Generating Functions and Figurate Numbers

Elements of the algebra $\mathbb{C}[[z]]$, with certain codified rules of manipulation. We then present the generating function of the Natural Numbers $\{1,2,3,4,5,6\ldots\}$. We note that $$\sum_{n=0}^{\infty}nz^n = z\frac{d}{dz}\left(\sum_{n=1}^{\infty}z^n\right),$$ (36) and therefore by virtue of (34) we have $$\sum_{n=0}^{\infty}nz^n = z\frac{d}{dz}\frac{1}{1-z} = \frac{z}{(1-z)^2},$$ (37) and therefore that $$\{1,2,3,4,5,6\ldots\} \to \frac{z}{(1-z)^2}.$$ (38) For those interested in deepening the topic rigorously, we refer to S.K. Lando, Lectures on Generating Functions, American Mathematical Society, 2003. Through these two generating functions we can easily obtain all the generating functions of figurate numbers. Indeed, the convolution of series is obtained through the multiplication of generating functions, so given a series $\{a_0, a_1, a_2, \ldots\}$ with generating function $f(z)$, if we want to obtain the generating function of the triangular development, i.e., $\{a_0, a_0+a_1, a_0+a_1+a_2, \ldots\}$, it suffices to multiply the function by $\frac{1}{1-z}$; i.e., $$\{a_0, a_0+a_1, a_0+a_1+a_2, \ldots\} \to f(z)\frac{1}{1-z}.$$ (39) This allows us to immediately obtain the generating functions of simplicial numbers (triangular, tetrahedral, pentatopic, etc.) which are nothing but the triangular development one of the other: • (dim. 1) Natural Numbers: $\{1,2,3,4,5\ldots\} \to \frac{z}{(1-z)^2}$; • (dim. 2) Triangular Numbers: $\{1,3,6,10,15\ldots\} \to \frac{z}{(1-z)^3}$; • (dim. 3) Tetrahedral Numbers: $\{1,4,10,20,35\ldots\} \to \frac{z}{(1-z)^4}$; • (dim. 4) Pentatopic Numbers: $\{1,5,15,35,70\ldots\} \to \frac{z}{(1-z)^5}$; • (dim. n) n-simplicial Numbers: $\left\{1, (n+1), \frac{1}{2}(n+1)(n+2), \ldots\right\} \to \frac{z}{(1-z)^{n+1}}$; If we then want to obtain the polygonal numbers previously treated, we can obtain them from the generating functions of triangular numbers through multiplication by $(1+(n-3)z)$ where $n$ identifies the $n$-th regular polygon. This means that the generating function for $n$-gonal numbers will be $$P_n(z) = \frac{z(1+(n-3)z)}{(1-z)^3}.$$ (40) We have in fact: • Triangular Numbers: $\frac{z}{(1-z)^3} \to \{1,3,6,10\ldots\}$ • Square Numbers: $\frac{z(1+z)}{(1-z)^3} \to \{1,4,9,16\ldots\}$ • Pentagonal Numbers: $\frac{z(1+2z)}{(1-z)^3} \to \{1,5,12,22\ldots\}$ • Hexagonal Numbers: $\frac{z(1+3z)}{(1-z)^3} \to \{1,6,15,28\ldots\}$ • ... The fact of having moved to the study of generating functions allows us to generalize these data even for geometric constructions of non-representable dimensions such as fractional or superior to the third. For example, we can easily identify numbers corresponding to simplicial numbers for non-integer but fractional dimensions such as fractal triangular numbers of dimension $0.5$, i.e., a set of points aggregated in a set with Hausdorff dimension between zero and one, such as the numbers $$\Delta_{1/2}: \frac{z}{(1-z)^{3/2}} \to \left\{1, \frac{3}{2}, \frac{15}{8}, \frac{35}{16}, \frac{315}{128}, \ldots\right\},$$ (41) those of dimension $1.5$ between the first and second dimension as $$\Delta_{3/2}: \frac{z}{(1-z)^{5/2}} \to \left\{1, \frac{5}{2}, \frac{35}{8}, \frac{105}{16}, \frac{1155}{128}, \ldots\right\},$$ (42) those of dimension $2.5$ between the second and third as $$\Delta_{5/2}: \frac{z}{(1-z)^{7/2}} \to \left\{1, \frac{7}{2}, \frac{63}{8}, \frac{231}{16}, \frac{3003}{128}, \ldots\right\}.$$ (43) We believe that these numbers, which we introduce for the first time, can play a role even in contemporary profane mathematics in the modeling of fractal surfaces, just as simplicial numbers play it in the modeling of integer-dimensional surfaces. To conclude, we want to report the generating functions of Platonic numbers, i.e.: • Tetrahedral Numbers: $f(z) = \frac{1}{(1-z)^4}$; • Octahedral Numbers: $f(z) = \frac{1+2z+z^2}{(1-z)^4}$; • Cubic Numbers: $f(z) = \frac{1+4z+z^2}{(1-z)^4}$; • Icosahedral Numbers: $f(z) = \frac{1+8z+6z^2}{(1-z)^4}$; • Dodecahedral Numbers: $f(z) = \frac{1+16z+10z^2}{(1-z)^4}$. We hope with these brief indications to have given scholars of mathematical symbolism precious tools to be able to use. Unfortunately, space does not allow us to deepen all the relationships between these classes of numbers and their generalizations. However, we reserve the right to deepen in the next issues of Mathesis and to resume the discourse undertaken here. Notes on the Sacred Triangle (3,4,5) and the Ternary (Solis) In a previous article (see the music of Platonic solids) we had treated the Sacred Triangle 3:4:5 or Pythagorean Triangle highlighting its chromatic and musical correspondences. Specifically, we had written that this symbol, jewel of classical antiquity, perfectly identified Theurgic Science, i.e., the constructive capacity to link celestial realities to terrestrial ones. The Sacred Triangle is an expression of the Pythagorean Theorem, which links an exterior dimension to another interior, orthogonal and incommensurable to the previous one. It links the visible to the invisible, symbolized by the two legs, through an always original synthesis identified by the hypotenuse. We had then noted how the vibratory frequencies of the three additive primary colors (i.e., Red, Green and Blue) are in reciprocal ratio of 3:4:5 exactly like the frequencies of that inverted chord that was then renamed Sacred Chord and which is C:F:A. To be more explicit: if the luminous frequency of a deep Red color can be identified at 400 THz, then, calculating the equivalent corresponding major leg, we obtain $$400 \times \frac{4}{3} \approx 533 \text{ THz},$$ (44) which is the frequency of an intense Green color and, instead, multiplying by $5/3$ we get 667 THz which is the frequency of a deep Blue. The three colors Red, Green and Blue are the primary colors.