β-reciprocal polynomials

ABSTRACT

A new class of polynomials p_n(x) known as β-reciprocal polynomials is defined. Given a parameter $\beta \in \mathbb{C}$ that is not a root of −1, we show that the only β-reciprocal polynomials are p_n(x) ≡ xⁿ. When β is a root of −1, other polynomials are possible. For example, the Hermite polynomials are i-reciprocal, $\mathrm{i}=\sqrt{-1}$.

KEYWORDS:

• Hermite polynomial
• normal distribution
• polynomials

Acknowledgements

The authors would like to thank the editor and the two referees for careful reading and for their comments which greatly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.