Determination of the Properties of (p, q)‐Sigmoid Polynomials and the Structure of Their Roots

Abstract

Nowadays, many mathematicians have great concern about p q -numbers, which are various applications, and have studied these numbers in many different research areas. We know that p q -numbers are different to q -numbers because of the symmetric property. We find the addition theorem, recurrence formula, and p q -derivative about sigmoid polynomials including p q -numbers. Also, we derive the relevant symmetric relations between p q -sigmoid polynomials and p q -Euler polynomials. Moreover, we observe the structures of appreciative roots and fixed points about p q -sigmoid polynomials. By using the fixed points of p q -sigmoid polynomials and Newton’s algorithm, we show self-similarity and conjectures about p q -sigmoid polynomials.