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In mathematics, Bender–Dunne polynomials are a two-parameter family of sequences of orthogonal polynomials studied by Carl M. Bender and Gerald V. Dunne. They may be defined by the recursion:

${\displaystyle P_{0}(x)=1}$,

${\displaystyle P_{1}(x)=x}$ ,

and for ${\displaystyle n>1}$:

${\displaystyle P_{n}(x)=xP_{n-1}(x)+16(n-1)(n-J-1)(n+2s-2)P_{n-2}(x)}$

where ${\displaystyle J}$ and ${\displaystyle s}$ are arbitrary parameters.

References

1. Bender, Carl M.; Dunne, Gerald V. (1988). "Polynomials and operator orderings". Journal of Mathematical Physics. 29 (8): 1727–1731. Bibcode:1988JMP....29.1727B. doi:10.1063/1.527869. ISSN 0022-2488. MR 0955168.
2. Bender, Carl M.; Dunne, Gerald V. (1996). "Quasi-exactly solvable systems and orthogonal polynomials". Journal of Mathematical Physics. 37 (1): 6–11. arXiv:hep-th/9511138. Bibcode:1996JMP....37....6B. doi:10.1063/1.531373. ISSN 0022-2488. MR 1370155. S2CID 28967621.

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