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# 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 \( P_n \), 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

Brenke, W. C. (1945), "On generating functions of polynomial systems", The American Mathematical Monthly 52: 297–301, doi:10.2307/2305289, ISSN 0002-9890, MR 0012720

Chihara, Theodore Seio (1968), "Orthogonal polynomials with Brenke type generating functions", Duke Mathematical Journal 35: 505–517, doi:10.1215/S0012-7094-68-03551-5, ISSN 0012-7094, MR 0227488

Chihara, Theodore Seio (1971), "Orthogonality relations for a class of Brenke polynomials", Duke Mathematical Journal 38: 599–603, doi:10.1215/S0012-7094-71-03875-0, ISSN 0012-7094, MR 0280757

Geronimus, J. (1947), "The orthogonality of some systems of polynomials", Duke Mathematical Journal 14: 503–510, doi:10.1215/S0012-7094-47-01441-5, ISSN 0012-7094, MR 0021151

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