Brenke–Chihara polynomials

In mathematics, Brenke polynomials are special cases of generalized Appell polynomials, and Brenke–Chihara polynomials are the Brenke polynomials that are also orthogonal polynomials.

Brenke (1945)¹ introduced sequences of Brenke polynomials P_n, which are special cases of generalized Appell polynomials with generating function of the form

${\displaystyle A(w)B(xw)=\sum _{n=0}^{\infty }P_{n}(x)w^{n}.}$

Brenke observed that Hermite polynomials and Laguerre polynomials are examples of Brenke polynomials, and asked if there are any other sequences of orthogonal polynomials of this form. Geronimus (1947)² found some further examples of orthogonal Brenke polynomials. Chihara (1968, 1971)³⁴ completely classified all Brenke polynomials that form orthogonal sequences, which are now called Brenke–Chihara polynomials, and found their orthogonality relations.