On the combinatorics of the Hirota $$D$$-operators.(English)Zbl 0868.35101

Summary: A generic formula is presented which relates the Hirota $$D$$-operators to simple combinatorics. Particular classes of partition polynomials (Bell-polynomials and generalizations) are found to play an important role in the characterization of bilinearizable equations. As a consequence it is shown that bilinear Bäcklund transformations for single-field bilinearizable equations linearize systematically into corresponding Lax-pairs.

MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 05E35 Orthogonal polynomials (combinatorics) (MSC2000) 33C80 Connections of hypergeometric functions with groups and algebras, and related topics
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