A New Special Function and Its Application in Probability

TL;DR

This paper introduces a new special function similar to the error function, providing accurate approximations for its CDF using Chebyshev polynomials and Padé approximants, with potential applications in various probability distributions.

Contribution

It presents a novel special function and develops accurate approximation methods for its CDF, extending the approach to other probability distributions.

Findings

Achieves an approximation accuracy of 10^{-6} in quadratic mean norm.

Provides closed-form approximations using Chebyshev polynomials and Padé approximants.

Demonstrates applicability to Maxwell–Boltzmann and normal distributions.

Abstract

In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chebyshev polynomials of the first kind and the error function. Also, we provide its series representation using Pad\'e approximant. We show convincing numerical evidence of an accuracy of $10^{-6}$ for the approximants in the sense of the quadratic mean norm. A similar approach may be applied to other probability distributions, for example, the Maxwell--Boltzmann distribution and the normal distribution, and we show its application using both of those distributions.