*3.6. Rogers-Szeg˝o-Chaos*

The Rogers-Szeg˝o polynomials and the wrapped normal distribution provide a convenient basis and random variable pairing for the linear combination in Equation (21). The Rogers-Szeg˝o polynomials in Equation (3) can be rewritten according to [39]

$$\phi\_{\mathfrak{n}}\left(-\frac{\vartheta}{\sqrt{q}},q\right) = \sum\_{k=0}^{n}(-1)^{n-k}\binom{n}{k}\_q q^{\frac{n-k}{2}}\vartheta^k,\tag{25}$$

where *q* is calculated based on the standard deviation of the unwrapped normal distribution: *q* = *e*<sup>−</sup>*σ*<sup>2</sup> . These polynomials satisfy the orthogonality condition

$$\frac{1}{2\pi} \int\_{-\pi}^{\pi} \phi\_{\text{ff}}\left(-\frac{\vartheta}{\sqrt{q}}, q\right) \overline{\phi\_{\text{fl}}\left(-\frac{\vartheta}{\sqrt{q}}, q\right)} \theta\_3\left(\frac{\theta}{2}, \sqrt{q}\right) d\theta = (q; q)\_{\text{fl}} \delta\_{\text{mm}} \frac{1}{2}$$

where *<sup>ϑ</sup>*3(*<sup>α</sup>*, *β*) is the theta function

$$\theta\_3(\alpha, \beta) = \sum\_{k=-\infty}^{\infty} \beta^{k^2} e^{2ik\alpha},\tag{26}$$

which is another form of the wrapped normal distribution. Note the distinction between the variables *ϑ*3 and *ϑ* = *e*.

For convenience, the inverse to the given theta function is

$$
\theta\_3^{-1}(\alpha, \beta) = 2\alpha + \pi + 2 \sum\_{k=1}^{\infty} \frac{\beta^{k^2} \sin(2k\alpha)}{k}.
$$

The inverse of the theta function is particularly useful if the cumulative distribution function (cdf) is required to draw random samples. The number of wrappings in Equation (26) significantly affects the results. For reference, the results presented in this work truncate the summation to ±1000.

Written out, the first five orders of this form of the Rogers-Szeg˝o polynomials are

$$\begin{aligned} \phi\_0 &= 1\\ \phi\_1 &= \theta - q^{1/2} \\ \phi\_2 &= \theta^2 - q^{1/2}(q+1)\theta + q\\ \phi\_3 &= \theta^3 - q^{1/2}(q^2+q+1)\theta^2 + q(q^2+q+1)\theta - q^{3/2} \\ \phi\_4 &= \theta^4 - q^{1/2}(q+1)(q^2+1)\theta^3 + q(q^2+1)(q^2+q+1)\theta^2 - q^{3/2}(q+1)(q^2+1)\theta + q^2 \end{aligned}$$

and are shown graphically in Figure 3. Because the polynomials are complex valued, the real and imaginary components are show, separately. In both cases, the polynomials are oscillatory, with the real component being symmetric about *θ* = 0, and the imaginary component being antisymmetric about *θ* = 0. Additionally, the amplitude of the oscillations increase both with increasing order and distance from *θ* = 0.

(**b**)

**Figure 3.** The zeroth through fourth Rogers-Szeg˝o polynomials with an unwrapped standard deviation of 0.1. (**a**) Real Component. (**b**) Imaginary Component.

The zeroth polynomial is one, as is standard; therefore, the difference between the two generating functions given in Equations (25) and (26) will only be apparent in the calculation of moments beyond the first.
