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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 Pn, which are special cases of generalized Appell polynomials with generating function of the form
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)[3][4] completely classified all Brenke polynomials that form orthogonal sequences, which are now called Brenke–Chihara polynomials, and found their orthogonality relations.
References
[edit]- ↑ Brenke, W. C. (June 1945). "On Generating Functions of Polynomial Systems". The American Mathematical Monthly. 52 (6): 297. doi:10.2307/2305289.
- ↑ Geronimus, J. (1947-06-01). "The orthogonality of some systems of polynomials". Duke Mathematical Journal. 14 (2). doi:10.1215/S0012-7094-47-01441-5. ISSN 0012-7094.
- ↑ Chihara, T. S. (1968-09-01). "Orthogonal polynomials with Brenke type generating functions". Duke Mathematical Journal. 35 (3). doi:10.1215/S0012-7094-68-03551-5. ISSN 0012-7094.
- ↑ Chihara, T. S. (1971-09-01). "Orthogonality relations for a class of Brenke polynomials". Duke Mathematical Journal. 38 (3). doi:10.1215/S0012-7094-71-03875-0. ISSN 0012-7094.
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