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) introduced sequences of Brenke polynomials Pn, which are special cases of generalized Appell polynomials with generating function of the form

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) found some further examples of orthogonal Brenke polynomials. Chihara (1968, 1971) completely classified all Brenke polynomials that form orthogonal sequences, which are now called Brenke–Chihara polynomials, and found their orthogonality relations.

References

This article is issued from Wikipedia - version of the Saturday, May 10, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.