Paper IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round,line join=round]
2 \node[] at (2.56000,4.63000) {denominator convolution};
3 \draw[fill=black!12,draw=black!40,line width=.35pt] (0.80000,3.27000) rectangle ++(0.88000,0.88000);
4 \node[] at (1.24000,3.71000) {$1$};
5 \draw[fill=black!0,draw=black!40,line width=.35pt] (1.68000,3.27000) rectangle ++(0.88000,0.88000);
6 \node[] at (2.12000,3.71000) {$0$};
7 \draw[fill=black!0,draw=black!40,line width=.35pt] (2.56000,3.27000) rectangle ++(0.88000,0.88000);
8 \node[] at (3.00000,3.71000) {$0$};
9 \draw[fill=black!0,draw=black!40,line width=.35pt] (3.44000,3.27000) rectangle ++(0.88000,0.88000);
10 \node[] at (3.88000,3.71000) {$0$};
11 \draw[fill=black!12,draw=black!40,line width=.35pt] (0.80000,2.39000) rectangle ++(0.88000,0.88000);
12 \node[] at (1.24000,2.83000) {$- \frac{1}{2}$};
13 \draw[fill=black!12,draw=black!40,line width=.35pt] (1.68000,2.39000) rectangle ++(0.88000,0.88000);
14 \node[] at (2.12000,2.83000) {$1$};
15 \draw[fill=black!0,draw=black!40,line width=.35pt] (2.56000,2.39000) rectangle ++(0.88000,0.88000);
16 \node[] at (3.00000,2.83000) {$0$};
17 \draw[fill=black!0,draw=black!40,line width=.35pt] (3.44000,2.39000) rectangle ++(0.88000,0.88000);
18 \node[] at (3.88000,2.83000) {$0$};
19 \draw[fill=black!12,draw=black!40,line width=.35pt] (0.80000,1.51000) rectangle ++(0.88000,0.88000);
20 \node[] at (1.24000,1.95000) {$\frac{1}{24}$};
21 \draw[fill=black!12,draw=black!40,line width=.35pt] (1.68000,1.51000) rectangle ++(0.88000,0.88000);
22 \node[] at (2.12000,1.95000) {$- \frac{1}{2}$};
23 \draw[fill=black!12,draw=black!40,line width=.35pt] (2.56000,1.51000) rectangle ++(0.88000,0.88000);
24 \node[] at (3.00000,1.95000) {$1$};
25 \draw[fill=black!0,draw=black!40,line width=.35pt] (3.44000,1.51000) rectangle ++(0.88000,0.88000);
26 \node[] at (3.88000,1.95000) {$0$};
27 \draw[fill=black!12,draw=black!40,line width=.35pt] (0.80000,0.63000) rectangle ++(0.88000,0.88000);
28 \node[] at (1.24000,1.07000) {$- \frac{1}{720}$};
29 \draw[fill=black!12,draw=black!40,line width=.35pt] (1.68000,0.63000) rectangle ++(0.88000,0.88000);
30 \node[] at (2.12000,1.07000) {$\frac{1}{24}$};
31 \draw[fill=black!12,draw=black!40,line width=.35pt] (2.56000,0.63000) rectangle ++(0.88000,0.88000);
32 \node[] at (3.00000,1.07000) {$- \frac{1}{2}$};
33 \draw[fill=black!12,draw=black!40,line width=.35pt] (3.44000,0.63000) rectangle ++(0.88000,0.88000);
34 \node[] at (3.88000,1.07000) {$1$};
35 \node[] at (5.79000,4.63000) {$b$};
36 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.35000,3.27000) rectangle ++(0.88000,0.88000);
37 \node[] at (5.79000,3.71000) {$1$};
38 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.35000,2.39000) rectangle ++(0.88000,0.88000);
39 \node[] at (5.79000,2.83000) {$\frac{1}{2}$};
40 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.35000,1.51000) rectangle ++(0.88000,0.88000);
41 \node[] at (5.79000,1.95000) {$\frac{5}{24}$};
42 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.35000,0.63000) rectangle ++(0.88000,0.88000);
43 \node[] at (5.79000,1.07000) {$\frac{61}{720}$};
44 \node[] at (7.00000,2.40000) {$=$};
45 \node[] at (8.09000,4.63000) {unit series};
46 \draw[fill=black!12,draw=black!40,line width=.35pt] (7.65000,3.27000) rectangle ++(0.88000,0.88000);
47 \node[] at (8.09000,3.71000) {$1$};
48 \draw[fill=black!0,draw=black!40,line width=.35pt] (7.65000,2.39000) rectangle ++(0.88000,0.88000);
49 \node[] at (8.09000,2.83000) {$0$};
50 \draw[fill=black!0,draw=black!40,line width=.35pt] (7.65000,1.51000) rectangle ++(0.88000,0.88000);
51 \node[] at (8.09000,1.95000) {$0$};
52 \draw[fill=black!0,draw=black!40,line width=.35pt] (7.65000,0.63000) rectangle ++(0.88000,0.88000);
53 \node[] at (8.09000,1.07000) {$0$};
54 \node[] at (10.85000,3.60000) {$b_1=\frac12$};
55 \node[] at (10.85000,2.45000) {$b_2=\frac14-\frac1{24}$};
56 \node[] at (10.85000,1.20000) {$b_3=\frac{61}{720}$};
57 \node[] at (6.50000,-0.30000) {$P_{2,n}(1)=(2n)!b_n:\qquad 1,\ 1,\ 5,\ 61$};
58 \end{tikzpicture}

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