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We first review and analyze the golden integral and its definitions and some properties. Then we introduce a new generalization of the Hermite polynomials via the golden exponential function (called Fibonacci-Hermite polynomials) and investigate several properties and relations. We derive some explicit and implicit summation formulas for mentioned polynomials. Then, we analyze derivative properties and provide a higher-order difference equation of the Fibonacci-Hermite polynomials. Moreover, we examine a recurrence relation and integral representation. In addition, we obtain some properties of Fibonacci-Bernstein polynomials. Lastly, we obtain a correlation between the Fibonacci-Hermite polynomials and the Fibonacci-Bernstein polynomials
Duran et al. (Wed,) studied this question.