$$ with $$f(0) = g(0) = 1$$, we give an elementary proof of the formula

$$\lim_{k\to\infty}(f(A/k)g)(B/k))^{k}=\exp (f'(0)A+g'(0)B)$$

For the case $$f = g = \exp$$, this is Lie’s famous product formula for matrices.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.254

Kenneth Lange

This note is devoted to a short, but elementary, proof of Hadamard’s determinant inequality.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.258

S. Kocak and M. Limoncu

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.259

Justin Tatch Moore

This note gives a “Zorn’s Lemma” style proof that any two bases in a vector space have the same cardinality.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.260

Marc Chamberland and Doron Zeilberger

This note offers a short, elementary proof of a result similar to Ptolemy’s theorem. Specifically, let $$d_{i, j}$$ denote the distance between $$P_{i}$$ and $$P_{j}$$ . Let $$n$$ be a positive integer and $$P_{i}$$ , for $$1\leq i \leq 2n$$, be cyclically ordered points on a circle. If

$$\sum_{i=1}^{n}\frac{1}{R_{2i}}=\sum_{i=1}^{n}\frac{1}{R_{2i-1}}$$

Then

$$\sum_{i=1}^{n}\frac{1}{R_{2i}}=\sum_{i=1}^{n}\frac{1}{R_{2i-1}}$$

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.263

Problems 11761-11767

Solutions 11613, 11620, 11623, 11624, 11625

To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.266

*Mathematics in Popular Culture: Essays on Appearances in Film, Fiction, Games, Television and Other Media*. Edited by Jessica K. Sklar and Elizabeth S. Sklar; Foreword by Keith Devlin.

Reviewed by Mark Kozek

To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.03.274